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Real Options SLS - Russian User Manual (2012)

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1. Figure 6 SLS Results with a 10 Step Lattice 1 4 Multiple Asset Super Lattice Solver MSLS The MSLS is an extension of the SLS in that the MSLS can be used to solve options with multiple underlying assets and multiple phases The MSLS allows the user to enter multiple underlying assets as well as multiple valuation lattices These valuation lattices can call to user defined custom variables Some examples of the types of options that the MSLS can be used to solve include e Sequential Compound Options two three and multiple phased sequential options e Simultaneous Compound Options multiple assets with multiple simultaneous options Chooser and Switching Options choosing among several options and underlying assets e Floating Options choosing between calls and puts e Multiple Asset Options D binomial option models User Manual
2. 5 5000 User Manual 45 Real Options Super Lattice Solver Real Options SLS Plain Vanilla Call Option I Single Asset Super Lattice Solver SLS 1000 E 123 a 8l
3. 3 Figure 18A Payoff Chatts Sensitivity Analysis Scenatio Tables Convergence Analysis Monte Carlo Risk Simulation Real Options SLS fise Language 5 0 9 9 amp 45 O 1 2 3 5 00 E m R amp D E E
4. 1 2 10 20 35 yara _ Max Asset Cost 0 Max Asset Cost 0 _ IF Asset Suboptimal Cost Max Asset Cost 0 OptionOpen Figure 69 SLS Results of a Call Option with Suboptimal Behavior User Manual 107 Real Options Super Lattice Solver Real Options SLS erican ons Assumptions Stock Price Strike Price Maturity in Years Risk free Rate 96 Dividends 96 Volatility 9 Suboptimal Exercise Multiple Underlying Stock Price Lattice 100 00 00 uboptimal Exercise Behavior m Intermediate Calculations Stepping Time at 1 0000 Up Step Size up 1 1052 Down Step Size down 0 9048 Risk neutral Probability prob 73 09 r Results 10 Step Lattice Results Generalized Black Scholes 100 Step Binomial Super Lattice Binomial Super Lattice Steps 10 Step Trinomial Super Lattice Trinomial Super Lattice Step
5. 0 80 1 2 10 20 35 Max Asset Asset Contraction Savings Call 359 359 Put 170 28 Max Asset Cost 0 Max Asset Contraction Savings OptionOpen 1005 1970 Max Asset Cost OptionOpen B OptionOpen pee Figure 28 A Customized Option to Contract with Changing Savings 58 Real Options Super Lattice Solver Real Options SLS User Manual 23 European Bermudan and Customized Expansion Options The Expansion Option values the flexibility to expand from a current existing state to a larger or expanded state Therefore an existing state or condition must first be present in o
6. 130 3154 Max Salvage OptionOpen Max Asset Cost OptionOpen B OptionOpen Figure 13 Customized Abandonment Option using SLS User Manual 24 Real Options Super Lattice Solver Real Options SLS User Manual A B D E IP 1 2 SUPER LATTICE SOLVER SINGLE ASSET 3 4 Option Type Custom Variables List 5 PV Underlying Asset 120 00 Variable Name Value Starting Steps B Annualized Volatility 7 Maturity Years 8 implementation Cost 0 00 9 Risk Free Rate 5 00 10 Dividend Yield 0 00 11 Lattice Steps 12 Terminal Equation MAX Asset Salvage 13 Intermediate Equation MAX Salvage G 14 Intermediate Equation During Blackout 15 Blackout Steps 18 Super Lattice Solver Result 20 Note This is the Excel version of the Super Lattice Solver useful when running simulations or when linking to and from other spreadsheets 21 Use this sample spreadsheet for your models You can simply click on File Save As to save as a different file and start using the model 22 For the option type set 0 American 1 European 2 Bermudan 3 Custom 23 The function used is
7. 100 100 ner 5 100 1 2 10 20 35 Max Asset Cost 0 Max Asset Cost 0 Vama 23 3975 Max Asset Cost OptionOpen Figure 4 Custom Equation Inputs In addition you can create an Audit Worksheet in Excel to view a sample 10 step binomial lattice by checking the box Generat
8. B c pieced 20 10 200 100 12000 User Manual 43 Real Options Super Lattice Solver Real Options SLS B ono User Manual 34 Real Options Super Lattice Solver Real Options SLS
9. 0 10 1 2 10 20 35 Max Asset Salvage Cost 0 POTUM 130 3154 Max Salvage OptionOpen Max Asset Cost OptonOpen OptionOpen OptonOpen Figure 24 Customized Abandonment Option 52 Real Options Super Lattice Solver Real Options SLS User Manual 22 Ametican European Bermudan and Customized Contraction Options Contraction Option evaluates the flexibility value of being able to reduce producti
10. Figure 9 MSLS Solution to a Simple Two Phased Sequential Compound Option The strategy tree for this option is seen in Figure 10 The project is executed in two phases the first phase within the first year costs 5 million while the second phase within two years but only after the first phase is executed and costs 80 million both in present value dollars The PV Asset of the project is 100 million NPV is therefore 15 million and faces 30 volatility in its cash flows see the Appendix on Volatility for the relevant volatility computations The computed strategic value using the MSLS is 27 67 million indicating that there is a 12 67 million in option value That is spreading out and staging the investment into two phases has significant value an expected value of 12 67 million to be exact Cash flow generating activities PV Asset 100M YearQ Year1l Year 2 Figure 10 Strategy tree for two phased sequential compound option User Manual 20 Real Options Super Lattice Solver Real Options SLS User Manual 1 5 Multinomial Lattice Solver The Muattinomial Lattice Soler MNLS is another module of the Real Options Super Lattice Solver software The MNLS applies multinomial lattices where multiple branches stem
11. 0 50 1 2 10 20 35 Max Asset Cost 0 Max Asset Cost OptionOpen OptionOpen Figure 39 Simple American Bermudan and European Options with Dividends Blackout Steps User Manual 71 Real Options Super Lattice Solver Real Options SLS User Manual 2 6 Basic American European and Bermudan Put Options The American and European Put Options without dividends are calculated using the SLS in Figure 40 The sample results of this calculation indicate the strategic value of the project s NPV and provide an option to sell the project within the specified Maturity in years There is a chance that the project value can significantly exceed the single point estimate of PVY Asset Value measur
12. 578 9030 565 8139 E pop Max Asset Cost OptionOpen OptionOpen Figure 30 American and European Options to Expand with Dividend Rate 62 Real Options Super Lattice Solver Real Options SLS Figure 31 Single Asset Super Lattice Solver American Option to Expand To change to European deselect Custom and select European JZ V 400 S 250 5 100
13. _ 708 2317 Max Asset Expansion Cost OptionOpen Max Asset Cost OptionOpen B OptionOpen El cons secco rome Figure 33 Customized Expansion Option 65 Real Options Super Lattice Solver Real Options SLS 2 4 Contraction Expansion and Abandonment Options The Contraction Expansion and Abandonment Option applies when a firm has three competing and mutually exclusive options on a single project to choose from at different times up to the time of expiration Be aware that this is a mutually exclusive set of options That is you cannot execute any combinations of expansion contraction or abandonment at the same time Only one option can be executed at any time That is for mutually exclusive options use a single model to compute the option value as seen in Figure 34 example file used Expand Contract Abandon American and European Option However if the options are non mutually exclusive calculate them individually in different models and add up the values for the total value of the strategy
14. If Asset Barrier Max Asset Cost OptionOpen OptionOpen Max Asset Cost OptionOpen B OptionOpen El Figure 64 Up and In Upper American Barrier Option 100 Real Options Super Lattice Solver Real Options SLS Upper Up and Out Call This option is live only when the asset value doesnt breach the upper barier J 100 7 80 7 5 100 1 2 10 20 35
15. o C 117 4220 Figure 34 Ametican European and Custom Options to Expand Contract and Abandon User Manual 66 Real Options Super Lattice Solver Real Options SLS Figure 35 illustrates a Bermudan Option with the same parameters but with certain blackout periods example file used Expand Contract Abandon Bermudan Option while Figure 36 example file used Expand Contract Abandon Customized Option 1 illustrates a more complex Custom Option where during some earlier period of vesting the option to expand does not exist yet perhaps the technology being developed is not yet mature enough in the early stages to be expanded into some spin off technology In addition during the post vesting period but prior to maturity the option to contract or abandon does not exist perhaps the technology is now being reviewed for spin off opportunities and so forth Finally Figure 37 uses the same example in Figure 36 but now the input parameters salvage value are allowed to change over time perhaps accounting for the increase in project asset or firm value if abandoned at different t
16. 250 5 100 1 2 10 20 35 Call Put 238 238 238 Max Asset Cast 0 238 638 7315 Max Asset Expansion Cost OptionOpen ysna cpoka Max Asset Asset Expansion Cost Max Asset Cost OptionOpen OptionOpen Figure 29 American and Eutopean Options to
17. Figure 47 Solving a Multi Phased Sequential Compound Option using MSLS User Manual 79 Real Options Super Lattice Solver Real Options SLS 2 10 Customizing Sequential Compound Options The Sequential Compound Option can be further complicated by adding customized options at each phase as illustrated in Figure 48 where at every phase there may be different combinations of mutually exclusive options including the flexibility to stop investing abandon and salvage the project in return for some value expand the scope of the project into another project e g spin off projects and expand into different geographical locations contract the scope of the project resulting in some savings or continue on to the next phase The seemingly complicated option can be very easily solved using MSLS as seen in Figure 49 example file used Multiple Phased Complex Sequential Compound Option In reality an R amp D project will yield intellectual property and patent rights that the firm caneasily license off Abandon PHASE Ill 22 In addition at any phase the project s development be slowed down Contract or accelerated Expand depending on the outcome of each phase PHASE II CONTRACT ABANDON END ABANDON o END START An NPV analysis cannot account for these options to make midcourse corrections over time when unce
18. 250 5 7 100 1 2 10 20 35 Max Asset Asset Expansion Cost Call Put 204 205 204 Max Asset Cost 0 205 570 4411 Max Asset Expansion Cost OptionOpen Max Asset Cost OptionOpen B O
19. This option to abandon can only be executed at expiration and not before V Eia M n 90 0 120 90 5 7 100 1 2 10 20 35 Max Asset Salvage Max Asset Cost 0 124 5054 OptionOpen
20. 1 2 10 20 35 Max Asset Cost 0 12 3113 12 3113 12 3113 Figure 38 Simple American Bermudan and European Options without Dividends User Manual 70 Real Options Super Lattice Solver Real Options SLS V V V 100 100 2 net 1 100
21. 1 2 10 20 35 31 9863 18 6183 Max Asset Cost OptionOpen Max Asset Cost OptionOpen 89 Real Options Super Lattice Solver Real Options SLS American Put Option with Mean Reverting Underlying Asset using Trinomial Lattice V lt A 100 100 _ 25
22. lt __ c 128 Bit AES E lt R amp D 10 00 1 Elegant 3D 1 Elegant 3D 2 Elegant 2D 3 Fuzzy Links 3D 4 Fuzzy Plain 3D 5 On Fire 3D 6 Clear Point 7 Hollow Point 3D User Defined Style 2 User Defined Style 3 E X Figure 18B Strategy Trees Real Options SLS User Manual 1 12 Key SLS Notes and Tips Here
23. Berno Figure 45 Solving Two Phased Sequential Compound Option using MSLS 77 Real Options Super Lattice Solver Real Options SLS User Manual 2 9 Multiple Phased Sequential Compound Options The Sequential Compound Option can similarly be extended to multiple phases with the use of MSLS A graphical representation of a multi phased or stage gate investment is seen in Figure 46 The example illustrates a multi phase project where at every phase management has the option and flexibility to either continue to the next phase if everything goes well or to terminate the project otherwise Based on the input assumptions the results in the MSLS indicate the calculated strategic value of the project while the NPV of the project is simply the Asset less all Implementation Costs in present values if implementing all phases immediately Therefore with the strategic option value of being able to defer and wait before implementing future phases because due to the volatility there is a possibility that the asset value will be significantly higher Hence the ability to wait before making the investment decisions in the future is the option value or the strategic value of the project less the NPV Figure 47 shows the results using the MSLS Notice that due to the backward induction process used the analytical convention is to start with the last phase and
24. 80 5 100 1 2 10 20 35 M M if Asset lt Bamier Max Asset Cost 0 0 Max Asset Cost 0 7 3917 If Asset Barrier Max Asset Cost OptionOpen OptionOpen Max Asset Cost OptionOpen p Figure 62 Down and In Lower American Barrier Option 96 Real Options Super Lattice Solver Real Options SLS II O Lower Barrier Down and Out Call This option is live only when the asset value doesnt breach the lower
25. Figure 22 European Abandonment Option with 100 Step Lattice Sometimes a Bermudan option is appropriate where there might be a vesting period or blackout period when the option cannot be executed For instance if the contract stipulates that for the 5 year abandonment buy back contract the airline customer cannot execute the abandonment option within the first 2 5 years This is shown in Figure 23 using a Bermudan option with a 100 step lattice on 5 years where the blackout steps are from 0 50 This means that during the first 50 steps as well as right now or step 0 the option cannot be executed This is modeled by inserting Op onOpen into the Intermediate Node Equation During Blackout and Vesting Periods This forces the option holder to only keep the option open during the vesting period preventing execution during this blackout period You can see that the American option is worth more than the Bermudan option which is wotth more than the European option in Figure 23 by virtue of each option type s ability to execute early and the frequency of execution possibilities User Manual 50 Real Options Super Lattice Solver Real Options SLS Figure 23 Single Asset Super Lattice Solver n SLS
26. 100 Boe 0 50 1 2 10 20 35 Max Asset Asset Expansion ExpandCost Asset Contraction ContractSavings Salvage Call 26 00 26 00 26 00 Max Asset Cost 0 26 00 115 6590 Max Asset Asset Expansion ExpandCost Max Asset Cost OptionOpen B Max Asset Contraction ContractSavings Salvage OptionOpen OptionOpen Figure 36 Custom Options with Mixed Expand Contract and Abandon Capabilities User Manual
27. Figure 8 Multiple Super Lattice Solver To illustrate the power of the MSLS a simple illustration is in order Click on Start Programs Real Options Valuation Real Options SLS Real Options SLS In the Main Screen click on New Multiple Asset Option Model and then select File Examples Simple Two Phased Sequential Compound Option Figure 9 shows the MSLS example loaded In this simple example a single underlying asset is created with two valuation phases User Manual 19 Real Options Super Lattice Solver Real Options SLS Figure 9 Multiple Asset Super Lattice Solver Simple Two Phased Sequential Compound Option 100 30 1 5 5 0 50 Max Phase2 Cost 0 Max Phase2 Cost OptionOpen Phase2 80 5 0 100 Max Underlying Cost 0 Max Underlying Cost OptionOpen
28. Exotic Financial Options Valuator Strategy Tree Real Options 5 1 2 Functions amp Options Valuator Figure 1 Single Super Lattice Solver SLS 11 Real Options Super Lattice Solver Real Options SLS User Manual 1 3 Single Asset SLS Examples To help you get started several simple examples are in order A simple European call option 15 computed in this example using SLS To follow along in the Main Screen click on New Single Asset Model and then click File Examples Plain Vanilla Call Option I This example file will be loaded into the SLS software as seen in Figure 2 The starting PV Underlying Asset or starting stock price is 100 and the Implementation Cost or strike price is 100 with a 5 year maturity The annualized risk free rate of return is 5 and the historical comparable or future expected annualized volatility is 10 Click RUN Alt R and a 100 step binomial lattice is computed with the results indicating a value of 23 3975 for both the European and American call options Benchmark values using Black Scholes and partial differential Closed Form American approximation models as well as standard plain vanilla Binomial American and Binomial European Call and Put Options with 1 000 step binomial lattices are also computed Notice that only the American and European Opt
29. Rep 1 21 A 30 10 00 10M 1 2 1 1 R amp D 5 10 00 GM D Uo has merces
30. 1 2 10 20 35 Max Asset Cost Cost Asset 0 Max Asset Cost 0 Max Asset Cost Cost Asset OptionOpen Max Asset Cost OptionOpen Figure 41 Ametican and European Exotic Chooser Option using SLS User Manual 74 Real Options Super Lattice Solver Real Options SLS more complex Option can be constructed using the MSLS as seen in Figure 42 example Multiple Asset Option Module file used Exozic Complex Floating European Chooser and Figure 43 example file used Exotic Complex Floating American Chooser In these examples the execution costs of the call versus put are set at different levels An interesting example of a Complex Chooser Option is a firm developing a new technology that is highly uncertain and risky The firm tries to hedge its downside as well as capitalize its upside by creating a Chooser Option That is the firm can decide to build the technology itself once the research and development phase is complete versus selling the
31. R D nien ee am mount Cocco pecus Real Options SLS 10 00 R D R amp D 10 00 lt 30 10M R amp D 10 00 40 00 lt
32. 1 2 10 20 35 Max Cost Asset 0 Max Asset Cost 0 13 1408 18 7595 Max Cost Asset OptionOpen Max Asset Cost OptionOpen sanpera Figure 55A 55B Comparing Mean Reverting Calls and Puts to Regular Calls and Puts Other items of interest in mean reverting options include higher lower the long term rate level the higher lower the call options The higher lower the long term rate level the lower higher the put options Finally be careful when modeling mean reverting options as higher lattice steps are usually required and certain combinations of reversion rates long term rate level and lattice steps may yield unsolvable trinomial lattices When this occurs the MNLS will return error messages User Manual 90 Real Options Super Lattice Solver Real Options SLS User Manual 2 15 Jump
33. 1 2 10 20 35 Max Cost Asset 0 Max Asset Cost 0 24 4213 Max Cost Asset OptionOpen Max Asset Cost OptionOpen B OptionOpen Figure 40 American and European Put Options using SLS 73 Real Options Super Lattice Solver Real Options SLS 2 7 Exotic Chooser Options Many types of user defined and exotic options can be solved using the SLS and MSLS For instance Figure 41 shows a simple Exotic Chooser Option example file used Exotic Chooser Option In this simple analysis the option holder has two options a call and a put Instead of having to purchase or obtain two separate options one single option is obtained which allows the option holde
34. PHASEG 483 2670 Figure 51 Solving a Multiple Investment Simultaneous Compound Option using MSLS User Manual 84 Real Options Super Lattice Solver Real Options SLS User Manual 213 American and European Options Using Trinomial Lattices Building and solving trinomial lattices is similar to building and solving binomial lattices complete with the up down jumps and risk neutral probabilities but it is more complicated due to more branches stemming from each node At the limit both the binomial and trinomial lattices yield the same result as seen in the following table However the lattice building complexity is much higher for trinomial or multinomial lattices The only reason to use a trinomial lattice is because the level of convergence to the correct option value is achieved more quickly than by using a binomial lattice In the sample table notice how the trinomial lattice yields the correct option value with fewer steps than it takes for a binomial lattice 1 000 as compared to 5 000 Because both yield identical results at the limit but t
35. If Asset Barrier Max Asset Cost 0 0 Max Asset Cost 0 23 6931 K Asset Barrier Max Asset Cost OptionOpen OptionOpen Max Asset Cost OptionOpen B OptionOpen Figure 65 Up and Out Upper American Barrier Option User Manual 101 Real Options Super Lattice Solver Real Options SLS User Manual 2 19 American and European Double Barrier Options and Exotic Barriers The Double Barrier Option is solved using the binomial lattice This model measures the strategic value of an option this applies to both calls and puts that comes either in the money or out of the money when the Asset Value hits either the artificial Upper or Lower Barriers Therefore an Up and In and Down and In option for both calls and puts indicates that the option becomes live if the asset value either hits the uppe
36. 51 5 American Option to Expand Contract and Abandon To make it European simple change INE to OptionOpen S w0 c 5 100 0 5 45 100 Boe Thu 2 4620 35 Max Asset Asset Expansion ExpandCost Asset Contraction ContractSavings Salvage Max Asset Cost 0
37. 0 10 100 0 39 1 2 10 20 35 Max Asset Cost Max Asset Cost 0 26 1821 IF Asset Suboptimal Cost Max Asset Cost 0 IF Asset Suboptimal Cost ForfeiturePost D T Max Asset Cost 0 1 ForfeiturePost DT OptionOpen 0 Max Asset Cost OptionOpen B 1 ForfeiturePre DT OptionOpen OptionOpen Figure 73 SLS Re
38. 2 10 2 1 2 10 20 35 Max Asset Cost 0 Max Asset Cost 0 31 9863 A 18 6183 ysna EE Max Asset Cost OptionOpen 34 6900 Max Asset Cost OptionOpen B Figure 58 Quadranomial Lattice Results on Jump Diffusion Options 92 Real Options Super Lattice Solver Real Options SLS User Manual 2 16 Dual Variable Rainbow Options Using Pentanomial Lattices The Dual Variable Rainbow Option for bo
39. _ Pesynerar ysna cpoka 116 0737 Max Asset Asset Expansion ExpandCost Max Asset Cost OptionOpen Max Asset Contraction ContractSavings Salvage OptionOpen OptionOpen Figure 37 Custom Options with Mixed Expand Contract and Abandon Capabilities with Changing Input Parameters User Manual 69 Real Options Super Lattice Solver Real Options SLS 2 5 Basic American European and Bermudan Call Options Figure 38 shows the computation of basic American European and Bermudan Options without dividends example file used Bas American European versus Bermudan Call Options while Figure 39 shows the computation of the same options but with a dividend yield Of course European Options can only be executed at termination and not before while in American Options early exercise is allowed versus a Bermudan Option where eatly exercise is allowed except during blackout or vesting periods Notice that the results for the three options without dividends are identical for simple call options but they differ when dividends exist When dividends are included the simple ca
40. 4 2 2 1 2 10 20 35 Max Asset Asset2 Cost 0 Max Asset Cost 0 c rainbow 61 7481 Max Asset Asset2 Cost OptionOpen Max Asset Cost OptionOpen B Figure 61 Pentanomial Lattice Solving a Dual Asset Rainbow Option 94 Real Options Super Lattice Solver Real Options SLS User Manual 2 17 American and European Lower Bartier Options The Lower Barrier Option measutes the strategic value of an option this applies to both calls and puts that comes either in the money or out of the money when the Asset Value hits an artificial Lower Barrer that is currently lower than the asset value Therefore a Down and In option for both calls and puts indicates that the o
41. 100 7 10 7 100 Boe 0 39 1 2 10 20 35 Max Asset Cost 0 Max Asset Cost 0 49 7310 Max Asset Cost 0 OptionOpen Max Asset Cost OptionOpen OptionOpen OptionOpen Coss sec prego Figure 67 SLS Results of a Vesting Call Option User Manual 105 Real Options Super Lattice Solver Real Options SLS erican on wi esting Requiremen Assumptions Intermediate Calculations Stock Price 100 00 Stepping T ime dt 1 0000 Strike
42. 1001 6361 1001 4524 Nevers Max Asset Cost OptionOpen ysna Figure 26 American and Eutopean Options to Contract with 100 Step Lattice User Manual 56 Real Options Super Lattice Solver Real Options SLS Bermudan Contraction Option where contraction cannot occur at certain times 4 S 1000 1000 EE 100 1 2 10 20 35 Max Asset Ass
43. 1 2 10 20 35 Call Put ysna cpoka 160 Max Asset Asset Expansion Cost 175 160 Max Asset Cost 0 176 550 0000 Max Asset Expansion Cost OptionOpen Max Asset Cost OptionOpen B sanpera OptionOpen Figure 31 Dividend Rate Optimal Trigger Value User Manual 63 Real Options Super Lattice Solver Real Options SLS User Manual Bermudan Option to Expand no expansion during cooling off period at the blackout steps 4 400
44. Max Asset Cost 0 31 9863 31 9863 jnOpen B OptionOpen Figure 53 10 Step Binomial Lattice Comparison Result User Manual 87 Real Options Super Lattice Solver Real Options SLS User Manual 2 14 American and European Mean Reversion Options Using Trinomial Lattices The Mean Reversion Option in MNLS calculates both the American and European options when the underlying asset value is mean reverting A mean reverting stochastic process reverts back to the long term mean value Long Term Rate Level at a particular speed of reversion Reversion Rate Examples of variables following a mean reversion process include inflation rates interest rates gross domestic product growth rates optimal production rates price of natural gas and so forth Certain variables such as these succumb to either natural tendencies or economic business conditions to revert to a long term level when the actual values stray too far above or
45. 90 5 100 1 2 10 20 35 Max Asset Salvage Max Asset Cost 0 125 4582 Max Salvage OptionOpen Max Asset Cost OptionOpen OptionOpen Figure 21 American Abandonment Option with 100 Step Lattice User Manual 49 Real Options Super Lattice Solver Real Options SLS Figure 22 Single Asset Super Lattice Solver SLS
46. Bermudan Abandonment Option with Blackout Vesting Period American gt Bermudan gt European option is i 120 90 4 5 100 0 50 1 2 10 20 35 Call Put 54 39 54 39 5 36 54 39 448 Max Asset Cost 0 5439 5 44 Max Asset Salvage Pes 125 3417 Max Salvage OptionOpen
47. 10 1 2 10 20 35 Max Asset Salvage Max Asset Cost 0 Max Salvage OptionOpen Max Asset Cost OptionOpen Figure 19 Simple American Abandonment Option User Manual 47 Real Options Super Lattice Solver Real Options SLS User Manual Assumptions PV Asset Value Implementation Cost Years Risk free Rate Dividends Volatility 95 Lattice Steps Option Type User Defined Inputs Option Valuation Audit Sheet Intermediate Computations Stepping Time dt Up Step Size up Down Step Size down Risk neutral Probability 25 00 Results 120 00 Terminal Max Asset Salvage In
48. c co C _ 100 2 2 2 1 2 10 20 35 0 Max Asset Cost 0 31 9863 Max Asset Cost OptionOpen Max Asset Cost OptionOpen ysna
49. V V 100 100 5 1000 Max Asset Cost 0 23 4187 _ 23 4187 Figure 3 SLS Comparing Results with Benchmarks Alternatively you can enter Terminal and Intermediate Node Equations for a call option to obtain the same results Notice that using 100 steps and creating your own Terminal Node Equation of Max Asset Cost0 and Intermediate Node Equation of Max Asser Cost OptionOpen will yield the same answer When entering your own equations make sure that Custom Option is first checked 13 Real Options Super Lattice Solver Real Options SLS User Manual W
50. sanpera OptionOpen OptionOpen El cous propo Figure 71 SLS Results of a Call Option accounting for Vesting and Suboptimal Behavior User Manual 109 Real Options Super Lattice Solver Real Options SLS User Manual Assumptions Stock Price Strike Price Maturity in Years Risk free Rate 6 Dividends 9 Volatility 96 Suboptimal Exercise Multiple Vesting in Years Underlying Stock Price Lattice Option Valuation Lattice zm uw 5 2 0 oenl NEM 12 13 35 112 Intermediate Calculations Stepping Time dt Up Step Size up Down Step Size down Risk neutral Probability prob Results 10 Step Lattice Results 10 61 Generalized Black Scholes 100 Step Binomial Super Lattice Binomial Super Lattice Steps 100 Steps 100 Step Trinomial Super Lattice 9 43 Trinomial Super Lattice Steps 100 Steps Figure 72 ESO Toolkit Results of a Call Option accounting for Vesting and Suboptimal Behavior Real Options Super Lattice Solver Real Options SLS 3 4 American ESO with Vesting Suboptimal Exercise Behavior Blackout Periods and Forteiture Rate This example now incorporates the element of forfeiture into the model as seen in Figure 73 example file used ESO Vesting Blackout Suboptimal Forfeiture This means that if the option is vested and the prevailing stock price exceeds the
51. i a ioris 29 1 10 Payoff Charts Tornado Convergence Scenario Sensitivity Analysis Monte Carlo Risk Simmlation 30 TAI ROV o ITE T 31 1 12 Key SENIN dnd uda 39 SECTION I REAL OPTIONS ANALYSIS 43 2 1 American European Bermudan and Customized Abandonment Options 44 2 2 American European Bermudan and Customized Contraction Options 53 2 3 American European Bermudan and Customized Expansion Options 59 2 4 Contraction Expansion and Abandonment DONE Qa taie 66 2 5 Basic American European and Bermudan Call 70 User Manual 4 Real Options Super Lattice Solver Real Options SLS 2 6 Basic American European and Bermudan Put 72 2 47 Exotic Chooser 2 s ode m e eed vedete ide tpe n Rep pde cette Fe d oven 74 2 0 Sequential Compound ODORE ao ee dass d e d pee er e i ta Pete d dedit 76 2 9 Multiple Phased Sequential Compound Options ee eese ttes 78 2 10 Customizing Sequential Compound Options esee esent 80 2 11 Patb Dependent Path Independent Mutually Exclusive Non Mutually Exclusive and Complex Combinatorial Nested OPNS ite cued deus vars vende sgh d get te re dius ork irte e caus red endete seas 82 2 12 Sinsultaneous Compound i ie eiretie 83 2 13 Ame
52. Max Asset Cost 0 Max Asset Cost 0 23 3975 Max Asset Cost OptionOpen Max Asset Cost OptionOpen ysna B sanpera Figure 11 Multinomial Lattice Solver 21 Real Options Super Lattice Solver Real Options SLS User Manual Figure 11 shows a sample call and put option computation using trinomial lattices Note that the results shown in Figure 11 using a 50 step lattice are equivalent to the results shown in Figure 2 using a 100 step binomial lattice In fact a trinomial lattice or any other multinomial lattice provides identical answers to the binomial lattice at the limit but convergence is achieved faster at lower steps Because both yield identical results at the limit but trinomials are much mote difficult to calculate and take a longer computation time in practice the binomial lattice is usually used instead Nonetheless using the SLS software the computation times are only seconds making this traditionally difficult to run model computable almost instantly However a trinomial is required only under one special circumstance when the underlying asset follows a mean reverting process
53. Simultaneous Compound Option for Two Phases Underlying 0B gt PhaseB 200 5 0 100 Max PhaseA CostO Max PhaseA Cost OptionOpen Phase 500 5100 Max Underlying Cost0 Max Underlying Cost OptionOpen IPHASEB 483 2670 3 Lee Figure 50 Solving a Simultaneous Compound Option using MSLS
54. if Asset lt LowerBanier Asset gt UpperBamier Max Asset Cost 0 0 Max Asset Cost 0 _ _ 41 9996 If Asset LowerBarrier Asset gt UpperBarrier Max Asset Cost OptionOpen OptionOpen Max Asset Cost OptionOpen ysna sanpera OptionOpen Figure 66 Up and In Down and In Double Bartier Option User Manual 103 Real Options Super Lattice Solver Real Options SLS SECTION EMPLOY EE STOCK OPTIONS User Manual 104 Real Options Super Lattice Solver Real Options SLS 3 1 American ESO with Vesting Period Figure 67 illustrates how an employee stock option ESO with a vesting period and blackout dates can be modeled Enter the blackout steps 0 39 Because the blackout dates input box has been used you will need to enter the Terminal Node Equation TE Intermediate Node Equation IE and Intermediate Node Equation During Vesting and Blackout Periods EV Enter Max Stock Strike O for the
55. sanpera Figure 52 Simple Trinomial Lattice Solution User Manual 86 Real Options Super Lattice Solver Real Options SLS SLS Plain Vanila American and European Call Options V S 100 7 100 5 10 1 1 5
56. Max Stock Strike 0 OptionOpen for the IE and OptonOpen for example file used ESO Vesting This means the option is executed or left to expire worthless at termination execute eatly or keep the option open during the intermediate nodes and keep the option open only and no executions are allowed during the intermediate steps when blackouts or vesting occurs The result is 49 73 Figure 67 which can be corroborated with the use of the ESO Valuation Toolkit Figure 68 ESO Valuation Toolkit is another software tool developed by Real Options Valuation Inc specifically designed to solve ESO problems following the 2004 FAS 123 In fact this software was used by the Financial Accounting Standards Board to model the valuation example in their final FAS 123 Statement in December 2004 Before starting with ESO valuations it is suggested that the user read Dr Johnathan Mun s book Valuing Employee Stock Options Wiley 2004 as a primer Figure 67 Single Asset Super Lattice Solver SLS Employee Stock Option with a vesting period V 100
57. Click on the 1 License Real Options SLS and select ACTIVATE then browse to the SLS license file that we sent you Click on the 2 License Functions amp Options Valuator and enter in the NAME and KEY combination we sent you 147 Real Options Super Lattice Solver
58. Real Options SLS B 4 Management Assumption Abproacb ett ttt ttt ttt ttti 128 B 5 Market Proxy IS NR RR T 133 Appendix C Technical Formulae Exotic Options Formulas eee 135 Black and Scholes Option Model European 135 Black and Scholes with Drift Dividend European Version 136 Black and Scholes with Future Payments European 137 Chooser Options 138 d UG 139 Compound Oprions 141 142 Generalized 143 OPENS ONT s 144 Tape Corfu mud Assets 145 Appendix D Quick Install and Licensing Guide 146 n 147 REESE 147 User Manual 6 Real Options Super Lattice Solver Real Options SLS User Manual SECTION I GETTING STARTED Single Asset Super Lattice Solver SLS Multiple Asset Super Lattice Solver MSLS Multinomial Lattice Solver MNLS Lattice Audit Sheet Lattice Maker SLS Excel Solution SLS Functions Payoff Charts Sensitivity Analysis Scenario Tables Convergence Analysis Monte Carlo Risk Simulation Strategy Trees Real Options Super Lattice Solver Real Options SLS User Manual 11 Introduction to the Super Lattice Software SLS The Real Options Super Lattice Software SLS comprises several modules including the Single Super Lattice Sol
59. e 5 10 2 Steps 5 7 9 are blackout steps the symbol means skip size e 5 14 3 Steps 5 8 11 14 are blackout steps e 5 6 3 Step 5 is a blackout step e 5 6 3 Step 5 1s a blackout step white spaces are ignored Identifiers An identifier is a sequence of characters that begins with a z A Z After the first character a z A Z 0 9 _ are valid characters in the sequence Note that space is not a valid character However it can be used if the variable is enclosed in a pair of curly braces Identifiers are case sensitive except for function names The following are some examples of valid identifiers myVariable MYVARIABLE _myVariable myVartiable myVariable This is a single variable Numbers A number can be an integer defined as one or more characters between 0 and 9 The following are some examples of integers 0 1 00000 12345 Another type of number is a rea number The following are some examples of real numbers 0 3 0 0 0 1 3 9 5 934 3E3 3 5E 5 0 2E 4 3 2 2 3 5e 5 Operator Precedence The operator precedence when evaluating the equations is shown below However if there are two terms with two identical precedence operators the expression is evaluated from left to right o Parenthesized expression has highest precedence L Not and unary minus e g 3 S ra Br cR o lt gt lt lt gt gt amp Mathematical Expre
60. 1 2 on Microsoft s historical stock prices Dependent Variable MSFT Method ML ARCH Date 02 25 05 Time 00 20 Sample adjusted 3 52 Included observations 50 after adjusting endpoints Convergence achieved after 67 iterations Bollerslev Wooldrige robust standard errors amp covariance Coefficient Std Error z Statistic Prob C 23 14431 1301024 17 78930 0 0000 D MSFT 1 0 456040 0 062391 7 309364 0 0000 AR 1 0 967490 0 027575 3508601 0 0000 Variance Equation C 0 151406 0 028717 5 272435 0 0000 ARCH 1 0 148308 0 053559 2 769061 0 0056 GARCH 1 0 735869 0 097780 7 525790 0 0000 GARCH 2 0 867066 0 083186 10 42325 0 0000 R squared 0 898576 Mean dependent var 24 48620 Adjusted R squared 0 884424 S D dependent var 1 290867 S E of regression 0 438849 Akaike info criterion 1 106641 Sum squared resid 8 281300 Schwarz criterion 1 374324 Log likelihood 20 66602 F statistic 63 49404 Durbin Watson stat 1 308287 Prob F statistic 0 000000 Inverted AR Roots 97 Figure B5 Sample GARCH Results 127 Real Options Super Lattice Solver Real Options SLS User Manual B 4 Management Assumption Approach A simpler approach is the use of Management Assumptions This approach allows management to get a rough volatility estimate without performing more protracted analysis This approach is also great for educating management what volatility is and how it works Mathematically and statistically the width or risk of a variable
61. CF will follow in Years 3 to 6 yielding sum of PV Asset of 100 CF discounted at say 9 7 discount or hurdle rate and the Volatility of these CFs is 30 At a 5 risk free rate the strategic value is calculated at 27 67 as seen in Figure 45 using a 100 step lattice which means that the strategic option value of being able to defer investments and to wait and see until mote information becomes available and uncertainties become resolved is worth 12 67M because the NPV is worth 15M 100 5M 85 In other words the Expected Value of Perfect Information is worth 12 67M which indicates that assuming market research can be used to obtain credible information to decide if this project is a good 76 Real Options Super Lattice Solver Real Options SLS User Manual one the maximum the firm should be willing to spend in Phase I is oz average no more than 17 67M 12 67 5M if PI is part of the market research initiative or simply 12 67M otherwise If the cost to obtain the credible information exceeds this value then it is optimal to take the risk and execute the entire project immediately at 85M The Multiple Asset module example file used is Simple Two Phased Sequential Compound Option In contrast if the volatility decreases uncertainty and risk are lower the strategic option value decreases In addition when the cost of waiting as described by the Dividend Rate as a percentage of the Asset Va
62. If Asset Barrier Max Asset Cost OptionOpen OptionOpen Max Asset Cost OptionOpen ysna sanpera OptionOpen sree Figure 63 Down and Out Lower American Barrier Option User Manual 97 Real Options Super Lattice Solver Real Options SLS User Manual 2 18 American and European Upper Bartier Options The Upper Barrier Option measures the strategic value of an option this applies to both calls and puts that comes either in the money or out of the money when the Asset Value hits an artificial Upper Barrier that is currently higher than the asset value Therefore an Up and In option for both calls and puts indicates that the option becomes live if the asset value hits the upper barrier Conversely for the Up and Out option the option is live only when the upper barrier is not breached This is very similar to the Lower Barrier Option but now the barrier is above the starting asset value and for a binding barrier option the implementation cost is typically lower than the upper barrier That is the upper barrier is usually gt implementation cost and the upper barrier is also gt starting asset value Examples of this option include contractual agreements whereby if the upper barrier is breached some event or clause is triggered The values of barrier options ar
63. Max Asset Cost OptionOpen B OptionOpen Figure 23 Bermudan Abandonment Option with 100 Step Lattice Sometimes the salvage value of the abandonment option may change over time To illustrate in the previous example of an acquisition of a startup firm the intellectual property will most probably increase over time because of continued research and development activities thereby changing the salvage values over time An example is seen in Figure 24 where there are five salvage values over the 5 year abandonment option This can be modeled by using the Custom Variables Type in the Variable Name Value and Starting Step and hit ENTER to input the variables one at a time as seen in Figure 24 s Custom Variables list Notice that the same variable name Salvage is used but the values change over time and the starting steps represent when these different values become effective For instance the salvage value 90 applies at step 0 until the next salvage value of 95 takes over at step 21 This means that for a 5 year option with a 100 step lattice the first year including the current period steps 0 to 20 will have a salvage value of 90 which then increases to 95 in the second year steps 21 to 40 and so forth Notice that as the value of the
64. SLSSingle Figure 14 Customized Abandonment Option using SLS Excel Solution The only difference is that in the Excel Solution the function cell B18 in Figure 14 has an added input specifically the Type If the option type value is set to 0 you get an American option 1 for European option 2 for Bermudan option and 3 for customized options Similarly the MSLS can also be solved using the SLS Excel Solver Figure 15 shows a complex multiple phased sequential compound option solved using the SLS Excel Solver The results shown ate identical to the results generated from the MSLS module example file Multiple Phased Complex Sequential Compound Option One small note of caution here is that if you add or reduce the number of option valuation lattices make sure you change the function s link for the MSLS Result to incorporate the right number of rows otherwise the analysis will not compute properly For example the default shows 3 option valuation lattices and by selecting the MSLS Results cell in the spreadsheet and clicking on Insert Function you will see that the function links to cells A24 H26 for these three rows for the OVLattices input in the function If you add another option valuation lattice change the link to A24 H27 and so forth You can also leave the list of custom variables as is The results will not be affected if these variables are not used in the custom equations Finally Figure 16 shows a Chang
65. to chart as well as its step size e g setting minimum as 20 and maximum as 200 with a step of 10 means to run the analysis for the values 20 30 40 180 190 200 and lattice steps the lower the lattice step number the faster the analysis runs but the less precise the results see the following discussion of Lattice Step Convergence for more details Click Update Chart D to obtain a new payoff chart E each time The default is to show a line chart but you can opt to choose area or bar charts and the generated chart and table can be copied and pasted into other applications or printed out as is G If you do not enter in any minimum and maximum values the software automatically picks some default test values for you the PV Underlying Asset is chosen by default and the typical hockey stick payoff chart will be displayed Finally there will be a warning message if any of the original input is zero requiring you to manually insert these minimum maximum and step size values in order to generate the payoff chart The Sensitivity tab H runs a quick static sensitivity of each input variable of the model one at a time and lists the input variables with the highest impact to the lowest impact You can control the option type lattice steps and sensitivity to test I The results will be returned in the form of a tornado chart J and sensitivity analysis table Tornado analysis captures the static impacts of each input v
66. 0985 Using this intermediate value perform a Monte Carlo simulation on the discounted cash flow model thereby simulating the individual cash flows and obtain the resulting forecast distribution of X As seen previously the sample standard deviation of the forecast distribution of X is the volatility estimate used in the real options analysis It is important to note that only the numerator is simulated while the denominator remains unchanged 122 Real Options Super Lattice Solver Real Options SLS User Manual The downside to estimating volatility this way is that the approach requires Monte Carlo simulation but the calculated volatility measure 15 a single digit estimate as compared to the Logarithmic Cash Flow or Stock Price Approach which yields a distribution of volatilities that in turn yield a distribution of calculated real options values The main objection to using this method is its dependence on the variability of the discount rate used For instance we can expand the X equation as follows n CF CF CF CF zt X In 1 D 1 D 1 D 1 D F 289 m m 1 D 1 D 1 D 1 D where D represents the constant discount rate used Here we see that the cash flow series CF for the numerator is offset by one period and the discount factors are also offset by one period Therefore by performing a Monte Carlo simulation on the cash flows alon
67. 18 Real Options Super Lattice Solver Real Options SLS The MSLS software has several areas including a Maturity and Comment area The Maturity value is a global value for the entire option regardless of how many underlying or valuation lattices exist The Comment field 1s for your personal notes describing the model you are building There is also a Blackout and Vesting Period Steps section and a Custom Variables list similar to the SLS The MSLS also allows you to create Audit Worksheets Notice too that the user interface is resizable e g you can click and drag the right side of the form to make it wider Multiple Asset Super Lattice Solver lt
68. Diftusion Options Using Quadranomial Lattices The Jump Difjusion Calls and Puts for both American and European options applies the Quadranomial Lattice approach This model is appropriate when the underlying variable in the option follows a jump diffusion stochastic process Figure 56 illustrates an underlying asset modeled using a jump diffusion process Jumps ate commonplace in certain business variables such as price of oil and price of gas where prices take sudden and unexpected jumps e g during a war The underlying variable s frequency of jump is denoted as its Jump Rate and the magnitude of each jump is its Jump Intensity Underlying Asset Time Figure 56 Jump Diffusion Process The binomial lattice is only able to capture a stochastic process without jumps e g Brownian Motion and Random Walk processes but when there is a probability of jump albeit a small probability that follows a Poisson distribution additional branches are required The quadranomial lattice four branches on each node is used to capture these jumps as seen in Figure 57 Figure 57 Quadranomial Lattice 91 Real Options Super Lattice Solver Real Options SLS User Manual Be awate that due to the complexity of the models some calculations with higher lattice steps may take slightly longer to compute Furthermore certain combinations of inputs may yield negative implied risk neutral probabilities and result in noncomputable latt
69. Equation Intermediate Equation Intermediate Equation Blackouts Underlying Asset Lattice 125 06 122 29 119 59 119 59 116 94 114 36 114 36 111 83 109 36 109 36 109 36 106 94 106 94 104 57 104 57 104 57 104 57 102 26 102 26 102 26 100 00 100 00 100 00 100 00 100 00 97 79 97 79 97 79 95 63 95 63 95 63 95 63 93 51 93 51 91 44 91 44 91 44 89 42 87 44 87 44 85 51 83 62 83 62 81 77 79 96 Option Valuation Lattice 45 33 42 81 37 97 27 18 25 25 23 40 23 03 21 26 19 22 7 91 6 74 Figure 5 SLS Generated Audit Worksheet User Manual 17 Real Options Super Lattice Solver Real Options SLS 515 Plain Vanila American and European Call Options ower number of steps Useful for testing convergence V
70. Expand with 100 Step Lattice User Manual 61 Real Options Super Lattice Solver Real Options SLS User Manual American Option to Expand To change to European deselect Custom and select European _ _ V V i Expansion 2 20 400 7 250 7 2 5 35 100 1 2 10 20 35 ysna cpoka Max Asset Asset Expansion Cost Max Asset Cost 0
71. With the same logic quadranomials and pentanomials yield identical results as the binomial lattice with the exception that these multinomial lattices can be used to solve the following different special limiting conditions Trinomials Results are identical to binomials and are most appropriate when used to solve mean reverting underlying assets Quadranomials Results are identical to binomials and are most appropriate when used to solve options whose underlying assets follow jump diffusion processes Pentanomials Results are identical to binomials and are most appropriate when used to solve two underlying assets that are combined called rainbow options e g price and quantity are multiplied to obtain total revenues but price and quantity each follows a different underlying lattice with its own volatility but both underlying parameters could be correlated to one another See the sections Mean Reverting Jump Diffusion and Rainbow Options for more details examples and results interpretation In addition just like in the single asset and multiple asset lattice modules you can customize these multinomial lattices using your own custom equations and custom variables L6 SLS Lattice Maker The Lattice Maker module is capable of generating binomial lattices and decision lattices with visible formulas in Excel spreadsheet it is compatible with Excel XP 2003 2007 and 2010 Figure 12 illustrates example option
72. are applicable for research and development investments or any other investments that have multiple stages The MSLS is required for solving Sequential Compound Options The easiest way to understand this option is to start with a two phased example as seen in Figure 44 In the two phased example management has the ability to decide if Phase should be implemented after obtaining the results from Phase 1 PI For example a pilot project or market research in PI indicates that the market is not yet ready for the product hence PII is not implemented All that is lost is the PI sunk cost not the entire investment cost of both PI and An example below illustrates how the option is analyzed PHASE Il PHASE I START o END Volatility Measure based on these END uncertain cash flows N 9 9 9 9 9 9 09 START 5 80M 30 35 40M 48M INVESTMENT CASH FLOW PERIOD PERIOD Figure 44 Graphical Representation of a Two Phased Sequential Compound Option The illustration in Figure 44 is valuable in explaining and communicating to senior management the aspects of an American Sequential Compound Option and its inner wotkings In the illustration the I investment of 5M in present value dollars in Year 1 is followed by Phase investment of 80 in present value dollars in Year 2 Hopefully positive net free cash flows
73. as before providing some additional sample exercises 59 Real Options Super Lattice Solver Real Options SLS Suppose a pharmaceutical firm is thinking of developing a new type of insulin that can be inhaled and the drug will directly be absorbed into the blood stream A novel and honorable idea Imagine what this means to diabetics who no longer need painful and frequent injections The problem is this new type of insulin requires a brand new development effort but if the uncertainties of the market competition drug development and FDA approval are high perhaps a base insulin drug that can be ingested is first developed The ingestible version is a required precursor to the inhaled version The pharmaceutical firm can decide to either take the risk and fast track development into the inhaled version buy an option to defer to first wait and see if the ingestible version works this precursor works then the firm has the option to expand into the inhaled version How much should the firm be willing to spend on performing additional tests on the precursor and under what circumstances should the inhaled version be implemented directly Suppose the intermediate precursor development work yields an NPV of 100M but at any time within the next 2 years an additional 50M can be further invested into the precursor to develop it into the inhaled version which will triple the NPV However after modeling the tisk of technical success and
74. at a time Therefore remember to scroll down the list of variables by clicking on the vertical scroll bat to access the rest of the variables If you are a new user of Real Options SLS or have upgraded from an older version do spend some time reviewing the Key SLS Notes and Tips starting on the next few pages to familiarize yourself with the modeling intricacies of the software Search for a Function Or select a category Real Options Valuation Select a function SLSBinomialAmericanPut SLSBinomialChangingVolatility SLSBinomialDown SLSBinomialEuropeanCall SLSBinomialEuropeanPut SLSBinomialProbability SLSBinomialAmericanCall P Asset Cost Maturity Riskfree Returns the American call option with dividends using the binomial approach PY Asset 100 00 Cost 100 00 SL5BinomialAmericanCall Maturity 1 P Asset Risk Free 5 Volatility 25 Dividend 0 Maturity Steps 100 Riskfree Volatility 0 25 Cost Result 12 31 v 12 31130972 Returns the American call option with dividends using the binomial approach PVAsset Formula result 12 31130972 STEN EEG eNet Figure 17 Excel s Equation Wizard User Manual 28 Real Options Super Lattice Solver Real Options SLS 1 9 Exotic Financial Options Valuator The Exotic Financial Options Valuator is a comprehensive calculator of more than 250 functions and mode
75. exogenous variables that are irrelevant when estimating the risks of the project In addition the market valuation of a large public firm depends on multiple interacting and diversified projects Finally firms are levered but specific projects are usually unlevered Hence the volatility used in a real options analysis Oro should be adjusted to discount this leverage effect by dividing the volatility in equity prices by 1 D E where D E is the debt to equity ratio of the public firm That is we have O ggumv O ko gi D 1 This approach can be used if there are market comparables such as sector indexes or industry indexes It is incorrect to state that a project s risk as measured by the volatility estimate is identical to the entire industry sector or the market There are a lot of interactions in the market such as diversification overreaction and marketability issues that a single project inside a firm is not exposed to Great care must be taken in choosing the right comparables as the major drawback of this approach is that it is sometimes hard to find the right comparable firms and the results may be subject to gross manipulation by subjectively including or excluding certain firms The benefit is its ease of use industry averages are used and requires little to no computation 134 Real Options Super Lattice Solver Real Options SLS Appendix C Technical Formulae Exotic Options Formulas Black and
76. firm s intellectual property increases over time the option valuation results also increase which makes logical sense You can also model User Manual 51 Real Options Super Lattice Solver Real Options SLS in blackout vesting periods for the first 6 months steps 0 10 in the blackout area The blackout period is very typical of contractual obligations of abandonment options where during specified periods the option cannot be executed a cooling off period Note that you may use on the keyboard to move from the variable name column to the value column and on to the starting step column However remember to hit ENTER on the keyboard to insert the variable and to create a new row so that you may enter a new variable User Manual Bermudan Abandonment Option with changing salvage values over time 4 J V 120 3 5 S 90 7 0 5 25 100
77. flows implementation cost risk free rate time to expiration years volatility cumulative standard normal distribution continuous dividend payout or opportunity cost Call sevo MEMO a gem e ovT pu E820 00 ar seva User Manual 136 Real Options Super Lattice Solver Real Options SLS Black and Scholes with Future Payments European Version Here cash flow streams may be uneven over time and we should allow for different discount rates risk free rate should be used for all future times perhaps allowing for the flexibility of the forward risk free yield curve Definitions of Variables 5 present value of future cash flows X implementation cost r risk free rate T time to expiration years volatility cumulative standard normal distribution 4 continuous dividend payout or opportunity cost CF cash flow at time 7 Computation S S CFe CF CFe z Y che i l 2 e Call 2S In S X qt o 2 T Xe In S X r q o 2 T cT cT In S X r q 60 2 T E In S X r q 60 IDT Put X S et AN H User Manual 137 Real Options Super Lattice Solver Real Options SLS Chooser Options Basic Chooset This is the payoff for a simple chooser option when lt Tz or it doesn t work In addition it is ass
78. from each node such as trinomials three branches quadranomials four branches and pentanomials five branches Figure 11 illustrates the MNLS module The module has a Basic Inputs section where all of the common inputs for the multinomials are listed Then there are four sections with four different multinomial applications complete with the additional required inputs and results for both American and European call and put options To follow along with this simple example in the Main Screen click on New Mutltinomial Option Model and then select Fik Examples Trinomial American Call Option and set dividend to 0 and then hit run Figure 11 Multinomial Lattice Solver American Call Option using a Trinomial Lattice Model v 2 1 2 10 20 35
79. generated using this module The illustration shows the module inputs you can obtain this module by clicking Create Lattice from the Main Screen and the resulting output lattice Notice that the visible equations are linked to the existing spreadsheet which means this module will come in handy when running Monte Carlo simulations or when used to link to and from other spreadsheet models The results can also be used as a presentation and learning tool to peep inside the analytical black box of binomial lattices Last but not least a decision lattice is also available with specific decision nodes indicating expected optimal times of execution of certain options in this module The results generated from this module are identical to those generated using the SLS and Excel functions but has the added advantage of a visible lattice lattices of up to 200 steps can be generated using this module 22 Real Options Super Lattice Solver Real Options SLS User Manual CO CO CO CO CO CO CO CO CO CO hJ hJ h h O hJ MJ S Fard A st ge gt Ex anion en cn en en le iso n 4 Co ho 4 On Cn 4 CO B C D E F G H J K L Customized Real Options Results Assumptions Asset Value a ea Basic Inputs Basic Option isk free Rate 22 Dividends 22 PV Asset Implementation Cost 100 Ma
80. its vendors who had agreed to take up the excess capacity and space of the firm At the same time the firm can scale back and lay off part of its existing workforce to obtain this level of savings in present values The results indicate that the strategic value of the project 15 1 001 71M using a 10 step lattice as seen in Figure 25 which means that the NPV currently is 1 000M and the additional 1 71 comes from this contraction option This result is obtained because contracting now yields 90 of 1 000M 50M or 950M which is less than staying in business and not contracting and obtaining 1 000M Therefore the optimal decision is to not contract immediately but keep the ability to do so open for the future Hence in comparing this optimal decision of 1 000M to 1 001 71M of being able to contract the option to contract is worth 1 71M This should be the maximum amount the firm is wiling to spend to obtain this option contractual fees and payments to the vendor counterparty In contrast if Savings were 200M instead then the strategic project value becomes 1 100M which means that starting at 1 000M and contracting 10 to 900M and keeping the 200 in savings yields 1 100M in total value Hence the additional option value is 0M which means that it is optimal to execute the contraction option immediately as there is no option value and no value to wait to contract So the value of executing now is 1 100M as compared to
81. steps are shown below The calculations assume a constant discount rate The cash flows are discounted all the way to Time 0 and again to Time 1 with the cash flows in Time 0 ignored sunk cost Then the values are summed and the following logarithmic ratio is calculated User Manual 121 Real Options Super Lattice Solver Real Options SLS User Manual Time Period SUM Y PVCF PVCF i 0 where PCF is the present value of future cash flows at different time periods 2 This approach is more appropriate for use in real options where actual assets and projects cash flows are computed and their corresponding volatility is estimated This is applicable for project and asset cash flows and can accommodate less data points However this approach requires the use of Monte Carlo simulation to obtain a volatility estimate This approach reduces the measurement risks of autocorrelated cash flows and negative cash flows Present Value at Time Cash Flows 0 Present Value at Time 1 E UN 100 00 100 1 0 1 125 113 64 125 z 125 00 125 1 0 1 1 0 1 E 78 51 95 _ 86 36 95 10 1 0 1 105 78 89 105 86 78 105 1 0 1 1 0 1 155 105 87 E 116 45 155 1 0 1 1 0 13 nm 90 65 DE 99 72 146 1 0 1 1 0 1 567 56 514 31 Figure B3 Log Approach In the example above X is simply 514 31 567 56 0
82. suboptimal threshold above the strike price the option will be summarily and suboptimally executed If vested but not exceeding the threshold the option will be executed only if the post vesting forfeiture occurs but the option is kept open otherwise This means that the intermediate step is a probability weighted average of these occurrences Finally when an employee forfeits the option during the vesting period all options are forfeited with a pre vesting forfeiture rate In this example we assume identical pre and post vesting forfeitures so that we can verify the results using the ESO Toolkit Figure 74 In certain other cases a different rate may be assumed Figure 73 Single Asset Super Lattice Sover SLS Employee Stock Option with vesting period suboptimal exercise behavior and forfeiture rates ee ov V V i 01 0 100 _ 55 Forfeiture 01 0 100
83. such as stocks in financial options Sometimes used for other traded assets such as price of oil and price of electricity The drawback is that DCF models with only a few cash flows will generally overstate the volatility and this method cannot be used when negative cash flows occur The benefits include its computational ease transparency and modeling flexibility of the method In addition no simulation is required to obtain a volatility estimate Logarithmic Present Value Returns Approach Used mainly when computing the volatility on assets with cash flows a typical application is in real options The drawback of this method is that simulation is required to obtain a single volatility and is not applicable for highly traded liquid assets such as stock prices The benefit includes the ability to accommodate certain negative cash flows and applies more rigotous analysis than the logarithmic cash flow returns approach providing a more accurate and conservative estimate of volatility when assets are analyzed Generalized Autoregressive Moving Average GARCH Models Used mainly for computing the volatility on liquid and tradable assets such as stocks in financial options Sometimes used for other traded assets such as price of oil and price of electricity The drawback is that a lot of data is required advanced econometric modeling expertise is required and this approach is highly susceptible to user manipulation The benefit is that rigorous stati
84. take the natural logarithms of these relative returns The standard deviation of these natural logarithm returns is the periodic volatility of the cash flow series The resulting periodic volatility from the sample dataset in Figure B1 is 25 58 This value will then have to be annualized No matter what the approach used the periodic volatility estimate used in a real options or financial options analysis has to be an annualized volatility Depending on the periodicity of the raw cash flow or stock price data used the volatility calculated should be converted into annualized values using Ov P where P is the number of periods in a year and o is the periodic volatility For instance if the calculated volatility using monthly cash flow data is 10 the annualized volatility is 1096412 35 Similarly P 15 365 or about 250 if accounting for trading days and not calendar days for daily data 4 for quarterly data 2 for semiannual data and 1 for annual data Notice that the number of returns in Figure B1 1s one less than the total number of periods That is for time petiods 0 to 5 we have six cash flows but only five cash flow relative returns This approach is valid and correct when estimating the volatilities of liquid and highly traded assets historical stock prices historical prices of oil and electricity and is less valid for computing volatilities in a real options world where the underlying asset generates cash flows This is be
85. the MSLS for each option Examples include the ability to enter Japan from Years 0 3 Australia in Years 3 6 and U K at any time between Years 0 10 Each entry 82 Real Options Super Lattice Solver Real Options SLS User Manual strategy is not mutually exclusive if you can enter more than one countty and are path dependent as they are time dependent Nested Combinatorial Options These are the most complicated and can take a combination of any of the four types above In addition the options are nested within one another in that the expansion into Japan must come only after Australia and cannot be executed without heading to Australia first In addition Australia and U K are okay but you cannot expand to U K and Japan e g certain trade restrictions anti trust issues competitive considerations strategic issues restrictive agreements with alliances and so forth For such options draw all the scenarios on a strategy tree and use IF AND OR and MAX statements in MSLS to solve the option That is if you enter into U K that s it but f you enter into Australia you can still enter into Japan or U K but not Japan and U K 2 12 Simultaneous Compound Options The Simultaneous Compound Option evaluates a project s strategic value when the value of the project depends on the success of wo or more investment initiatives executed simultaneously in time The Sequential Compound Option evaluates these investments in stages o
86. the option ty expiration date for the option on the option years expiration for the underlying option years Computation First solve for the critical value of I using NC DELE IX q 0 127 zl oT 1 _ AL ED r q 0 I2YT o4 5 t Solve recursively for the value I above and then input it into 2 T 1 CIS is on cati s HR e 2 1 n1 4 0 I2 za of ed 2 2 xera BEP Imm 19057 off S Is 2 qto hei en User Manual 141 Real Options Super Lattice Solver Real Options SLS Forward Start Options Definitions of Variables 5 present value of future cash flows X implementation cost r risk free rate time when the forward start option begins years time to expiration of the forward start option years volatility cumulative standard normal distribution q continuous dividend payout Computation ae sereo EUe 2 1 zl e T t Ind a r q 0 2 05 1 x oT t In 1 a r q 60 2 1 T t oT 1 Os In a 9 0 12 oT 1 where is the multiplier constant Se eel Put setenta setenta Note If the option starts at X percent out of the money that is will be 1 X If it starts at the money will be 1 0 and 7 X if in the money User Manual 142 Real Options Super Lattice Solver Real Options SLS Gen
87. the strategic project value of 1 100M there is no additional option value and the contraction should be executed immediately That 53 Real Options Super Lattice Solver Real Options SLS User Manual 15 instead of asking the vendor to wait the firm is better off executing the contraction option now and capturing the savings Other applications include shelving an R amp D project by spending a little to keep it going but reserving the right to come back to it should conditions improve the value of synetgy in a merger and acquisition where some management personnel are let go to create the additional savings reducing the scope and size of a production facility reducing production rates a joint venture or alliance and so forth To illustrate here are some additional quick examples of the contraction option as before providing some additional sample exercises for the rest of us latge oil and gas company is embarking on a deep sea drilling platform that will cost the company billions to implement A DCF analysis is run and the NPV is found to be 500M over the next 10 years of economic life of the offshore rig The 10 year risk free rate is 5 and the volatility of the project is found to be at an annualized 45 using historical oil prices as a proxy If the expedition is highly successful oil prices are high and production rates are soaring then the company will continue its operations However if things are not
88. 0 Frequency L gt e a 50 2S PoQk EoE EB E So E Eo S S S E S E OR DQ OH NOTH x OO QN OG O o O H x OG 9 SA SG She RO PST ren PLNS FREE pee GO o dq 4 amp 4 ON DCA E EY Praia E Ere RU gre gan er S ES Sioa Log Returns Figure B12 Probability Distribution of Microsoft s Log Relative Returns B 5 Market Proxy Approach An often used not to mention abused and misused method in estimating volatility applies to publicly available market data That is for a particular project under review set of market comparable firms publicly traded stock prices are used These firms should have functions markets risks and geographical locations similar to those of the project under review Then using closing stock prices the standard deviation of natural logarithms of relative returns is calculated The methodology is identical to that used in the logarithm of cash flow returns approach previously alluded to The problem with this method is the assumption that the risks inherent in comparable firms are identical to 133 Real Options Super Lattice Solver Real Options SLS User Manual the risks inherent in the specific project under review The issue is that a firm s equity ptices are subject to investor overreaction and psychology in the stock market as well as countless other
89. 24 Real Options Super Lattice Solver Real Options SLS User Manual AJ D E H 24 Log Present Value Approach HE Input Parameters Results 9 Discount Rate Cash Flow Present Value Cash Flow 328 24 10 Discount Rate Impl Cost Present Value Impl Cost 189 58 11 Rate Net Present Value 138 67 12 2002 2003 2004 2005 2006 18 Revenue 500 00 22 Cost of Revenue 200 00 26 Gross Profit 60 00 120 00 180 00 240 00 300 00 31 Depreciation Expense 5 00 5 00 5 00 5 00 5 00 35 Interest Expense 3 00 39 Income Before Taxes 30 00 68 00 106 00 144 00 182 00 40 Taxes 3 00 6 80 10 60 14 40 18 20 41 Income After Taxes 27 00 61 20 95 40 129 60 163 80 42 Non Cash Expenses 12 00 46 Cash Flow 39 00 73 20 107 40 141 60 175 80 48 Implementation Cost 25 00 25 00 50 00 50 00 75 00 50 Volatility Estimates Logarithmic PV Approach 51 PV 0 39 00 63 65 81 21 93 10 100 51 52 PV 1 73 20 93 39 107 07 115 59 53 Static PV 0 39 00 63 65 81 21 93 10 100 51 54 Variable X 0 0307 55 Volatility Simulate Figure B4 Log Present Value Approach Now that you understand the mechanics of computing volatilities this way we need to explain why we did what we did Merely understanding the mechanics is insufficient in justifying the approach or explaining the ratio
90. 38 73 M less 550M is 88 73M the value of the ability to defer and to wait and see before executing the expansion option The example file used is Expansion American and European Option Increase the dividend rate to say 2 and notice that both the American and European Expansion Options are now worth less and that the American Expansion Option is worth more than the European Expansion Option by virtue of the American Option s ability for early execution Figure 30 The dividend rate implies that the cost of waiting to expand to defer and not execute the opportunity cost of waiting on executing the option and the cost of holding the option is high then the ability to defer reduces In addition increase the Dividend Rate to 4 9 and see that the binomial lattice s Custom Option result reverts to 550 the static expand now scenario indicating that the option is worthless Figure 31 This result means if the cost of waiting as a proportion of the asset value as measured by the dividend rate is too high then execute now and stop wasting time deferring the expansion decision Of course this decision can be reversed if the volatility is significant enough to compensate for the cost of waiting That is it might be worth something to wait and see if the uncertainty is too high even if the cost to wait is high Other applications of this option simply abound To illustrate here are some additional quick examples of the contraction option
91. 43 Multiple Asset Super Lattice Solver 1 Exotic Complex Floating American Chooser Option either a call or put option Underlying 60 25 5 0 100 Max Underlying Cost 0 Max Underlying Cost OptionOpen 5 0 100 Max Cost Underlying 0 Max Cost Underlying OptionOpen 5 0 100 Max CallOption PutOption 0 Max CallOption PutOption O COMBINATION 16 8675 m Figure 43 Complex American Exotic Chooser Option using MSLS User Manual 75 Real Options Super Lattice Solver Real Options SLS User Manual 2 8 Sequential Compound Options Sequential Compound Options
92. 5 Single Asset Super Lattice Solver SLS American and European Contraction Options V V 1000 S 1000 5 7 Max Asset Asset Contraction Savings Max Asset Cost 0 359 171 55 09 0 50 0 10 Put 359 138 38 359 170 28 359 138 32 10017133 1001 5629 El Figure 25 A Simple Ametican and European Options to Cont
93. 68 Real Options Super Lattice Solver Real Options SLS Figure 37 Single Asset Super Lattice Solver Customized Expansion Contraction and Abandonment Options with changing salvage values A J V 100 100 5 100 0 50 1 2 10 20 35 2600 Max Asset Asset Expansion ExpandCost Asset Contraction Contract Savings Salvage 26 00 26 00 Max Asset Cost 0 2600
94. B of free hard drive space To install the software make sure that your system has all the prerequisites Windows XP Windows Vista Windows 7 and beyond Excel XP Excel 2003 Excel 2007 Excel 2010 and beyond NET Framework 2 0 and beyond administrative rights 512MB of RAM or more and 100MB of free hard drive space Install the Real Options SLS software by either using the installation CD or going to the following web location www tealoptionsvaluation com clicking on Downloads and selecting Real Options SLS You can either select to download the FULL version assuming you have already purchased the software and have received the permanent license keys and the instructions to permanently license the software or a TRIAL version The trial version 15 exactly the same as the full version except that it expires after 14 days during which you would need to obtain the full license to extend the use of the software Install the software by following the onscreen prompts If you have the trial version and wish to obtain the permanent license visit www realoptionsvaluation com and click on the Purchase link left panel of the web site and complete the purchase order If you ate purchasing or have already purchased the software simply download and install the software 146 Real Options Super Lattice Solver Real Options SLS User Manual There are two licenses required to run Real Options SLS The first is a license for the Real Options S
95. Equal 39 Real Options Super Lattice Solver Real Options SLS User Manual OptionOpen at Terminal Nodes SLS or MSLS If OpzionOpen is specified as the Terminal Node equation the value will always evaluate to Not a Number error NaN This is clearly a user error as OpZiorOpen cannot apply at the terminal nodes Unspecified interval of custom variables If a specified interval with a custom vatiable has no value the value is assumed zeto For example suppose a model exists with 10 steps where a custom variable mylar of value 5 starts at step 6 exists This specification means myVar will be substituted with the value 5 from step 6 onwards However the model did not specify the value of myVar from steps 0 to 5 In this situation the value of mylar is assumed to be 0 for steps 5 Compatibility with SLS 1 0 Super Lattice Solver 2010 has a user interface similar to the previous version with the exception that SLS MSLS MNLS and Lattice Maker are all integrated into one Main Screen The data files created in SLS 1 0 can be loaded in SLS However because SLS includes advanced features that do not exist in the previous version the models created in SLS 1 0 may not run in SLS without some minor modifications The following lists the differences between SLS 1 0 and SLS o variable in SLS 1 0 has been replaced by in SLS Therefore SLS still recognizes as a special variable and
96. LS software single asset lattice models multiple assets and multiple phased models multinomial lattices and the lattice maker The second is a license for the Exotic Financial Valuator and the SLS Functions accessible inside Excel To license your software follow the simple steps below Preparation 1 Start Real Options SLS click on Start Programs Real Options Valuation Real Options SLS Real Options SLS Click on the 1 License Real Options SLS link and you will be provided with your HARDWARE ID this starts with the prefix SLS and should be between 12 and 20 digits Write this information down or copy it by selecting the identification number right click your mouse and select Copy and then Paste it in an e mail to us Click on the 2 License Functions amp Options Valuator link and write down or copy the HARDWARE FINGERPRINT it should be an 8 digit alphanumeric code Purchase a license at www realoptionsvaluation com by clicking on the Purchase link E mail admin realoptionsvaluation com these two identification numbers and we will send you your license file and license key Once you receive these please install the license using the steps below Installing Licenses 1 Save the SLS license file to your hard drive the license file we sent you after you purchased the software and then start Real Options SLS click on Start Programs Real Options Valuation Real Options SLS Real Options SLS
97. Lattice Solver useful when running simulations or when linking to and from other spreadsheets Use this sample spreadsheet for your models You can simply click on File Save As to save as a different file and start using the model Because this is an Excel solution the correlation function is not supported and is linked to an empty cell Figure 15 Complex Sequential Compound Option using SLS Excel Solver Real Options SLS Assumptions Results PV Asset Generalized Black Scholes Implementation Cost S 5 10 Step Super Lattice Maturity in Years i Super Lattice Steps 10Steps Vesting in Years Dividend Rate Additional Assumptions Year Risk free Volatility 1 00 20 00 Please be aware that by applying multiple 200 changing volatilities over time non recombining 3 00 20 00 lattice is required which increases the 400 20 0096 computation time significantly In addition only 7 smaller lattice steps may be computed The S 5 0096 5 00 5 00 100 200 300 400 500 600 900 10 00 10 00 Figure 16 Changing Volatility and Risk Free Rate Option 1 8 SLS Functions The software also provides a series of SLS functions that are directly accessible in Excel To illustrate its use start the SLS Functions by clicking on Szart Programs Real Options User Manual Valuation Real Options SLS SLS Functions and Excel will start Wh
98. PV is 100M a 35 volatility implies that 90 of the time the NPV will be less than 144 85M and that only 10 best case scenario of the time will the true NPV exceed this value Now that you understand the mechanics of estimating volatilities this way again we need to explain why we did what we did Merely understanding the mechanics is insufficient in justifying the approach or explaining the rationale why we analyzed it the way we did Hence let us look at the assumptions required and explain the rationale behind them e Assumption 1 We assume that the underlying distribution of the asset fluctuations is normal We can assume normality because the distribution of the final nodes on a super lattice is normally distributed In fact the Brownian Motion equation shown earlier requires a random standard normal distribution In addition a lot of distributions will converge to the normal distribution anyway a Binomial distribution becomes normally distributed when number of trials increase a Poisson distribution also becomes normally distributed with a high average rate a Triangular distribution is a normal distribution with truncated upper and lower values an so forth and it is not possible to ascertain the shape and type of the final NPV distribution if the DCF model is simulated with many different types of distributions e g revenues are Lognormally distributed and are negatively correlated to one another over time while operating e
99. Price Up Step Size up 1 6487 Maturity in Years Down Step Size down 0 6065 Risk Free Rate 96 Risk Neutrai Probability prob 39 6996 Dividends 96 volatility 96 Results Vesting in Years 10 Step Lattice Results Generalized Black Scholes American Closed Form Approx 100 Step Binomial Super Lattice Binomial Super Lattice Steps 100 Steps wv 10 Step Trinomial Super Lattice 44 95 Trinomial Super Lattice Steps 1055 v 9001 71 5459 82 5459 82 3311 55 2008 55 2008 55 1218 25 3311 55 2008 55 Underlying Stock Price Lattice 1218 25 1218 25 448 17 164 87 16487 _ 16487 100 00 100 00 100 00 100 00 60 65 60 65 60 65 60 65 36 79 2231 2231 821 183 _ 188 Vesting Calculation 3 11 067 oe 9 icons Option Valuation Lattice 1118 25 638 91 181 29 a 8 91 348 17 86 87 2248 Figure 68 ESO Valuation Toolkit Results of a Vesting Call Option User Manual 106 Real Options Super Lattice Solver Real Options SLS 3 2 American ESO with Suboptimal Exercise Behavior This example shows how suboptimal exercise behavior multiples can be included into the analysis and how the custom variables list can be used as seen in Figure 69 example file used ESO Subop
100. REAL OPTIONS SUPER LATTICE SOLVER JSER MANUAL REAL OPTIONS VALUATION INC o Real Options SLS A This manual and the software described in it are furnished under license and may only be used or copied in accordance with the terms of the end user license agreement Information in this document is provided for informational purposes only is subject to change without notice and does not represent a commitment as to merchantability or fitness for a particular purpose by Real Options Valuation Inc No part of this manual may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying and recording for any purpose without the express written permission of Real Options Valuation Inc Materials based on copyrighted publications by Dr Johnathan Mun Ph D MBA MS BS CRM CFC FRM MIFC Founder and CEO Real Options Valuation Inc and creator of the software Written designed and published in the United States of America Microsoft is a registered trademark of Microsoft Corporation in the U S and other countries Other product names mentioned herein may be trademarks and or registered trademarks of the respective holders Copyright 2005 2012 Dr Johnathan Mun All rights reserved Real Options Valuation Inc 4101F Dublin Blvd Ste 425 Dublin California 94568 U S A Phone 925 271 4438 Fax 925 369 0450 admin realoptionsvaluation com www tisksimulator com www tealoptionsvalua
101. Scholes Option Model European Version This 15 the famous Nobel Prize winning Black Scholes model without any dividend payments It is the European version where an option can only be executed at expiration and not before Although it is simple enough to use care should be taken in estimating its input variable assumptions especially that of volatility which is usually difficult to estimate However the Black Scholes model is useful in generating ballpark estimates of the true real options value especially for more genetic type calls and puts For more complex real options analysis different types of exotic options are required Definitions of Variables 5 present value of future cash flows X implementation cost r risk free rate T time to expiration years volatility cumulative standard normal distribution Computation sof 0 0 ar ERD ovT ES In S X r o 2 T In S X r o 2 T Put Xe 5 User Manual 135 Real Options Super Lattice Solver Real Options SLS Black and Scholes with Drift Dividend European Version This is a modification of the Black Scholes model and assumes a fixed dividend payment rate of q in percent This can be construed as the opportunity cost of holding the option rather than holding the underlying asset Definitions of Variables 5 X 4 Computation present value of future cash
102. Sep 04 27 53 27 57 26 74 27 51 51599880 2461 0 0008 20 52 19 7 Sep 04 27 29 27 51 27 14 2749 51935175 24 59 0 0139 21 30 20 30 Aug 04 27 30 27 68 26 85 27 11 45125980 24 25 0 0127 21 25 21 23 Aug 04 2727 27 67 27 09 27 46 40526880 24 56 0 0123 22 29 22 16 Aug 04 27 03 27 50 26 89 27 20 52571740 2426 0 0066 22 29 23 9 Aug 04 27 26 27 75 26 86 27 02 51244080 2410 0 0041 22 42 24 2 Aug 04 28 27 28 55 27 06 27 14 56739100 24 20 0 0488 22 42 25 26 Jul 04 28 36 28 81 28 13 2849 65555220 2541 0 0163 21 9796 26 19 Jul 04 27 62 29 89 27 60 28 03 114579322 25 00 0 0198 22 11 27 12 Ju 04 2767 28 36 27 25 27 48 57970740 24 51 0 0138 22 02 28 6 Jul 04 28 32 28 33 27 55 27 86 61197249 24 85 0 0250 22 04 29 28 Jun 04 2860 28 84 28 17 28 57 66214339 2548 0 0000 22 07 30 21 Jun 04 28 22 28 66 27 81 28 57 82202478 25 48 0 0079 22 30 31 14 Jun 04 26 55 28 50 26 53 28 35 97727643 25 28 0 0574 22 48 Figure B2 Computing Microsoft s 1 Year Annualized Volatility Clearly there are advantages and shortcomings to this simple approach This method is vety easy to implement and Monte Carlo simulation is not required to obtain a single point volatility estimate This approach is mathematically valid and is widely used in estimating volatility of financial assets However for real options analysis there are several caveats that deserve closer attention When cash flows are negative over certain time periods the relative returns will hav
103. a node all subsequent child nodes are also selected this allows you to move the entire tree starting from that selected node or if you wish to select only that node you may have to click on the empty background and click back on that node to select it individually Also you can move individual nodes or the entire tree started from the selected node depending on the current setting right click or in the Edit menu you can select to move nodes individually or together The following are some quick descriptions of the things that can be customized and configured in the node properties user interface It is simplest to try different settings for each of the following to see its effects in the Strategy Tree o Name o Value o Excel Link Notes Insert Above or Below a Node Show in Model Name Value Notes Local Color versus Global Color Label Inside Shape o Branch Event Name Select Real Options Global Elements are all customizable including elements of the Strategy Tree s Background Connection Lines Option Nodes Terminal Nodes and Text Boxes For instance the following settings can be changed for each of the elements o Font settings on Name Value Notes Label Event names o Node Size minimum and maximum height and width Borders line styles width color o Shadow colors and whether to apply a shadow or not o Global Color o Global Shape The Edit menu s View Data Requirements Window command open
104. aluable in this case e high tech disk drive manufacturer is thinking of acquiring a small startup firm with a new micro drive technology a super fast and high capacity pocket hard drive that may revolutionize the industry The startup is for sale and its asking price is 50M based on an NPV fair market value analysis some third party valuation consultants have performed The manufacturer can either develop the technology themselves or acquire this technology through the purchase of the firm The question is how much is this firm worth to the manufacturer and is 50M a good price Based on internal analysis by the manufacturer the NPV of this micro drive is expected to be 45M with a cash flow volatility of 40 and it would take another 3 years before the micro drive technology is successful and goes to market Assume that the 3 year risk free rate is 5 In addition it would cost the manufacturer 45M in present value to develop this drive internally If using an NPV analysis the manufacturer should build it themselves However if you include an abandonment option analysis whereby if this specific micro drive does not work the startup still has an abundance of intellectual property patents and proprietary technologies as well as physical assets buildings and manufacturing facilities that can be sold in the market at up to 40 The abandonment option together with the NPV yields 51 83 making buying the startup worth more than develo
105. and 2 0 academic research and previous valuation consulting experience at KPMG Consulting 5 A nonrecombining binomial lattice bifurcates splits into two every step it takes so starting from one value it branches out to two values on the first step 21 two becomes four in the second step 22 and four becomes eight in the third step 23 and so forth until the 1 000th step 21000 or over 10301 values to calculate and the world s fastest supercomputer won t be able to calculate the result within our lifetimes 113 Real Options Super Lattice Solver Real Options SLS Convergence in Binomial Lattice Steps 17 20 4 17 10 4 17 00 16 90 Black Scholes 16 80 4 Option Value 16 70 16 60 16 50 1 10 100 1000 10000 Lattice Steps Black Scholes Result 12 336 Binomial 5 Step Lattice 12 795 Binomial 10 Step Lattice 12 093 Binomial 20 Step Lattice 12 213 Binomial 50 Step Lattice 12 287 Binomial 100 Step Lattice 12 313 Binomial 1 000 Step Lattice 12 336 Figure A1 Convergence of the Binomial Lattice Results to Closed Form Solutions User Manual 114 Real Options Super Lattice Solver Real Options SLS User Manual Appendix B Volatility Estimates There are several ways to estimate the volatility used in the option models Logarithmic Cash Flow Returns Approach or Logarithmic Stock Price Returns Approach Used mainly for computing the volatility on liquid and tradable assets
106. are only interested in the volatility of revenues In fact if the proportions remain constant the volatilities computed are identical e g revenues of 100 200 300 400 500 versus a 10 proportional EBITDA of 10 20 30 40 50 yields identical 20 80 volatilities Finally taking the oil and gas example a step further computing the volatility of revenues assuming no other market risks exist below this revenue line in the DCF is justified because this firm may have global operations with different tax conditions and financial leverages different ways of funding projects The volatility should only apply to market risks and not private risks how good a negotiator the CFO is on getting foreign loans or how shrewd your CPAs are in creating offshore tax shelters Now that you understand the mechanics of computing volatilities this way we need to explain why we did what we did Merely understanding the mechanics is insufficient in justifying the approach or explaining the rationale why we analyzed it the way we did Hence let us look at the steps undertaken and explain the rationale behind them e Step 1 Collect the relevant data and determine the periodicity and time frame You can use forecast financial data cash flows from a DCF model comparable data comparable market data such as sector indexes and industry averages or historical data stock prices or price of oil and electricity Consider the periodicity and time frame
107. are s Main Screen After installing the software the user can access the SLS Main Screen by clicking on Start Programs Real Options Valuation Real Options SLS Real Options SLS From this Main Screen you can run the Single Asset model Multiple Asset model Multinomial model Lattice Maker and Advanced Exotic Financial Options Valuator open example models and open an existing model You can move your mouse over any one of the items to obtain a short description of what that module does You may also purchase or install a newly obtained permanent license from this main screen by clicking on each of the two license links at the bottom Finally Real Options SLS supports 7 languages including English Chinese Spanish Japanese Italian German and Portuguese and you can change the language using the droplist on the main screen To access the SLS Functions SLS Excel Solutions or a sample Volatility computation file go to Start Programs Real Options Valuation Real Options SLS and select the relevant module EIE Real Options Valuation SLS 2012 lt
108. are some noteworthy changes from the previous version and interesting tips on using Real Options SLS The User Manual is accessible within SLS MSLS or MNLS For instance simply start the Real Options SLS software and create a new model or open an existing SLS MSLS or MNLS model Then click on He p User Manual Example Files are accessible directly in the SLS Main Screen when in the SLS MSLS or MNLS models you can access the example files at Fe Examples ness License information can be obtained in SLS MSLS or MNLS at About A Variable List is available in SLS MSLS and MNLS by going to Heip Variable List Specifically the following are allowed variables and operators in the custom equations boxes Asset The value of the underlying asset at the current step in currency Cost The implementation cost in currency o Dividend The value of dividend in percent o Maturity The years to maturity in years OptionOpen The value of keeping the option open formerly in version 1 0 o RiskFree The annualized risk free rate in percent o Step The integer representing the current step in the lattice Volatility The annualized volatility in percent Subtract o l Not lt gt Not equal amp Multiply o Divide o Power lt gt lt gt Comparisons
109. ariable on the outcome of the option value by automatically perturbing each input some preset amount captures the fluctuation on the option value s result and lists the resulting perturbations ranked from the most significant to the least The results are shown as a sensitivity table with the starting base case value the perturbed input upside and downside the resulting option value s upside and downside and the absolute swing or impact The precedent variables are ranked from the highest impact to the lowest impact The tornado chart illustrates this data in graphic form Green bars in the chart indicate a positive effect while red bars indicate a negative effect on the option value For example Implementation Cost s red bar is on the right side indicating a negative effect of investment cost in other words for a simple call option implementation cost option strike price and option value are negatively correlated The opposite is true for PV 30 Real Options Super Lattice Solver Real Options SLS Scenario Analysis Lattice Step Convergence Analysis Monte Carlo Simulation User Manual Underlying Asset stock price where the green bar is on the right side of the chart indicating a positive correlation between the input and output The Scenario tab runs a two dimensional scenario of two input variables L based on the selected option type and lattice steps M and returns a scenario analysis table N of the resultin
110. ate our using the data back that far Step 2 Compute relative returns Relative returns are used in geometric averages while absolute returns are used in arithmetic averages To illustrate suppose you purchase an asset or stock for 100 You hold it for one petiod and it doubles to 200 which means you made 100 absolute returns You get greedy and keep it for one more petiod when you should have sold it and obtain the capital gains The next petiod the asset goes back down to 100 which means you lost half the value or 50 absolute returns Your stockbroker calls you up and tells you that you made an average of 25 returns in the two periods the arithmetic average of 100 and 50 is 25 You started with 100 and ended up with 100 You clearly did not make a 25 return Thus an atithmetic average will over inflate the average when fluctuations occur fluctuations do occut in the stock market or for your real options project otherwise your volatility is very low and there s no option value and hence no point in doing an options analysis A geometric average is a better way to compute the return The computation is seen below and you can cleatly see that as part of the geometric average calculation relative returns are computed That is if 100 goes to 200 the relative return is 2 0 and the absolute return 15 100 or when 100 goes down to 90 the relative return is 0 9 anything less than 1 0 is a loss 10 absolute retu
111. barrier 4 100 S 80 5 7 100 1 2 10 20 35 IK Asset Barrier Max Asset Cost 0 Max Asset Cost 0 42 1937
112. below this level For instance monetary and fiscal policy will prevent the economy from significant fluctuations while policy goals tend to have a specific long term target rate or level Figure 54 illustrates a regular stochastic process dotted red line versus a mean reversion process solid line Clearly the mean reverting process with its dampening effects will have a lower level of uncertainty than the regular process with the same volatility measure J j 2 4 4 Long run mean value vi v Underlying Asset Mean reverting process Time Figure 54 Mean Reversion in Action Figure 55 shows the call and put results from a regular option modeled using the Trinomial Lattice versus calls and puts assuming a mean reverting MR tendency of the underlying asset using the Mean Reverting Trinomial Lattice Several items are worthy of attention e The MR Call lt regular Call because of the dampening effect of the mean reversion asset The MR asset value will not increase as high as the regular asset value e Conversely the MR Put gt regular Put because the asset value will not rise as high indicating that there will be a higher chance that the asset value will hover 88 Real Options Super Lattice Solver Real Options SLS User Manual around the PV Asset and a higher probability it will be below the PV Asset making the put option more valuable e With the dampening effect the MR Call and MR P
113. bining Two Binomial Lattice 93 Real Options Super Lattice Solver Real Options SLS Figure 61 shows an example Dual Asset Rainbow Option example file used MMNLS Dual Asset Rainbow Option Pentanomial Lattice Notice that a high positive correlation will increase both the call option and put option values This is because if both underlying elements move in the same direction there is a higher overall portfolio volatility price and quantity can fluctuate at high high and low low levels generating a higher overall underlying asset value In contrast negative correlations will reduce both the call option and put option values for the opposite reason due to the portfolio diversification effects of negatively correlated variables Of course correlation here is bounded between 1 and 1 inclusive User Manual Figure 61 Multinomial Lattice Solver American Rainbow Option using Pentanomial Lattice 7 2
114. can be measured through several different statistics including the range standard deviation 0 variance coefficient of variation and percentiles Figure B6 illustrates two different stocks historical prices The stock depicted as a dark bold line is clearly less volatile than the stock with the dotted line The time series data from these two stocks can be redrawn as a probability distribution as seen in Figure B7 Although the expected value of both stocks are similar their volatilities and hence their risks are different The x axis depicts the stock prices while the y axis depicts the frequency of a particular stock price occurring and the area under the curve between two values is the probability of occurrence The second stock dotted line in Figure has a wider spread a higher standard deviation 07 than the first stock bold line in Figure The width of Figure B7 s x axis is the same width from Figure B6 s y axis One common measure of width is the standard deviation Hence standard deviation is a way to measure volatility The term volatility is used and not standard deviation because the volatility computed is not from the raw cash flows or stock prices themselves but from the natural logarithm of the relative returns on these cash flows or stock prices Hence the term volatility differentiates it from a regular standard deviation Stock prices Time Figure B6 Volatility 128 Real Options Super Lattice So
115. cause to obtain valid results many data points are required and in modeling real options the cash flows generated using a DCF model may only be for 5 to 10 periods In contrast a large number of historical stock prices or oil prices can be downloaded and analyzed With smaller data sets this approach typically overestimates the volatility Cash Flow Relative Natural Logarithm of Cash Flows Returns Cash Flow Returns X 100 125 125 100 1 25 In 125 100 0 2231 95 95 125 0 76 In 95 125 0 2744 105 105 95 1 11 In 105 95 0 1001 155 155 105 1 48 In 155 105 0 3895 146 146 155 0 94 In 146 155 0 0598 Figure B1 Log Cash Flow Returns Approach 116 Real Options Super Lattice Solver Real Options SLS User Manual The volatility estimate is then calculated as 2 Y x 25 58 n li volatility where is the number of Xs and is the average X value To further illustrate the use of this approach Figure B2 shows the stock prices for Microsoft downloaded from Yahoo Finance a publicly available free resource You can follow along the example by loading the example file Start Programs Real Options Valuation Real Options Super Lattice Solver Volatility Estimates and select the worksheet tab Log Cash Flow Approach The data in columns A to G in Figure B2 are downloaded from Yahoo The formula in cell I3 is simply LN G3 G4 to compute the natural logarith
116. d and can be retrieved for future use The results will not be saved because you may accidentally delete or change an input and the results will no longer be valid In addition re running the super lattice computations will only take a few seconds and it is always advisable for you to re run the model when opening an old analysis file You may also enter Blackout Steps These are the steps on the super lattice that will have different behaviors than the terminal or intermediate steps For instance you can enter 7000 as the lattice steps 0400 as the blackout steps and some Blackout Equation e g OptionOpen This means that for the first 400 steps the option holder can only keep the option open Other examples include entering 7 3 5 10 if these are the lattice steps where blackout periods occur You will have to calculate the relevant steps within the lattice where the blackout exists For instance if the blackout exists in years 1 and 3 on a 10 year 10 step lattice then steps 1 3 will be the blackout dates This blackout step feature comes in handy when analyzing options with holding periods vesting periods or periods where the option cannot be executed Employee stock options have blackout and vesting periods and certain contractual real options have periods during which the option cannot be executed e g cooling off periods or proof of concept periods If equations are entered into the Terminal Node Equation box and American Euro
117. drilling to obtain more oil to sell at the higher price which will cost another 50M thereby increasing the NPV by 20 The economic life of this platform is 10 years and the risk free rate for the corresponding term is 5 Is obtaining this slightly larger platform worth it Using the SLS the option value is worth 827 12M when applying 100 step lattice Therefore the option cost of is worth it However this expansion option will not be worth it if annual dividends exceed 0 75 or 7 5M year this is the annual net revenues lost by waiting and not drilling as a percentage of the base case NPV Figure 32 shows a Bermudan Expansion Option with certain vesting and blackout steps while Figure 33 shows a Customized Expansion Option to account for the expansion factor changing over time Of course other flavors of customizing the expansion option exist including changing the implementation cost to expand and so forth User Manual 60 Real Options Super Lattice Solver Real Options SLS Bona samem O American Option to Expand To change to European deselect Custom and select European V S 400
118. e invest if the NPV is positive In such a case although you obtain zeto value for the option the analytical interpretation is significant A zero or very low value is indicative of an optimal decision not to wait 78 Real Options Super Lattice Solver Real Options SLS PHASES IV X The analysis can be extended to multiple PHASE phases the software accommodates up d to 10 phases where the success of 60M Phase depends on the success of sa Phase which in turn depends on the PHA SE II success of Phase END START o END TOY Spreading investments out to several phases will reduce the risk of future investments A END regular NPV will not yield reasonable results because at any checkpoint management can pull the plug onthe project Figure 46 Graphical Representation of a Multi Phased Sequential Compound Option vapers 5 Sequential Compound Option for Multiple Phases
119. e Audit Worksheet For instance loading the example file 14 Real Options Super Lattice Solver Real Options SLS User Manual Plain Vanilla Call Option I and selecting the box creates a worksheet as seen in Figure 5 There are several items that should be noted about this audit worksheet e The audit worksheet generated will show the first 10 steps of the lattice regardless of how many you enter That is if you enter 1 000 steps the first 10 steps will be generated If a complete lattice is required simply enter 10 steps in the SLS and the full 10 step lattice will be generated instead The Intermediate Computations and Results are for the Super Lattice based on the number of lattice steps entered and not based on the 10 step lattice generated To obtain the Intermediate Computations for 10 step lattices simply re run the analysis inputting 70 as the lattice steps This way the Audit Worksheet generated will be for a 10 step lattice and the results from SLS will now be comparable Figure 6 e The worksheet only provides values as it is assumed that the user was the one who entered the terminal and Intermediate Node Equations hence there is really no need to recreate these equations in Excel again The user can always reload the SLS file and view the equations or print out the form if required by clicking on Fie Prini The software also allows you to save or open analysis files That is all the inputs in the software will be save
120. e Covariance and Correlation method accounting for a specific VaR percentile land holding period Single Input Parameters Horizon Days 10 00 Percentile 0 90 Input3 Input4 Input5 Input Input7 Input InputS Input10 Input 11 Input 12 Input 13 Multiple Series Input Parameters Values are SPACE separated Rows are SEMICOLON separated Amounts Daily Volatility Correlations Figure 18 Exotic Financial Options Valuator User Manual 29 Real Options Super Lattice Solver Real Options SLS Payoff Chart Tornado Sensitivity Analysis User Manual 1 10 Payoff Charts Tornado Convergence Scenario Sensitivity Analysis Monte Carlo Risk Simulation The main Single Asset SLS module also comes with payoff charts sensitivity tables scenario analysis and convergence analysis Figure 18A To run these analyses first create a new model or open and run an existing model e g from the first tab Options SLS click File Examples and select Plain Vanilla Call Option I then hit Run to compute the option value and click on any one of the tabs To use these tools you need to first have a model specified in the main Options SLS tab Here are brief explanations of these tabs and how to use their corresponding controls as shown in Figure 18A The Payoff Chart tab A allows you to generate a typical option payoff chart where you have the ability to choose the input variable to chart B by entering some minimum and maximum values C
121. e negative values and the natural logarithm of a negative value does not exist Hence the volatility measure does not fully capture the possible cash flow downside and may produce erroneous results In addition autocorrelated cash flows estimated using time series forecasting techniques or cash flows following a static growth rate will yield volatility estimates that are erroneous Great care should be taken in such instances This flaw is neutralized in larger datasets that only carry positive values such as historical stock prices or price of oil or electticity This approach is valid and correct as computed in Figure B2 for liquid and traded assets with a lot of historical data The reason why this approach is not valid for computing the volatility of cash flows in a DCF for the purposes of real options analysis is because of the lack of data For instance the following annualized cash flows 100 200 300 400 500 would yield a volatility of 20 80 as compated to the following annualized cash flows 100 200 400 800 1600 which would yield a volatility of 0 versus the following cash flows 100 200 100 200 100 200 which yields 75 93 All these cash flow streams seem fairly deterministic and yet provided very different volatilities In addition the third set of negatively autocorrelated cash flows should actually be less volatile due to its predictive cyclical nature and reversion back to a base level but its volatility is computed
122. e normal distribution 4 continuous dividend payout ctitical value solved recursively Z intermediate variables Z and 42 Computation First solve recursively for the critical J value as below BEP C enm n gGAH t XQe 009 In1 X r q 0 2 T 0 oT t oT t Ind X q r o7 2 T 1 OAT t _ 1 4 2 t co dT oT t Then using the J value calculate restora _ In S 1 r qt o7 2 t ct d and d d ovt User Manual 139 Real Options Super Lattice Solver Real Options SLS In S X r7 q 6 12 T ay In S X r q o7 2 T a P 4t T and 4t Tp Option Value Se O d y p Ads y oT p Se O d y 302 X O d y a4T i p User Manual 140 Real Options Super Lattice Solver Real Options SLS Compound Options on Options The value of a compound option is based on the value of another option That is the underlying vatiable for the compound option is another option Again solving this model requires programming capabilities Definitions of Variables 5 present value of future cash flows r risk free rate volatility cumulative standard normal distribution q continuous dividend payout I ctitical value solved recursively Q cumulative bivariate normal distribution Xi strike for underlying X strike for the option on
123. e to reallocate and reroute their existing portfolio of planes globally and an excess plane on the tarmac is very costly The airline can sell the excess plane in the secondary market where smaller regional carriers buy used planes but the price uncertainty 15 very high and is subject to significant volatility of say 45 and may fluctuate wildly between 10M and 25M for this class of aircraft The aircraft manufacturer can reduce the airline s risk by providing a buy back provision or abandonment option where at anytime within the next five years the manufacturer agrees to buy back the plane at a guaranteed residual salvage ptice of 20M at the request of the airline The corresponding risk free rate for the next five years is 5 This reduces the downside risk of the airline and hence reduces its risk chopping off the left tail of the price fluctuation distribution and shifting the expected value to the right This abandonment option provides risk reduction and value enhancement to the airline Apphing the abandonment option in SLS using 100 step binomial lattice this option is worth 3 52M If the airline is the smarter counterparty and calculates this 45 Real Options Super Lattice Solver Real Options SLS User Manual value and gets this buy back provision for free as part of the deal the aircraft manufacturer has just lost over 10 of its aircraft value that it left on the negotiation table Information and knowledge is highly v
124. e typically lower than standard options as the barrier option will have value within a smaller price range than the standard option The holder of a barrier option loses some of the traditional option value and therefore a barrier option should sell at a lower price than a standard option An example would be a contractual agreement whereby the writer of the contract can get into or out of certain obligations if the asset or project value breaches barrier The Up and In Upper American Barrier Option has slightly lower value than a regular American call option as seen in Figure 64 This is because some of the option value when the asset is less than the barrier but greater than the implementation cost is lost Clearly the higher the upper barrier the lower the up and in barrier option value will be as more of the option value is lost due to the inability to execute when the asset value is below the barrier example file used Barrier Option Up and In Upper Barrier Call For instance e When the upper barrier is 110 the option value is 41 22 e When the upper barrier is 120 the option value is 39 89 In contrast an Up and Out Upper American Barrier Option is worth a lot less because this barrier truncates the option s upside potential Figure 65 shows the computation of such an option Clearly the higher the upper barrier the higher the option value will be example file used Barrier Option Up and Out Upper Barrier Call For i
125. e versus performing a Monte Carlo simulation on both cash flow variables as well as the discount rate will yield very different X values The main critique of this approach is that in a real options analysis the variability in the present value of cash flows is the key driver of option value and not the variability of discount rates used in the analysis Modifications to this method include duplicating the cash flows and simulating only the numerator cash flows thereby providing different numerator values but a static denominator value for each simulated trial while keeping the discount rate constant In fact when running this approach it might be advisable to set the discount rate as static risk free rate simulate the DCF and obtain the volatility then reset the discount rate back to its original value Figure B4 illustrates an example of how this approach can be implemented easily in Excel To follow along open the example file l o atify Computations and select the worksheet tab Log Present Value Approach The example shows sample DCF model where the cash flows row 46 and implementation costs row 48 are computed separately This is done for several reasons The first is to separate the market risks revenues and associated operating expenses from the private risks cost of implementation of course only if it makes sense to separate them as there might be situations where the implementation cost is subject to market risk as well H
126. each time its value changes and you can specify when the appropriate salvage value becomes effective For instance in a 10 year 100 step super lattice problem where there are two salvage values 100 occurring within the first 5 years and increases to 150 at the beginning of Year 6 you can enter two salvage variables with the same name 100 with a starting step of 0 and 150 with a starting step of 51 Be careful here as Year 6 starts at step 51 and not 61 That is for a 10 year option with a 100 step lattice we have Steps 1 10 Year 1 Steps 11 20 Year 2 Steps 21 30 Year 3 Steps 31 40 Year 4 Steps 41 50 Year 5 Steps 51 60 Year 6 Steps 61 70 Year 7 Steps 71 80 Year 8 Steps 81 90 Year 9 and Steps 91 100 Year 10 Finally incorporating 0 as a blackout step indicates that the option cannot be executed immediately A Custom Vatiable s name must be a single continuous word 16 Real Options Super Lattice Solver Real Options SLS Option Valuation Audit Sheet Assumptions Intermediate Computations PV Asset Value 100 00 Stepping Time at 0 0500 Implementation Cost 100 00 Up Step Size up 1 0226 Maturity Years 5 00 Down Step Size down 0 9779 Risk free Rate 5 00 Risk neutral Probability 0 5504 Dividends 0 00 Volatility 10 00 Results Lattice Steps 100 Auditing Lattice Result 10 steps 23 19 Option Type European Super Lattice Results 23 40 Terminal
127. ed by the present value of all uncertain future cash flows discounted at the risk adjusted rate of return or be significantly below it Hence the option to defer and wait until some of the uncertainty becomes resolved through the passage of time is worth more than executing immediately The value of being able to wait before executing the option and selling the project at the Implementation Cost in present values is the value of the option The NPV of executing immediately is simply the Implementation Cost less the Asset Value 0 The option value of being able to wait and defer selling the asset only if the condition goes bad and becomes optimal for selling is the difference between the calculated result total strategic value and the NPV or 24 42 for the American Option and 20 68 for the European Option The American put option is worth more than the European put option even when no dividends exist contrary to the call options seen previously For simple call options when no dividends exist it is never optimal to exercise early However it may sometimes be optimal to exercise early for put options regardless of whether dividend yields exist In fact a dividend yield will decrease the value of a call option but increase the value of a put option This is because when dividends are paid out the value of the asset decreases Thus the call option will be worth less and the put option will be worth more The higher the dividend yield the earlier
128. edge import export issues and so forth If successful the firm will agree to give this small Chinese manufacturer 20 of its net income as remuneration for their services plus some startup fees The question is how much is this option to contract worth that is how much should the firm be willing to pay on average to cover the initial startup fees plus the costs of this proof of concept stage A contraction option valuation result using SLS shows that the option is worth 1 59M assuming a 5 risk free rate for the 1 year test period So as long as the total costs for a pilot test costs less than 1 59 it is optimal to obtain this option especially if it means potentially being able to save over 20M Figure 25 illustrates a simple 10 step Contraction Option while Figure 26 shows the same option using 100 lattice steps example file used is Contraction American and European Option vesting Figure 27 illustrates a 5 year Bermudan Contraction Option with a 4 year period blackout steps of 0 to 80 out of a 5 year 100 step lattice where for the first 4 years the option holder can only keep the option open and not execute the option example file used is Contraction Bermudan Option Figure 28 shows a customized option where there is a blackout period and the savings from contracting change over time example file used is Contraction Customized Option These results are for the aeronautical manufacturing example Figure 2
129. eeds Abandonment Option with changing salvage values over time 4 V 120 90 2 5 7 100 0 10 1 2 10 20 35 NS Max Asset Salvage Max Asset Cost 0
130. en in Excel you can click on the function wizard icon or simply select an empty cell and click on Insert Function While in Excel s equation wizard select the ALL category and scroll down to the functions starting with the SLS prefixes Here you will see a list of SLS functions that are ready for use in Excel Figure 17 shows the Excel equation wizard Start the Excel Functions module and select the ALL category when in Excel s function wizatd then scroll down to access the SLS functions You may have to check your macro security settings before starting in Excel XP 2003 click on Tools Macro Security and make sure it is set to Medium or below as well as in Excel 2007 2010 click on the large Office Button on the top left corner of Excel click on Excel Options Trust Center Trust Center Settings Add Ins uncheck all 3 options then click on Macro Settings and select Enable All Macros and check Trust Access to the VBA project click OK Suppose you select the first function SLSB nomialAmericanCall and hit OK Figure 17 shows how the function can be linked to an existing Excel model The values in cells B1 27 Real Options Super Lattice Solver Real Options SLS to B7 can be linked from other models or spreadsheets can be created using VBA mactos or can be dynamic and changing as in when running a simulation Note Be aware that certain functions require many input variables and Excel s equation wizard can only show 5 variables
131. endent Mutually Exclusive Non Mutually Exclusive and Complex Combinatorial Nested Options Sequential Compound Options are path dependent options where one phase depends on the success of another in contrast to path independent options like those solved using SLS Figure 49 shows that in a complex strategy tree at certain phases different combinations of options exist These options can be mutually exclusive ot non mutually exclusive In all these types of options there might be multiple underlying assets e g Japan has a different risk return or profitability volatility profile than the U K or Australia You can build multiple underlying asset lattices this way using the MSLS and combine them in many various ways depending on the options The following are some examples of path dependent versus path independent and mutually exclusive versus non mutually exclusive options Path Independent and Mutually Exclusive Options Use the SLS to solve these types of options by combining all the options into a single valuation lattice Examples include the option to expand contract and expand These are mutually exclusive if you cannot both expand into a different country while at the same time abandoning and selling the company These are path independent if there are no restrictions on timing that is you can expand contract and abandon at any time within the confines of the maturity period Path Independent and Non Mutual
132. er option If the lower barrier is below the implementation cost then the option will be worthless under all conditions It is when the lower barrier level is between the implementation cost and starting asset value that the option is potentially worth something However the value of the option is dependent on volatility Using the same parameters in Figure 62 and changing the volatility and risk free rates the following examples illustrate what happens e volatility of 75 the option value is 4 34 volatility of 25 the option value is 3 14 e volatility of 5 the option value is 0 01 The lower the volatility the lower the probability that the asset value will fluctuate enough to breach the lower barrier such that the option will be executed By balancing volatility with the threshold lower barrier you can create optimal trigger values for barriers In contrast the Lower Barrier Option for Down and Out Call option is shown in Figure 63 Here if the asset value breaches this lower barrier the option is worthless but is only valuable when it does not breach this lower barrier As call options have higher 95 Real Options Super Lattice Solver Real Options SLS User Manual values when the asset value is high and lower value when the asset is low this Lower Barrier Down and Out Call Option is hence worth almost the same as the regular American option The higher the barrier the lower the value of the lowe
133. eralized Black Scholes Model Definitions of Variables 5 present value of future cash flows X implementation cost r risk free rate T time to expiration years volatility cumulative standard normal distribution b carrying cost 4 continuous dividend payout Computation cT s B Catt BELO Gto ar eraf 987050 ar ECT In S X b 60 2 T 5 0 2 T Put a eee Notes b 0 Putures options model b r q Black Scholes with dividend payment b r Simple Black Scholes formula b r r Foreign currency options model User Manual 143 Real Options Super Lattice Solver Real Options SLS User Manual Options on Futures The underlying security is a forward or futures contract with initial price Here the value of F is the forward or futures contract s initial price replacing 5 with F as well as calculating its present value Definitions of Variables X implementation cost futures single point cash flows r risk free rate T time to expiration years volatility cumulative standard normal distribution q continuous dividend payout Computation Call real nee eraf MEO ar ovT cT X c 22s revo 0870s 144 Real Options Super Lattice Solver Real Options SLS Two Correlated Assets Option The payoff on an option depends on w
134. ere we assume that implementation cost is subject to only private risks and will be discounted at a tisk free or at the cost of money close to the risk free rate of return to discount it for time value of money The market risk cash flows are discounted at a market risk adjusted rate of return which can also be seen as discounting at 5 risk free rate to account for time value of money and discounted again at the market risk premium of 10 for risk or simply discounted one time at 15 As discussed in Chapter 2 if you do not separate the market and private risks you end up discounting the private risks heavily and making the DCF a lot more profitable than it actually is i e if the costs that should be discounted at 5 are discounted at 15 the NPV will be inflated By separately discounting these cash flows the present value of cash flows and implementation costs can be computed cells H9 and H10 The difference will of course be the NPV The separation here is also key because from the Black Scholes 123 Real Options Super Lattice Solver Real Options SLS User Manual equation below the call option is computed as the present value of net benefits discounted at some risk adjusted rate of return or the starting stock price 5 times the standard normal probability distribution less the implementation cost or strike price CX discounted at the risk free rate and adjusted by another standard normal probability distribution If
135. et Contraction Savings Call Put 399 200 259 Max Asset Cost 0 259 1001 5682 Max Asset Contraction Savings OptionOpen Max Asset Cost OptionOpen OptionOpen Figure 27 A Bermudan Option to Contract with Blackout Vesting Periods User Manual 57 Real Options Super Lattice Solver Real Options SLS User Manual Figure 28 Single Asset Super Lattice Customized Contraction Option with changing savings amounts and blackout steps 4 1000 7 1000 5 100
136. et for analysis TIZ Real Options Super Lattice Solver Real Options SLS Lv Ie ev ql l J IE Te dE dE ET 1 Downloaded Weekly Historical Stock Prices of Microsoft Volatility Computations LN Relative Moving Average 2 Date Open High Low Close Volume Adj Close Ret ms Volatilities 3 27 04 27 01 27 10 26 68 26 72 52388840 2664 0 0108 17 87 4 20 Dec 04 27 01 27 17 26 78 27 01 77413174 26 93 0 0019 17 84 5 13 04 27 10 27 40 26 80 26 96 108628300 2688 0 0045 17 85 6 6 Dec 04 27 10 27 44 26 91 27 08 83312720 27 00 0 0055 18 00 One Year Annualized Volatility 29 Nov 04 26 64 2744 26 61 27 23 83103200 27 15 0 0235 18 13 8 22 Nov 04 26 75 26 82 26 10 26 60 61834599 26 52 0 0098 18 03 Average 21 89 9 15 Nov 04 27 34 27 50 26 84 26 86 75375960 26 78 0 0011 18 1096 Median 22 30 10 8 Nov 04 29 18 30 20 29 13 29 97 109385736 26 81 0 0223 18 20 11 1 Nov 04 28 16 29 36 27 96 29 31 85044019 26 22 0 0468 18 28 12 25 Oct 04 2767 28 54 27 55 27 97 70791679 25 02 0 0084 17 71 13 18 Oct 04 28 07 28 89 27 58 27 74 74671318 24 81 0 0092 17 80 14 11 Oct 04 2820 28 27 27 80 27 99 48396360 2504 0 0000 19 68 15 4 Oct 04 28 44 28 59 27 97 27 99 52998320 2504 0 0091 19 69 16 27 Sep 04 27 17 28 32 27 04 28 25 61783760 2527 0 0346 19 68 17 20 Sep 04 2744 27 74 27 07 27 29 59162520 2441 0 0082 19 62 18 13
137. g option values based on the various combinations of inputs The Convergence tab shows the option results from 5 to 5000 steps where the higher the number of steps the higher the level of precision granularity in lattices increases where at some point the results of the lattice converge and once convergence is achieved no additional lattice steps are required The number of steps is set by default from 5 to 5000 but you can select the option type and number of decimals to show and the convergence chart is displayed depending on your selection You can also copy or print the table with the chart as required The Simulation tab allows you to run Monte Carlo risk simulations on the real options lattice model The input vatiables are listed in the bottom grid To set an assumption click on the ADD or EDIT button specific to the input variable row in the grid An Assumption Properties window will appear for you to select the relevant probability distribution and to set the required distributional parameters Click RUN when ready and the simulation will execute and once completed you can select the one ot two tail confidence interval and either enter the relevant X values to recover their respective probability confidence interval or enter in the certainty percentage and obtain the options value confidence interval remember to hit on the keyboard after entering the desired values in otder to activate the computations The nu
138. g target over time put in several custom variables named Barrier with the different values and starting lattice steps User Manual 99 Real Options Super Lattice Solver Real Options SLS User Manual Upper Barrier Option Up and In Call This option is live only when the asset value breaches the upper barrier J V 100 80 5 100 1 2 10 20 35 if Asset gt Bamier Max Asset Cost 0 0 Max Asset Cost 0
139. going all the way back to the first phase the Multiple Asset module s example file used Sequential Compound Option for Multiple Phases In NPV terms the project is worth 500 However the total strategic value of the stage gate investment option is worth 41 78 This means that although on an NPV basis the investment looks bad but in reality by hedging the risks and uncertainties through sequential investments the option holder can pull out at any time and not have to keep investing unless things look promising If after the first phase things look bad pull out and stop investing and the maximum loss will be 100 Figure 47 and not the entire 1 500 investment If however things look promising the option holder can continue to invest in stages The expected value of the investments in present values after accounting for the probabilities that things will look bad and hence stop investing versus things looking great and hence continuing to invest is worth an average of 41 78M Notice that the option valuation result will always be greater than or equal to zero e g try reducing the volatility to 5 and increasing the dividend yield to 8 for all phases When the option value is very low or zero this means that it is not optimal to defer investments and that this stage gate investment process is not optimal here The cost of waiting is too high high dividend or that the uncertainties in the cash flows are low dow volatility henc
140. hen entering your own equations make sure that Custom Option is first checked Figure 4 illustrates how this analysis is done Notice that the value 23 3975 in Figure 4 agrees with the value in Figure 2 The Terminal Node Equation is the computation that occurs at maturity while the Intermediate Node Equation is the computation that occurs at all periods prior to maturity and is computed using backward induction The term OptionOpen represents keeping the option open and is often used in the Intermediate Node Equation when analytically representing the fact that the option is not executed but kept open for possible future execution Therefore in Figure 4 the Intermediate Node Equation Max Asset Cost OptionOpen represents the profit maximization decision of either executing the option or leaving it open for possible future execution In contrast the Terminal Node Equation of Max Asset Cost 0 represents the profit maximization decision at maturity of either executing the option if it is in the money or allowing it to expire worthless if it is at the money or out of the money Figure 4 Single Asset Super Lattice Solver SLS Plain Vanila American and European Call Options lower number of steps Useful for testing convergence
141. hether the other correlated option is in the money This is the continuous counterpart to a correlated quadranomial model Definitions of Variables 5 present value of future cash flows X implementation cost r risk free rate T time to expiration years volatility cumulative bivariate normal distribution function p correlation 7 6 between the two assets di continuous dividend payout for the first asset 42 continuous dividend payout for the second asset Computation In S Xj r q 05 2 T JT In S Xy r q 0 2 T Call e c T GT po NT p Xeo In X r q 0 2 T 5 X r q 0 2 T c T c NT Pass ero 6 HO ee W S X 02 2 gt jp c NT c JT In X r q 02 2 T o JT In S Xj r q 01 IDT S e T c NT po NT User Manual 145 Real Options Super Lattice Solver Real Options SLS User Manual Appendix D Quick Install and Licensing Guide This section is the quick install guide for more advanced users For a more detailed installation guide please refer to the next section The SLS software requires the following minimum requirements e Windows XP or Vista and beyond e Excel XP or Excel 2003 or Excel 2007 or Excel 2010 and beyond NET Framework 2 0 or higher e Administrative rights during installation only e 512MB of RAM more 2GB recommended e 100M
142. ice In that case make sure the inputs are correct e g Jump Intensity has to exceed 1 where 1 implies no jumps check for erroneous combinations of Jump Rates Jump Sizes and Lattice Steps The probability of a jump can be computed as the product of the Jump Rate and time step 07 Figure 58 illustrates a sample Quadranomial Jump Diffusion Option analysis example file used Jump Diffusion Calls and Puts Using Quadranomial Lattices Notice that the Jump Diffusion call and put options are worth more than regular calls and puts This is because with the positive jumps 10 probability per year with an average jump size of 1 50 times the previous values of the underlying asset the call and put options are worth more even with the same volatility If a real options problem has more than 2 underlying assets either use the MSLS and or Risk Simulator to simulate the underlying asset s trajectories and capture their interacting effects in a DCF model Figure 58 Multinomial Lattice Solver Call Option with Trinomial Trinomial Mean Reversion and Jump Diffusion Models V e v pese 100
143. imes example file used Expand Contract Abandon Customized Option 11 Figure 35 Single Asset Super Lattice Solver JURE EIE 515 Bermudan Option to Expand Contract and Abandon where there is a cooling off period blackout step periods V E 100 5 j 100 5 7 100 0 80 1 2 10 20 35 Max Asset Asset Expansio
144. in depth real options analysis using this software This manual will not cover some of the fundamental topics already discussed in the book Note The 1st edition of Real Options Analysis Tools and Techniques published in 2002 shows the Real Options Analysis Toolkit software an older precursor to the Super Lattice Solver also created by Dr Johnathan Mun The Real Options Super Lattice Solver supersedes the Real Options Analysis Toolkit by providing the following enhancements and is introduced in Rea Options Analysis 2nd edition 2005 e Runs 100X faster and is completely customizable and flexible e All inconsistencies computation errors and bugs have been fixed and verified e Allows for changing input parameters over time customized options e Allows for changing volatilities over time e Incorporates Bermudan vesting and blackout periods and Customized Options e Has flexible modeling capabilities in creating or engineering your own customized options e Includes general enhancements to accuracy precision and analytical prowess e Includes more than 250 exotic options models closed form exotic multinomial lattice As the creator of both the Super Lattice Solver and Real Options Analysis Toolkit ROAT software the author suggests that the reader focuses on using the Super Lattice Solver as it provides many powerful enhancements and analytical flexibility over its predecessor ROAT The SLS software requires the following mi
145. ing Volatility and Changing Risk free Rate Option In this model the volatility and risk free yields are allowed to change over time and a non recombining lattice is required to solve the option In most cases it is recommended that you create option models without the changing volatility term structure because getting a single volatility is difficult enough let alone a series of changing volatilities over time If different volatilities that are uncertain need to be modeled run a Monte Carlo simulation on volatilities instead This model should only be used when the volatilities are modeled robustly are rather certain and change over time The same advice applies to a changing risk free rate term structure 25 Real Options Super Lattice Solver Real Options SLS MULTIPLE SUPER LATTICE SOLVER MULTIPLE ASSET amp MULTIPLE PHASES Maturity Years MSLS Result 134 0802 Blackout Steps Correlation EE Underlying Asset Lattices Custom Variables Lattice Name PV Asset Volatility Name Value Starting Steps Underlying 100 00 100 00 abord mr es Irc E o es Contract 09 Lo nl y I c a ee i Y Po See Se Option Valuation Lattices Lattice Name Cost Riskfree Dividend Steps Terminal Equation Intermediate Equation Intermediate Equation for Blackout Note This is the Excel version of the Multiple Super
146. ing asset value yields a result of 120M or 0M for the option This result means that the option or contract is worthless because the safety net is set so low that it will never be utilized Conversely setting the 2 http www treas gov offices domestic finance debt management interest rate yield hist html 44 Real Options Super Lattice Solver Real Options SLS User Manual salvage level to thrice the prevailing asset value or 360M would yield a result of 360M and the results indicate 360M which means that there is no option value there is no value in waiting and having this option or simply execute the option immediately and sell the asset if someone is willing to pay three times the value of the project right now Thus you can keep changing the salvage value until the option value disappears indicating the optimal trigger value has been reached For instance if you enter 166 80 as the salvage value the abandonment option analysis yields a result of 166 80 indicating that at this price and above the optimal decision is to sell the asset immediately At any lower salvage value there is option value and at any higher salvage value there will be no option value This breakeven salvage point is the optimal trigger value Once the market price of this asset exceeds this value it is optimal to abandon Finally adding a Dividend Rate the cost of waiting before abandoning the asset c the annualized taxes and maintena
147. intellectual property of the technology both at different costs To further complicate matters you can use the MSLS to easily and quickly solve the situation where building versus selling off the option each has a different volatility and time to choose Figure 42 Multiple Asset Super Lattice Solver 1 Exotic Complex Floating European Chooser Option can be either call or put option i Tu Underlying 60 25 i gt CallOption 55 5 0 100 Max Underlying Cost 0 OptionOpen PutOption 65 5 0 100 Max Cost Underlying ionOpen Combination 0 Bl 10 SACRI poet COMBINATION 16 6035 Figure 42 Complex European Exotic Chooser Option using MSLS Figure
148. ion as the volatility is similar to the Log Cash Flow Returns approach If the sums of the present values of the cash flows are fluctuating between positive and negative values during the simulation you can again move up the DCF model and use items like EBITDA and net revenues as proxy variables for computing volatility Another alternative volatility estimate is to combine both approaches if enough data exists That is from a DCF with many cash flow estimates compute the PV Cash Flows for periods 0 1 2 3 and so forth Then compute the natural logarithm of the relative returns of these PV Cash Flows The standard deviation is then annualized to obtain the volatility This is of course the preferred method and does not require the use of Monte Carlo simulation but the drawback is that a longer cash flow forecast series is required B 3 GARCH Approach Another approach is the GARCH Generalized Autoregressive Conditional Heteroskedasticity model which can be utilized to estimate the volatility of any time series data GARCH models ate used mainly in analyzing financial time series data in order to ascertain its conditional variances and volatilities These volatilities are then used to value the options as usual but the amount of historical data necessary for a good volatility estimate remains significant Usually several dozens and even up to hundreds of data points are required to obtain good GARCH estimates In addition GARCH model
149. ions are selected and the computed results are for these simple plain vanilla American and European Call Options Figure 2 Single Asset Super Lattice Solver 515 Plain Vanila American and European Call Options lower number of steps Useful for testing convergence V V 100 c 100 5 100 23 3975 23 3975 Figure 2 SLS Results of a Simple European a
150. ity used in options analysis is annualized for several reasons The first reason is that all other inputs are annualized inputs e g annualized risk free rate annualized dividends and maturity in years Second if a cash flow or stock price stream of 10 to 20 to 30 that occurs in three different months versus three different days has very different volatilities Clearly if it takes days to double or triple your asset value that asset is a lot more volatile All these have to be common sized in time and be annualized Finally the Brownian Motion stochastic equation has the values O4j t That is suppose we have a 1 year option modeled using a 12 step lattice then is 1 12 If we use monthly data compute the monthly volatility and use this as the input this monthly volatility will again be partitioned into 12 pieces GJ t Therefore we need to first annualize the volatility to an annual volatility multiplied by the square root of 12 input this annual volatility into the model and let the model partition the volatility multiplied by the square toot of 1 12 into its periodic volatility This is why we annualize volatilities in Step 5 B 2 Volatility Estimates Logarithmic Present Value Returns The Logarithmic Present Value Returns Approach to estimating volatility collapses all future cash flow estimates into two present value sums one for the first time period and another for the present time Figure B3 The
151. ive equations useful for running Monte Carlo simulations with the Risk Simulator software an Excel add in risk based simulation forecasting and optimization software also developed by Real Options Valuation Inc or for linking to and from other spreadsheet models The lattices generated also include decision lattices where the strategic decisions to execute certain options and the optimal timing to execute these options are shown The Advanced Exotic Financial Options Valuator is a comprehensive calculator of more than 250 functions and models from basic options to exotic options e g from Black Scholes to multinomial lattices to closed form differential equations and analytical methods for valuing exotic options as well as other options related models such as bond options volatility computations delta gamma hedging and so forth This valuator complements the ROV Risk Modeler and ROV Valuator software tools with more than 800 functions and models also developed by Real Options Valuation Inc ROV which are capable of running at extremely fast speeds handling large datasets and linking into existing ODBC compliant databases e g Oracle SAP Access Excel CSV and so forth The SLS Excel Solution implements the SLS and MSLS computations within the Excel environment allowing users to access the SLS and MSLS functions directly in Excel This feature facilitates model building formula and value linking and embedding as well as r
152. latility as 50M 100M 50M 39 0296 Inverse 0 10 x 100M 1 2815x 100M Volatility User Manual 129 Real Options Super Lattice Solver Real Options SLS Frequency Best Case Scenario 150M 10 probability NPV of Project Expected NPV 100 90th percentile Figure B8 Going from Probability to Volatility This implies that the volatility is a symmetrical measure That is at an expected NPV of 100M 50 increase is equivalent to 150M while a 50 decrease is equivalent to 50M And because the normal distribution is assumed as the underlying distribution this symmetry makes perfect sense So now by using this simple approach if you obtain a volatility estimate of 39 02 you can explain to management by stating that this volatility is equivalent to saying that there 15 a 10 probability the NPV will exceed 150M Through this simple analysis you have converted probability into volatility using the equation above where the latter is a lot easier for management to understand Conversely if you model this in Excel you can convert from volatility back into probability Figures B9 and B10 illustrate this approach Open the example file Vo atility Estimates and select the worksheet tab Volatility fo Probability to follow along A B C D E 6 H J K E M N 0 1 2 c Frequency 3 Probability to Volatility Best Case Scenario 4 5 Expected NPV of
153. ll option values for American 2 Bermudan gt European in most basic cases as seen in Figure 39 insert a 5 dividend rate and blackout steps of 0 50 Of course this generality can be applied only to plain vanilla call options and do not necessatily apply to other exotic options e g Bermudan options with vesting and suboptimal exercise behavior multiples tend to sometimes carry a higher value when blackouts and vesting occur than regular American options with the same suboptimal exercise parameters Figure 38 Single Asset Super Lattice Solver SLS American European and Bermudan Basic Call Options without Dividends V 100 2 100 7 1 7 100
154. looking too good oil prices are low or moderate and production is only decent it is very difficult for the company to abandon operations why lose everything when net income is still positive although not as high as anticipated and not to mention the environmental and legal ramifications of simply abandoning an oil rig in the middle of the ocean Hence the oil company decides to hedge its downside risk through an American Contraction Option The oil company was able to find a smaller oil and gas company a former partner on other explorations to be interested in a joint venture The joint venture is structured such that the oil company pays this smaller counterparty a lump sum tight now for a 10 year contract whereby at any time and at the oil company s request the smaller counterparty will have to take over all operations of the offshore oil rig 1 taking over all operations and hence all relevant expenses and keep 30 of the net revenues generated The counterparty is in agreement because it does not have to partake in the billions of dollars required to implement the rig 1n the first place and it actually obtains some cash up front for this contract to assume the downside risk The oil company is also in agreement because it reduces its own risks if oil prices are low and production is not up to par and it ends up saving over 75M in present value of total overhead expenses which can then be reallocated and invested somewhere else In thi
155. ls from basic options to exotic options e g from Black Scholes to multinomial lattices to closed form differential equations and analytical methods for valuing exotic options as well as other options related models such as bond options volatility computations delta gamma hedging and so forth Figure 18 illustrates the valuator You can click on the Load Sample Values button to load some samples to get started Then select the Mode Category left panel as desired and select the Mode right panel you wish to run Click COMPUTE to obtain the result Note that this valuator complements the ROV Risk Modeler and ROV Valuator softwate tools with more than 800 functions and models also developed by Real Options Valuation Inc ROV which are capable of running at extremely fast speeds and handling large datasets and linking into existing ODBC compliant databases e g Oracle SAP Access Excel CSV and so forth Finally if you wish to access these 800 functions including the ones in this Exotic Financial Options Valuator tool use the ROV Modeling Toolkit software instead where in addition to having access to these functions and more you can run Monte Carlo simulation on your models using ROV s Risk Simulator softwate Two Asset Cash or Nothing Up Down Two Asset Correlation Call alysis Value at Risk Volatility Portfolio Risk and Retums Writer Extendible Put Option Model Description Computes the Value at Risk using the Varianc
156. lue increases it is better not to defer and wait that long Therefore the higher the dividend rate the lower the strategic option value For instance at an 8 dividend rate and 15 volatility the resulting value reverts to the NPV of 15M which means that the option value is zero and that it is better to execute immediately as the cost of waiting far outstrips the value of being able to wait given the level of volatility uncertainty and risk Finally if risks and uncertainty increase significantly even with a high cost of waiting e g 7 dividend rate at 30 volatility it is still valuable to wait This model provides the decision maker with a view into the optimal balancing between waiting for more information Expected Value of Perfect Information and the cost of waiting You can analyze this balance by creating strategic options to defer investments through development stages where at every stage the project is reevaluated as to whether it is beneficial to proceed to the next phase Based on the input assumptions used in this model the Sequential Compound Option results show the strategic value of the project and the NPV is simply the Asset less both phases Implementation Costs In other words the strategic option value is the difference between the calculated strategic value minus the NPV It is recommended for your consideration that the volatility and dividend inputs are varied to determine their interactions specifically
157. lver Real Options SLS Frequency e Probability area under the curve p 2 Stock price Figure B7 Standard Deviation However for the purposes of explaining volatility to management we relax this terminological difference and on a very high level state that they are one and the same for discussion purposes Thus we can make some management assumptions in estimating volatilities For instance starting from an expected NPV the mean value you can obtain an alternate NPV value with its probability and get an approximate volatility For instance say that a project s NPV is expected to be 100M Management further assumes that the best case scenario exceeds 150M if everything goes really well and that there is only a 10 probability that this best case scenario will hit Figure B8 illustrates this situation If we assume for simplicity that the underlying asset value will fluctuate within a normal distribution we can compute the implied volatility using the following equation Percentile Value Mean Volatility Inverse of the Percentile x Mean For instance we compute the volatility of this project as Volatility S 50M 8100M 50M Inverse 0 90 100 12815x 100M Where the Inverse of the Percentile can be obtained by using Excel s NORMSINV 0 9 function Similarly if the worst case scenario occurring 10 of the time will yield an NPV of 50M we compute the vo
158. ly Exclusive Options Use the SLS to solve these types of options by running each of the options that are non mutually exclusive one at a time in SLS Examples include the option to expand your business into Japan U K and Australia These are not mutually exclusive if you can choose to expand to any combinations of countries e g Japan only Japan and U K U K and Australia and so forth These are path independent if there are no restrictions on timing that is you can expand to any country at any time within the maturity of the option Add the individual option values and obtain the total option value fot expansion Path Dependent and Mutually Exclusive Options Use the MSLS to solve these types of options by combining all the options into one valuation lattice Examples include the option to expand into the three countries Japan U K and Australia However this time the expansions are mutually exclusive and path dependent That is you can only expand into one country at a time but at certain periods you can only expand into certain countries e g Japan is only optimal in three years due to current economic conditions export restrictions and so forth as compared to the U K expansion which can be executed right now Path Dependent and Non Mutually Exclusive Options Use MSLS to solve these These are typically simple Sequential Compound Options with multiple phases If more than one non mutually exclusive option exists re run
159. mber of simulation trials seed values decimals and corresponding simulation statistics are also available on the page For more technical details on running simulations or to better understand probability distributions and simulation statistics please refer to Modeling Risk Second Edition Wiley 2010 by Dr Johnathan Mun or review the Risk Simulator software see the software s user manuals hands on examples and getting started guides 1 11 ROV Strategy Tree The ROV Strategy Trees module Figure 18B is available from the main SLS user interface and is used to create visually appealing representations of strategic real options This module is used to simplify the drawing and creation of strategy trees but is not used for the actual teal options valuation modeling use the Real Options SLS software modules for actual modeling purposes The following are some main quick getting started tips and procedures in using this intuitive tool e There are 11 localized languages available in this module and the current language can be changed through the Language menu e Insert Option nodes or insert Terminal nodes by first selecting any existing node and then clicking on the option node icon square box or terminal node icon triangle box or use the Insert menu 31 Real Options Super Lattice Solver Real Options SLS User Manual Modity Individual Option and Terminal Node properties by double clicking on a node Sometimes when you click on
160. mic value of the relative returns week over week and is copied down the entire column The formula in cell J3 is STDEV I3 I54 SORT 52 which computes the annualized by multiplying the square root of the number of weeks in a year volatility by taking the standard deviation of the entire 52 weeks of the year 2004 data The formula in cell J3 is then copied down the entire column to compute a moving window of annualized volatilities The volatility used in this example is the average of a 52 week moving window which covers two years of data That is cell L8 s formula is AVERAGE J3 54 where cell J54 has the following formula STDEV 154 1105 SQRT 52 and of course row 105 is January 2003 This means that the 52 week moving window captures the average volatility over a 2 year petiod and smoothes the volatility such that infrequent but extreme spikes will not dominate the volatility computation Of course a median volatility should also be computed If the median is far off from the average the distribution of volatilities is skewed and the median should be used otherwise the average should be used Finally these 52 volatilities can be fed into Monte Carlo simulation Rak Simulator software and the volatilities themselves can be simulated 6 Go to http finance yahoo com and enter a stock symbol e g MSFT Click on Quotes Historical Prices and select Weekly and select the period of interest You can then download the data to a spreadshe
161. n ExpandCost Asset Contraction ContractSavings Salvage Max Asset Cast 0 116 8171 Max Asset Expansion ExpandCost Asset Contraction Contract Savings Salvage OptionOpen Max Asset Cost OptionOpen ysna OptionOpen OptionOpen Crs emer rsen Figure 35 Bermudan Option to Expand Contract Abandon User Manual 67 Real Options Super Lattice Solver Real Options SLS Figure 36 Single Asset Super Lattice Solver J V V 100 100 5
162. nale why we analyzed it the way we did Hence let us look at the steps undertaken and explain the rationale behind them Step 1 Compute the present values at times 0 and 1 and sum them The theoretical price of a stock is the sum of the present values of all future dividends for non dividend paying stocks we use market replicating portfolios and comparables and the funds to pay these dividends are obtained from the company s net income and free cash flows The theoretical value of a project or asset is the sum of the present value of all future free cash flows or net income Hence the price of a stock is equivalent to the price or value of an asset the NPV Thus the sum of the present values at time 0 is equivalent to the stock price of the asset at time 0 the value today The sum of the present value of the cash flows at time 1 is equivalent to the stock price at time 1 or a good proxy for the stock price in the future We use this as a proxy because in most DCF models cash flow forecasts only a few periods Hence by running Monte Carlo simulation we are changing all future possibilities and capturing the uncertainties in the DCF inputs This future stock price is hence a good proxy of what may happen to the future stream of cash flows remember that sum of the present value of future cash flows at time 1 included in its computations all future cash flows from the DCF thereby capturing future fluctuations and uncertainties Thi
163. nce fees that have to be paid if you keep the asset and not sell it off measured as a percentage of the present value of the asset will decrease the option value Hence the breakeven trigger point where the option becomes worthless can be calculated by successively choosing higher dividend levels This breakeven point again illustrates the trigger value at which the option should be optimally executed immediately but this time with respect to a dividend yield That is if the cost of carry or holding on to the option or the option s leakage value is high that is if the cost of waiting is too high don t wait and execute the option immediately Other applications of the abandonment option include buy back lease provisions in a contract guaranteeing a specified asset value asset preservation flexibility insurance policies walking away from a project and selling off its intellectual property purchase price of an acquisition and so forth To illustrate here are some additional quick examples of the abandonment option and sample exercises for the rest of us e An aircraft manufacturer sells its planes of a particular model in the primary market for say 30M each to various airline companies Airlines are usually risk adverse and may find it hard to justify buying an additional plane with all the uncertainties in the economy demand price competition and fuel costs When uncertainties become resolved over time airline carriers may hav
164. nd American Call Option 12 Real Options Super Lattice Solver Real Options SLS User Manual The benchmark results use both closed form models Black Scholes and Closed Form Approximation models and 1 000 step binomial lattices on plain vanilla options You can change the steps to 7000 in the basic inputs section to verify that the answers computed are equivalent to the benchmarks as seen in Figure 3 Notice that of course the values computed for the American and European options are identical to each other and identical to the benchmark values of 23 4187 as it is never optimal to exercise a standard plain vanilla call option early if there are no dividends Be aware that the higher the lattice step the longer it takes of course to compute the results It is advisable to start with lower lattice steps to make sure the analysis is robust and then progressively increase lattice steps to check for results convergence See Appendix A on convergence criteria on lattices for more details about binomial lattice convergence as to how many lattice steps are required for a robust option valuation Figure 3 Single Asset Super Lattice Solver 51 5 Plain Vanila American and European Options lower number of steps Useful for testing convergence
165. ne after another over time while the simultaneous option evaluates these options in concurrence Clearly the sequential compound is worth more than the simultaneous compound option by virtue of staging the investments Note that the simultaneous compound option acts like a regular execution call option Hence the American Call Option is a good benchmark for such an option Figure 50 shows how Simultaneous Compound Option can be solved using the MSLS example file used Simple Two Phased Simultaneous Compound Option Similar to the sequential compound option analysis the existence of an option value implies that the ability to defer and wait for additional information prior to executing is valuable due to the significant uncertainties and risks as measured by Volatility However when the cost of waiting as measured by the Dividend Rate is high the option to wait and defer becomes less valuable until the breakeven point where the option value equals zero and the strategic project value equals the NPV of the project This breakeven point provides valuable insights for the decision maker into the interactions between the levels of uncertainty inherent in the project and the cost of waiting to execute The same analysis can be extended to Multiple Investment Simultaneous Compound Options as seen in Figure 51 example file used Phased Simultaneous Compound Option 83 Real Options Super Lattice Solver Real Options SLS
166. nimum requirements e Windows XP Vista Windows 7 and beyond e Excel XP Excel 2003 Excel 2007 or Excel 2010 Framework 2 0 or later 9 Real Options Super Lattice Solver Real Options SLS User Manual e Administrative rights for software installation e Minimum 512MB of RAM 1GB recommended e 80MB of free hard drive space The software will work on most foreign operating systems such as foreign language Windows or Excel and the SLS software has been tested to work on most international Windows operating systems with just a quick change in settings by clicking on Szart Control Panel Regional and Language Options Select English United States This change is required because the numbering convention is different in foreign countries e g one thousand dollars and fifty cents is written as 1 000 50 in the United States versus 1 000 50 in certain European countries To install the software make sure that your system has all the prerequisites described above If you require NET Framework 2 0 please browse the software installation CD and install the file named dotnet x20 exe ot if you do not have the installation CD you can download the file from the following web location www tealoptionsvaluation com attachments dotnetfx20 exe You need to first install this software before proceeding with the SLS software installation Note that NET 2 0 works in parallel with NET 1 1 and you do not and should not uninstall
167. nstance e When the upper barrier is 110 the option value is 23 69 e When the upper barrier is 120 the option value is 29 59 98 Real Options Super Lattice Solver Real Options SLS Finally note the issues of nonbinding barrier options Examples of nonbinding options ate e Up and Out Upper Barrier Calls when the Upper Barrier lt Implementation Cost then the option will be worthless e Up and In Upper Barrier Calls when Upper Barrier lt Implementation Cost then the option value reverts to a simple call option Examples of Upper Barrier Options are contractual options Typical examples are e A manufacturer contractually agrees not to sell its products at prices higher than a pre specified upper barrier price level e A client agrees to pay the market price of a good or product until a certain amount and then the contract becomes void if it exceeds some price ceiling Figures 64 and 65 illustrate American Barrier Options To change these into European Barrier Options set the Intermediate Node Equation Nodes to OptionOpen In addition for certain types of contractual options vesting and blackout periods can be imposed For solving such Bermudan Barrier Options keep the same Intermediate Node Equation as the American Barrier Options but set the Intermediate Node Equation During Blackout and Vesting Periods to OptionOpen and insert the corresponding blackout and vesting period lattice steps Finally if the Barrier is a changin
168. of the data In using forecast and comparable data your choices are limited to what is available or what models have been built and are typically annual quarterly or monthly data usually for a limited amount of time When using historical data your choices are more varied Typically daily data has too much random fluctuations and white noise that may erroneously impact the volatility computations Monthly quarterly and annual historical data are spread too far out and all the fluctuations inherent in the time series data may be smoothed out The optimal periodicity is weekly data if available Any intraday and intraweek fluctuations are smoothed out but weekly fluctuations are still inherent in the dataset Finally the time frame of the historical data is also important Periods of extreme events need to be carefully considered e g dot com bubble global recession depression terrorist attacks That is if these are actual events that will recur and hence are not outliers but 119 Real Options Super Lattice Solver Real Options SLS User Manual RES Period 1 End Value Period 2 End Value Period n End Value 200 Y 100 Geometric Average part of the undiversifiable systematic risk of doing business In Figure B2 s example above a 2 year cycle was used Clearly if the option has a 3 year matutity then a 3 year cycle should be considered with the exception that data is not available or if certain extreme events mitig
169. on output or to contract the scale and scope of a project when conditions are not as amenable thereby reducing the value of the asset or project by a Contraction Factor but at the same time creating some cost Savings As an example suppose you work for a large aeronautical manufacturing firm that is unsure of the technological efficacy and market demand for its new fleet of long range supersonic jets The firm decides to hedge itself through the use of strategic options specifically an option to contract 10 of its manufacturing facilities at any time within the next 5 years i e the Contraction Factor is 0 9 Suppose that the firm has a current operating structure whose static valuation of future profitability using a discounted cash flow model in other words the present value of the expected future cash flows discounted at an appropriate market risk adjusted discount rate is found to be 1 000M Asse Using Monte Carlo simulation you calculate the implied volatility of the logarithmic returns of the asset value of the projected future cash flows to be 30 The risk free rate on a riskless asset 5 year U S Treasury Note with zero coupons is found to be yielding 5 Further suppose the firm has the option to contract 10 of its current operations at any time ovet the next 5 years thereby creating an additional 50 million in savings after this contraction These terms are arranged through a legal contractual agreement with one of
170. one version in preference to the other You should have both versions running concurrently on your computer fot best performance Next install the SLS software by either using the installation CD or going to the following web location www realoptionsvaluation com clicking Downloads and selecting Real Options SLS You can either select to download the FULL version assuming you have already purchased the software and have received the permanent license keys and the instructions to permanently license the software or a TRIAL version The trial version is exactly the same as the full version except that it expires after 10 days during which you would need to obtain the full license to extend the use of the software Install the software by following the onscreen prompts If you have the trial version and wish to obtain the permanent license visit www realoptionsvaluation com and click on the Purchase link left panel of the web site and complete the purchase order You will then receive the pertinent instructions on installing the permanent license See Appendix D and E for additional installation details and Appendix F for licensing instructions Please visit www realoptionsvaluation com and click on FAQ and DOWNLOADS for any updates on instalation instructions troubleshooting issues 10 Real Options Super Lattice Solver Real Options SLS User Manual 12 Single Asset Super Lattice Solver Figure 1 illustrates the SLS softw
171. ontinue _ 000 000 Continue Continue Endl oo 0 00 Continue End Figure 12 Lattice Maker Module and Worksheet Results with Visible Equations L7 SLS Excel Solution SLS MSLS and Changing Volatility Models in Excel The SLS software also allows you to create your own models in Excel using customized functions This is an important functionality because certain models may require linking from other spreadsheets or databases run certain Excel macros and functions or certain inputs need to be simulated or inputs may change over the course of modeling your options This Excel compatibility allows you the flexibility to innovate within the Excel spreadsheet environment Specifically the sample worksheet solves the SLS MSLS and Changing Volatility model To illustrate Figure 13 shows a Customized Abandonment Option solved using SLS from the Single Asset Module click on File Examples Abandonment Customized Option The same 23 Real Options Super Lattice Solver Real Options SLS problem can be solved using the SLS Excel Solution by clicking on Start Programs Real Options Valuation Real Options SLS Excel Solution The sample solution is seen in Figure 14 Notice that the results the same using the SLS versus the SLS Excel Solution file You can use the template provided by simply clicking on Fie Save As in Excel and use the new file for your own modeling n
172. pean or Bermudan Options are chosen the Terminal Node Equation you entered will be the one used in the super lattice for the terminal nodes However for the intermediate nodes the American option will assume the same Terminal Node Equation plus the ability to keep the option open the European option will assume that the option can only be kept open and not executed while the Bermudan option will assume that during the blackout lattice steps the option will be kept open and cannot be executed you also wish to enter the Intermediate Node Equation the Custom Option should be first chosen otherwise you cannot use the Intermediate Node Equation box The Custom Option result will use all the equations you have entered in the Terminal Intermediate and Intermediate with Blackout sections 15 Real Options Super Lattice Solver Real Options SLS User Manual The Custom Variables list is where you can add modify or delete custom variables the variables that ate required beyond the basic inputs For instance when running an abandonment option you will require the salvage value You can add this in the Custom Variables list provide it a name a variable s name must be a single word without spaces the appropriate value and the starting step when this value becomes effective For example if you have multiple salvage values 1 if salvage values change over time you can enter the same variable name e g sa vage several times but
173. ping the technology internally and making the purchase price of 50M worth it Figure 19 shows the results of a simple abandonment option with a 10 step lattice as discussed previously while Figure 20 shows the audit sheet that is generated from this analysis Figure 21 shows the same abandonment option but with a 100 step lattice To follow along open the Single Asset SLS example file Abandonment American Option Notice that the 10 step lattice yields 125 48 while the 100 step lattice yields 125 45 indicating that the lattice results have achieved convergence The Terminal Node Equation is Max Asset Salvage which means the decision at maturity is to decide if the option should be executed selling the asset and receiving the salvage value or not to execute holding on to the asset The Intermediate Node Equation used is Max Salvage OptionOpen indicating that before maturity the decision is either to execute early in this American option to abandon and receive the salvage value or to hold on to the asset and hence hold on to and keeping the option open for potential future execution denoted simply as OptionOpen Figure 22 shows the European version of the abandonment option where the Intermediate Node Equation is simply OpsonOpen as early execution is prohibited before maturity Of course being only able to execute the option at maturity is worth less 124 5054 compared to 125 4582 than being able to exercise earlier The example file
174. ption becomes live if the asset value hits the lower barrier Conversely a Down and Out option is live only when the lower barrier is not breached Examples of this option include contractual agreements whereby if the lower barrier is breached some event or clause is triggered The value of a barrier option is lower than standard options as the barrier option will be valuable only within a smaller price range than the standard option The holder of a barrier option loses some of the traditional option value and therefore such options should be worth less than a standard option An example would be a contractual agreement whereby the writer of the contract can get into or out of certain obligations if the asset or project value breaches a barrier Figure 62 shows a Lower Barrier Option for a Down and In Call Notice that the value is only 7 3917 much lower than a regular American call option of 42 47 This is because the barrier is set low at 90 This means that all of the upside potential that the regular call option can have will be reduced significantly and the option can only be exercised if the asset value falls below this lower barrier of 90 example file used Barrier Option Down and In Lower Barrier Call To make such a Lower Barrier option binding the lower barrier level must be below the starting asset value but above the implementation cost If the barrier level is above the starting asset value then it becomes an upper barri
175. ptionOpen Figure 32 Bermudan Expansion Option 64 Real Options Super Lattice Solver Real Options SLS User Manual EZ Figure 33 Single Asset Super Lattice Solver _ Custom Bermudan Option to Expand with changing rates of expansion over time and blackout periods 4 V 400 250 5 100 Boe 0 80 1 2 10 20 35 Max Asset Asset Expansion Cost Max Asset Cost 0
176. r barrier option will be example file Barrier Option Down and Out Lower Barrier Call For instance e Ata lower barrier of 90 the option value is 42 19 e Ata lower barrier of 100 the option value is 41 58 Figures 62 and 63 illustrate American Barrier Options To change these into European Barrier Options set the Intermediate Node Equation Nodes to OptionOpen In addition for certain types of contractual options vesting and blackout periods can be imposed For solving such Bermudan Barrier Options keep the same Intermediate Node Equation as the American Barrier Options but set the Intermediate Node Equation During Blackout and Vesting Periods to OptionOpen and insert the corresponding blackout and vesting period lattice steps Finally if the Barrier is a changing target over time put in several custom variables named Barrier with the different values and starting lattice steps Figure 62 Single Asset Super Lattice Solver 515 Lower Barrier Down and In Call This option is live only when the asset value breaches the lower bamer 100
177. r or lower barrier Conversely for the Up and Out and Down and Out option the option is live only when neither the upper nor lower barrier is breached Examples of this option include contractual agreements whereby if the upper barrier is breached some event or clause is triggered The value of barrier options is lower than standard options as the barrier option will have value within a smaller price range than the standard option The holder of a barrier option loses some of the traditional option value and therefore should sell it at a lower price than a standard option Figure 66 illustrates an American Up and In Down and In Double Barrier Option This is a combination of the Upper and Lower Barrier Options shown previously The same exact logic applies to this Double Barrier Option Figure 66 illustrates the American Barrier Option solved using the SLS To change these into a European Barrier Option set the Intermediate Node Equation Nodes to OptionOpen In addition for certain types of contractual options vesting and blackout periods can be imposed For solving such Bermudan Barrier Options keep the same Intermediate Node Equation as the American Barrier Options but set the Intermediate Node Equation During Blackout and Vesting Periods to OpsonOpen and insert the corresponding blackout and vesting period lattice steps Finally if the Barrier is a changing target over time put in several custom variables named Barrier with the different val
178. r to choose whether the option will be a call or a put thereby reducing the total cost of obtaining two separate options For instance with the same input parameters in Figure 41 the American Chooser Option is worth 6 7168 as compared to 4 87 for the call and 2 02 for the put 6 89 total cost for two separate options Figure 41 Single Asset Super Lattice Solver 515 American amp European Chooser choose between Call and Put value exceeds Call Put due to ability to choose i 15 15 1 5 4 100 Boe
179. ract with 10 Step Lattice User Manual 55 Real Options Super Lattice Solver Real Options SLS Bias C V V 1000 1000 5 100 1 2 10 20 35 Max Asset Asset Contraction Savings Max Asset Cost 0
180. rder to use the expansion option That is there must be a base case to expand upon If there is no base case state then the simple Execution Option calculated using the simple Ca Option is more appropriate where the issue at hand is whether or not to execute a project immediately or to defer execution As an example suppose a growth firm has a static valuation of future profitability using a discounted cash flow model in other words the present value of the expected future cash flows discounted at an appropriate market risk adjusted discount rate that is found to be 400 million Asse Using Monte Carlo simulation you calculate the implied Volatility of the logarithmic returns on the assets based on the projected future cash flows to be 35 The Rzs Free Rate on a tiskless asset 5 year U S Treasury Note with zero coupons for the next 5 years is found to be 7 Further suppose that the firm has the option to expand and double its operations by acquiring its competitor for a sum of 250 million Implementation Cost at any time over the next 5 years Maturity What is the total value of this firm assuming that you account for this expansion option The results in Figure 29 indicate that the strategic project value is 638 73 M using a 10 step lattice which means that the expansion option value is 88 73M This result is obtained because the net present value of executing immediately is 400M x 2 250 or 550M Thus 6
181. rical volatility for this entire period to be 36 So the computation is close enough such that we can use this approach for management discussions This is why the normality assumption and using a regular standard deviation as a proxy are sufficient Assumption 3 We used a standard normal calculation to impute the volatility As we ate assuming that the underlying distribution is normal we can compute the volatility by using the standard normal distribution The standard normal distribution Z score is such that X X Z this means that O Z and because we normalize the volatility as a percentage 0 we divide this by the mean to obtain xc Zu In layman s terms we have Volatility Percentile Value Mean Inverse of the Percentile x Mean Again the inverse of the percentile is obtained using Excel s function NORMSINV 132 Real Options Super Lattice Solver Real Options SLS User Manual Distribution of Microsoft Stock Prices 350 300 250 200 150 100 50 Carne 11 nr Frequency pos e d s e I eo e umi N N Er Q Q X N hi a 19 eo e e eo e e A A e m N N N e e eo e e e A A e e A A A A A A Bin Figure B11 Probability Distribution of Microsoft s Stock Price Since 1986 Distribution of Microsoft Stock Log Returns 250 20
182. rican and European Options Using Trinomial 85 2 14 American and European Mean Reversion Options Using Trinomial Lattices eee 88 2 15 Jump Diffusion Options Using Quadranomial Lattices eee eee eene tentent tette ttes 91 2 16 Dual Variable Rainbow Options Using Pentanomial Lattices eese netten ttt ttes 93 2 17 American and European Lower Barrier Options 95 2 18 American and European Upper Barrier Options eee 98 2 19 American and European Double Barrier Options and Exotic Barriers eee 102 SECTION III EMPLOYEE STOCK OPTIONSS ee eeeeeeee sesenta 104 3T American ESO with Vesting Period ce NERONE RR RR NER RE ERR de ie 105 4 2 American ESO with Suboptimal Excercise Bebavior eese eee tette tete 107 3 3 American ESO with Vesting and Suboptimal Exercise Bebavior eese teens 109 3 4 American ESO with Vesting Suboptimal Exercise Behavior Blackout Periods and Forfeiture 111 Appendix As Lattice Convergence 113 Appendix B Volatility Estimates 115 B 1 Volatility Estimates Logarithmic Cash Flow Returns Stock Price Returns Approach 116 B 2 Volatility Estimates Logarithmic Present Value Returns eee teet tette tette tenente 121 BO GARCELApproadh ii E d tede adag R R ERA 126 User Manual 5 Real Options Super Lattice Solver
183. rinomials are much more difficult to calculate and take a longer computation time the binomial lattice is usually used instead However a trinomial is required only when the underlying asset follows a mean reverting process An illustration of the convergence of trinomials and binomials can be seen in the following example Steps 5 10 100 1 000 5 000 Binomial Lattice 30 73 29 22 29 72 29 77 29 78 Trinomial Lattice 29 22 29 50 29 75 29 78 29 78 Figure 52 shows another example using the Multinomial Option The computed American Call is 31 99 using a 5 step trinomial and is identical to a 10 step binomial lattice seen in Figure 53 Therefore due to the simpler computation and the speed of computation the SLS and MSLS use binomial lattices instead of trinomials or other multinomial lattices The only time a trinomial lattice is truly useful is when the underlying asset of the option follows a mean reversion tendency In that case use the MNLS module instead When using this MNLS module just like in the single asset lattices you can modify and add in your own customized equations and variables and the concepts are identical to that of the SLS examples throughout this user manual 85 Real Options Super Lattice Solver Real Options SLS ires Meeeraumeiae American Option using a Trinomial Lattice Model Tun gm
184. rns Thus to avoid over inflating the computations we use relative returns in Step 2 Period 1 Start Value Period 2 Start Value Period n Start Value 100 A 200 Step 3 Compute natural logarithm of the relative returns The natural log is used for two reasons The first is to be comparable to the exponential Brownian Motion stochastic process That is recall that a Brownian Motion is written as os g Gosoc or S To compute the volatility 0 used in an equivalent computation regardless of whether it is used in simulation lattices or closed form models because these three approaches require the Brownian Motion as a fundamental assumption a natural log is used The exponential of a natural log cancels each other out in the above equation Second in computing the geometric average relative returns were used then multiplied and taken to the root of the number of periods By taking a natural log of a root n we reduce the root n in the geometric average equation This is why natural logs are used in Step 3 Step 4 Compute the sample standard deviation to obtain the periodic volatility sample standard deviation is used instead of a population standard deviation because your dataset might be small For larger datasets the sample standard deviation converges to the population standard deviation so it is always safer to use the sample standard deviation Of course the sample standard deviation seen below is simpl
185. rtainty becomes resolved END Figure 48 Complex Multi Phased Sequential Compound Option Vutgle Phased Complies Sequertis Compound Option 1 1 Salvage Figure 49 Complex Multi Phased Sequential Compound Option using MSLS User Manual 80 Real Options Super Lattice Solver Real Options SLS User Manual To illustrate Figure 49 s MSLS path dependent sequential option uses the following inputs Phase 3 Terminal Intermediate Steps Phase 2 Terminal Intermediate Steps Phase 1 Terminal Intermediate Steps Max Underlying Expansion Cost Underlying Salvage Max Undetlying Expansion Cost Salvage OptionOpen 50 Max Phase3 Phase3 Contract Savings Salvage 0 Max Phase3 Contract Savings Salvage OptionOpen 30 Max Phase2 Salvage 0 Max Salvage OptionOpen 10 81 Real Options Super Lattice Solver Real Options SLS User Manual 2 11 Path Dependent Path Indep
186. s 100 Steps Y 37 94 10 Steps m 222155 20138 18221 164 87 122 14 164 87 149 18 149 18 149 18 134 99 134 99 134 99 110 52 110 52 100 00 122 14 100 00 7408 7408 122 14 122 14 110 52 110 52 110 52 100 00 100 00 74 08 67 03 9048 9044 9048 9048 9048 3187 8187 8187 8187 74 08 i 60 65 67 03 60 65 60 65 54 88 49 66 54 88 44 93 171 83 145 96 2255 255 101 38 101 38 90 05 88 34 8221 Option Valuation Lattice 7942 7644 69 75 69 55 65 67 5870 4918 6048 1 5599 4892 39 86 3166 2214 15 39 10 70 0 00 0 00 000 000 0 00 250 0 00 0 00 0 00 0 00 0 00 0 00 0 00 000 000 0 00 0 00 000 000 0 00 0 00 Figure 70 ESO Toolkit Results of a Call Option accounting for Suboptimal Behavior User Manual 108 Real Options Super Lattice Solver Real Options SLS 3 3 American ESO with Vesting and Suboptimal Exercise Behavior Next we have the ESO with vesting and suboptimal exercise behavior This is simply the extension of the previous two examples Again the result of 9 22 Figure 71 is verified using the ESO Toolkit as seen in Figure 72 example file used ESO Vesting with Suboptimal Behavior Employee Stock Option with vesting period and suboptimal exerci
187. s a docked window on the tight of the Strategy Tree such that when an option node or terminal node is selected the properties of that node will be displayed and can be updated directly This provides an alternative to double clicking on a node each time Example Files are available in the File menu to help you get started on building Strategy Trees Protect File from the File menu allows the Strategy Tree to be encrypted with up to a 256 bit password encryption Be careful when a file is being encrypted because if the password is lost the file can no longer be opened 32 Real Options Super Lattice Solver Real Options SLS e Capturing the Screen or printing the existing model can be done through the File menu The captured screen can then be pasted into other software applications e Add Duplicate Rename Delete a Strategy Tree be performed through right clicking the Strategy Tree tab or the Edit menu e You can also Insert a File Link and Insert a Comment on any option or terminal node or Insert Text or Picture anywhere in the background or canvas area e You can Change Existing Styles or Manage and Create Custom Styles of your Strategy Tree this includes size shape color schemes and font size color specifications of the entire Strategy Tree E Plain Vanilla Call Option 1 Single Asset Super Lattice Solver
188. s are very difficult to run and interpret and require great facility with econometric modeling techniques GARCH is a term that incorporates a family of models can take on a variety of forms known as GARCH 4 where and 4 are positive integers which define the resulting GARCH model and its forecasts For instance a GARCH 1 1 model takes the form of AXyte 2 2 D a a fo 126 Real Options Super Lattice Solver Real Options SLS User Manual where the first equation s dependent variable jj is a function of exogenous variables x with an error term The second equation estimates the variance squared volatility at time 7 which depends on a historical mean news about volatility from the previous period measured as a lag of the squared residual from the mean equation Er and volatility from the previous period 0 7 The exact modeling specification of a GARCH model is beyond the scope of this book and will not be discussed Suffice it to say that detailed knowledge of econometric modeling model specification tests structural breaks and error estimation is required to run a GARCH model making it less accessible to the general analyst The other problem with GARCH models is that the model usually does not provide a good statistical fit That is it is impossible to predict say the stock market and of course equally if not harder to predict a stock s volatility over time Figure B5 shows a GARCH
189. s example the contraction option using a 100 step lattice is valued to be 14 24M using SLS This means that the maximum amount that the counterparty should be paid should not exceed this amount Of course the option analysis can be further complicated by analyzing the actual savings on a present value basis For instance if the option is exercised within the first 5 years the savings is 872M but if exercised during the last 5 years then the savings is only 50M The revised option value is now 10 57M A manufacturing firm is interested in outsourcing its manufacturing of children s toys to a small province in China By doing so it will produce overhead savings of over 20M in present value over the economic life of the toys However outsourcing this internationally will mean lower quality control delayed shipping problems added importing costs and assuming the added risks of unfamiliarity with the local business practices In addition the firm will only consider outsourcing only if the quality of the workmanship in this Chinese firm is up to the stringent quality standards it requires The NPV of this 54 Real Options Super Lattice Solver Real Options SLS particular line of toys is 100 with a 25 volatility The firm s executives decide to purchase a contraction option by locating a small manufacturing firm in China spending some resources to try out a small scale proof of concept thereby reducing the uncertainties of quality knowl
190. s is why we perform Step 1 when we compute volatilities using the Log Present Value Returns Approach 125 Real Options Super Lattice Solver Real Options SLS User Manual e Step 2 Calculate the intermediate variable X This X variable is identical to the logarithmic relative returns in the Log Cash Flow Returns Approach It is simply the natural logarithm of the relative returns of the future stock price using the sum of present values at time 1 as a proxy from the current stock the sum of present values at time 0 We then set the sum of present values at time 0 as static because it is the base case and by definition of a base case the values do not change The base case can be seen as the NPV of the project s net benefits and is assumed to be the best estimate of the project s net benefit value It is the future that is uncertain and fluctuates hence we simulate the DCF model and allow the numerator of the X variable to change during the simulation while keeping the denominator static as the base case e Step 3 Simulate the model and obtain the standard deviation as volatility This approach requires that the model be simulated This makes sense because if the model is not simulated means that there is no uncertainties in the project or asset and hence the volatility is equal to zero You would only simulate when there are uncertainties hence you obtain a volatility estimate The rationale for using the sample standard deviat
191. s used are Abandonment American Option and Abandonment European Option 3 See the section on Expansion Option for more examples on how this startup s technology can be used as a platform to further develop newer technologies that can be worth lot more than just the abandonment option 46 Real Options Super Lattice Solver Real Options SLS For example the airline manufacturer in the previous case example can agree to a buy back provision that can be exercised at any time by the airline customer versus only at a specific date at the end of five years the former American option will clearly be worth more than the latter European option Else SLS This American Abandonment Option can be executed at any time up to and including expiration J V S 120 5 90 0 5 25
192. se behavior onion J 20 20 0 10 50 100 0 39 1 2 10 20 35 Max Asset Cost 0 Max Asset Cost 0 9 2178 IF Asset Suboptimal Cost Max Asset Cost 0 OptionOpen Max Asset Cost OptionOpen ysna B
193. ssion The following shows some examples of valid equations usable in the custom equations boxes Review the rest of the user manual recommended texts and example files for more illustrations of actual options equations and functions used in SLS o Max Asset Cost 0 o Max Asset Cost OptionOpen o 135 o 12 24 12 24 36 48 41 Real Options Super Lattice Solver Real Options SLS User Manual 3 ABS 3 3 MAX 1 2 3 4 MIN 1 2 3 4 SQRT 3 ROUND 3 LOG 12 IF a gt 0 3 4 returns 3 if a gt 0 else 4 ABS 3 MAX a b c MIN d e a gt b IF a gt 0 b lt 0 3 4 IF c lt gt 0 3 4 IFdF a lt 3 4 5 lt gt 4 a a b MAX My Cost 1 My Cost 2 Asset 2 Asset 3 This concludes a quick overview and tour of the software You are now equipped to start using the SLS software in building and solving real options financial options and employee stock options problems These applications are introduced starting the next section However it is highly recommended that you first review Dr Johnathan Mun s Real Options Analysis Tools and Techniques Second Edition Wiley 2006 for details on the theory and application of real options 42 Real Options Super Lattice Solver Real Options SLS SECTION II REAL OPTIONS ANALYSIS User Manual 43 Real Options Super Lattice Solver Real Options SLS User Manual 2 1 American European Bermudan and Customi
194. stical analysis is performed to find the best fitting volatility curve providing different volatility estimates over time Management Assumptions and Guesses Used for both financial options and real options The drawback is that the volatility estimates are very unreliable and are only subjective best guesses The benefit of this approach is its simplicity this method is very easy to explain to management the concept of volatility both in execution and interpretation Market Proxy Comparables or Indices Used mainly for comparing liquid and non liquid assets as long as comparable market sector or industry specific data are available The drawback is that it is sometimes hard to find the right comparable firms and the results may be subject to gross manipulation by subjectively including excluding certain firms The benefit is its ease of use 115 Real Options Super Lattice Solver Real Options SLS Time Period N User Manual B 1 Volatility Estimates Logarithmic Cash Flow Returns Stock Price Returns Approach The Logarithmic Cash Flow Returns or Logarithmic Stock Price Returns Approach calculates the volatility using the individual future cash flow estimates comparable cash flow estimates or historical prices generating their corresponding logarithmic relative returns as illustrated in Figure B1 Starting with a series of forecast future cash flows historical prices convert them into relative returns Then
195. sting Forfeiture Suboptimal Behavior and Blackout Periods 112 Real Options Super Lattice Solver Real Options SLS User Manual Appendix A Lattice Convergence The higher the number of lattice steps the higher the precision of the results Figure A1 illustrates the convergence of results obtained using a BSM closed form model on a European call option without dividends and comparing its results to the basic binomial lattice Convergence 15 generally achieved at between 500 1 000 steps Due to the high number of steps required to generate the results software based mathematical algorithms are used For instance a nonrecombining binomial lattice with 1 000 steps has a total of 2 109 nodal calculations to perform making manual computation impossible without the use of specialized algorithms 5 Figure A1 also illustrates the binomial lattice results with different steps and notes the convergence of the binomial for a simple European call option using the Black Scholes model This proprietary algorithm was developed by Dr Johnathan Mun based on his analytical work with FASB in 2003 2004 his books Valuing Employee Stock Options Under the 2004 FAS 123 Requirements Wiley 2004 Real Options Analysis Tools and Techniques Wiley 2002 Real Options Analysis Course Wiley 2003 Applied Risk Analysis Moving Beyond Uncertainty Wiley 2003 creation of his software Real Options Analysis Toolkit versions 1 0
196. sults of a Call Option accounting for Vesting Forfeiture Suboptimal Behavior and Blackout Periods User Manual 111 Real Options Super Lattice Solver Real Options SLS User Manual Customized tion Assumptions Stock Price Strike Price Maturity in Years Risk free Rate 96 Dividends 96 Volatility Suboptimal Exercise Multiple 1 80 Vesting in Years Forfeiture Rate 100 00 100 00 10 00 5 50 4 00 45 00 4 00 10 00 r Additional Assumptions Year Volatility 10 00 25 00 37 45 26 18 100 Steps v Results Generalized Black Scholes 100 Step Super Lattice Super Lattice Steps Lune Lo e Lossenm HS Year Risk free 96 10 00 5 5096 10 00 5 5096 10 00 5 5096 10 00 5 5096 10 00 5 5096 10 00 5 5096 10 00 5 5096 10 00 5 5096 10 00 5 5096 10 00 5 5096 Please be aware that by applying multiple changing volatilities over time non recombining lattice is required which increases the computation time significantly In addition only smaller lattice steps may be computed When many volatilifies over time and many lattice steps are required use Monte Carlo simulation on the volatilities and run the Basic or Advanced Custom Option module instead For additional steps use ESO Function Figure 74 ESO Toolkit Results after accounting for Ve
197. surance policy That is if the asset or project value increases above its current value the firm may decide to continue funding the project or sell it off in the market at the prevailing fair market value Alternatively if the value of the asset or project falls below the 90M threshold the firm has the right to execute the option and sell off the asset to the counterparty at 90M In other words a safety net of sorts has been erected to prevent the value of the asset from falling below this salvage level Thus how much is this safety net or insurance policy worth One can create competitive advantage in negotiation if the counterparty does not have the answer and you do Further assume that the 5 year Treasury Note Risk Free Rate zero coupon is 5 from the U S Department of Treasury The American Abandonment Option results in Figure 19 show a value of 125 48M indicating that the option value is 5 48M as the present value of the asset is 120M Hence the maximum value one should be willing to pay for the contract on average is 5 48 This resulting expected value weights the continuous probabilities that the asset value exceeds 90M versus when it does not where the abandonment option is valuable Also it weights when the timing of executing the abandonment is optimal such that the expected value is 5 48M In addition some experimentation can be conducted Changing the salvage value to 30M this means a 90M discount from the start
198. termediate Max Salvage Auditing Lattice Result 10 steps Super Lattice Result 10 steps 0 5000 1 1934 0 8380 0 5272 125 48 125 48 Name salvage Value SSS SS a Ss e Starting Step 0 Underlying Asset Lattice 203 94 170 89 170099 17089 17089 17089 20 _ mero Ss i 493 59 413 61 346 59 34659 290 43 290 43 Option Valuation Lattice 290 43 243 43 243 37 204 30 702 93 589 03 493 59 413 61 290 43 203 94 170 89 146 01 145 36 125 48 108 49 107 41 97 95 9713 96 03 94 57 90 00 91 44 90 88 90 00 90 00 90 00 105 93 90 00 90 00 90 00 90 00 48 90 00 Figure 20 Audit Sheet for the Abandonment Option Real Options Super Lattice Solver Real Options SLS American Abandonment Option can be executed at any time up to and including expiration a 7 120
199. th American and European options requires the Pentanomial Lattice approach Rainbows on the horizon after a rainy day comprise various colors of the light spectrum and although rainbow options aren t as colorful as their physical counterparts they get their name from the fact that they have two or more underlying assets rather than one In contrast to standard options the value of a rainbow option is determined by the behavior of two or more underlying elements and by the correlation between these underlying elements That is the value of a rainbow option is determined by the performance of two or more underlying asset elements This particular model is appropriate when there are two underlying variables in the option e g Price of Asset and Quantity where each fluctuates at different rates of volatilities but at the same time might be correlated Figure 59 These two variables are usually correlated in the real world and the underlying asset value is the product of price and quantity Due to the different volatilities a pentanomial or five branch lattice is used to capture all possible combinations of products Figure 60 Be aware that certain combinations of inputs may yield an unsolvable lattice with negative implied probabilities If that result occurs a message will appear Try a different combination of inputs as well as higher lattice steps to compensate P3 Q3 01 04 quantity 05 02 06 Figure 60 Pentanomial Lattice Com
200. the Asset 100 00 6 Alternate Best Case Scenario NPV 144 85 Best Case Scenario 7 Percentile of Best Case Scenario 90 00 150M 8 7 9 Implied Volatility Estimate 10 probability 10 11 12 ETT gt NPV of Project 13 ecte 100M 14 15 90th percentile 16 Frequency 17 18 Probability to Volatility Worst Case Scenario 18 20 Expected NPV of the Asset 100 00 Worst Case Scenario 21 Alternate Worst Case Scenario NPV 50 00 50M 22 Percentile of Worst Case Scenario 10 0096 23 24 Implied Volatility Estimate 25 26 i 27 Ep SW NPV of Project 28 ecte 29 100M 30 7 31 10th percentile Figure B9 Excel Probability to Volatility Model User Manual 130 Real Options Super Lattice Solver Real Options SLS User Manual Probability to Volatility Best Case Scenario Expected NPV of the Asset 100 00 Alternate Best Case Scenario NPV 144 85 Percentile of Best Case Scenario 90 00 Implied Volatility Estimate Goal Seek Figure B10 Excel Volatility to Probability Model Figure B9 allows you to enter the expected NPV and the alternate values best case and wotst case as well as its corresponding percentiles That is given some probability and its value we can impute the volatility Conversely Figure B10 shows how you can use Excel s Goal Seek function click on Tools Goal Seek in Excel to find the probability from a volatility For instance say the project s expected N
201. the call option should be exercised and the later the put option should be exercised The put option can be solved by setting the Terminal Node Equation as Max Cost 45 0 as seen in Figure 40 example file used Plain Vanilla Put Option Puts have a similar result as calls in that when dividends are included the basic put option values for American 2 Bermudan gt European in most basic cases You can confirm this by simply setting the Dividend Rate at 3 and Blackout Steps at 0 80 and re tunning the SLS module 72 Real Options Super Lattice Solver Real Options SLS User Manual O x American Put Option Make it European by setting INE OptionOpen or deselect Custom and select European Tun ORL V 100 100 5 100 Boe
202. timal Behavior and steps was changed to 100 in this example The is the same as the previous example but the IE assumes that the option will be suboptimally executed if the stock price in some future state exceeds the suboptimal exercise threshold times the strike price Notice that the IEV is not used because we did not assume any vesting or blackout periods Also the Suboptimal exercise multiple vatiable is listed on the customs variable list with the relevant value of 1 85 and a starting step of 0 This means that 1 85 is applicable starting from step 0 in the lattice all the way through to step 100 The results again are verified through the ESO Toolkit Figure 70 Figure 69 Single Asset Super Lattice Solver SLS Employee Stock Option with suboptimal exercise multiples A 100 m 100 10 100
203. tion com Real Options kw Valuation Real Options SLS User Manual PREFACE Welcome to the Real Options Super Lattice Solver Software Welcome to the Real Options Super Lattice Solver SLS software This software has several modules including e Single Super Lattice Solver SLS e Multiple Super Lattice Solver MSLS e Multinomial Lattice Solver MNLS e Lattice Maker e 515 Excel Solution e SLS Functions e ROV Strategy Tree These modules embrace the financial concepts of options as applied to real or physical assets For example when you purchase a call option on an underlying stock you are purchasing the right but not the obligation to buy a share of stock at a set cost or strike price When the time comes to buy the stock or exercise your option either at or before maturity you exercise the option if the stock price is higher than the strike price of your option Exercising the option means purchasing the stock at the strike price and selling it at the higher market price to make a profit less any taxes transaction costs and premiums paid to obtain the option However if the price is less than the strike price you don t buy the stock and your only losses are the transaction costs and premiums The future is difficult to predict and may be wrought with uncertainty and risk You cannot know for certain whether a specific stock will increase or decrease in value This is the beauty of options You ma
204. to be the highest The second cash flow stream seems more risky User Manual 118 Real Options Super Lattice Solver Real Options SLS User Manual than the first set due to larger fluctuations but has a volatility of 0 Therefore be careful when applying this method to small datasets When applied to stock prices and historical data that are nonnegative this approach is easy and valid Howevet if used on teal options assets the DCF cash flows may very well take on negative values returning an error in your computation 1 log of a negative value does not exist However there are certain approaches you can take to avoid this error The first is to move up your DCF model from free cash flows to net income to operating income EBITDA and even all the way up to revenues and prices whete all the values are positive If doing it this way then care must be taken such that all other options and projects are modeled this way for comparability s sake Also this approach is justified in situations where the volatility risk and uncertainty stem from a certain variable above the line is used For instance the only critical success factor for an oil and gas company is the price of oil price and the production rate quantity where both are multiplied to obtain revenues In addition if all other items in the DCF are proportional ratios e g operating expenses are 25 of revenues or EBITDA values ate 10 of revenues and so forth then we
205. turity Years Volatility 4 Combination Options Lattice Steps Option Type Risk free 4 Expansion Factor Dividend 2 ansion Implementation Cost Expansion Factor Lattice Steps 10 Contraction Factor Contraction Saving Abandonment Salvage Maturity Years Contraction Factor Abandonment Salvage American Option European Option Show Formulae Underieing Asset Lattice 100 00 10733 11549 123 63 132 69 1424 15285 16405 176 07 16897 20281 sest 9347 10000 10733 11519 123 63 13269 14241 15285 6036 6543 7022 75436 5580 6096 65 43 Opticon Valfoaticn Lattice 2319 205 3358 _ 39 77 4654 5446 6236 7127 8094 9143 10281 Continue Continue Continue Continue Continue _ __ Continue Continue Continue _ Execute 1544 1935 2399 2935 3543 4221 4964 5772 665i 7607 Continue Continue Continue Continue Continue Continue Continue Continue Continue Execute 833 1177 1533 1968 2486 3086 3757 4488 5285 6 L SL aot aeol 1599 0 00 000 000 Centinue Continue Continue Continue Centinue oo 0 00 Continue Continue Continue Continue _ End oof ooof 000 Continue Continue C
206. ues and starting lattice steps Exotic Barrier Options exist when other options are combined with barriers For instance an option to expand can only be executed if the PV Asset exceeds some threshold or a contraction option to outsource manufacturing can only be executed when it falls below some breakeven point Again such options can be easily modeled using the SLS 102 Real Options Super Lattice Solver Real Options SLS Double Barrier Up amp In Down amp In Call This option is live only when the asset value breaches either barrier _ __ 4 100 80 5 4 100 1 2 10 20 35
207. umed that the holder has the right to choose either a call or a put with the same strike price at time and with the same expiration date T2 For different values of strike prices at different times we need a complex variable chooser option Definitions of Vatiables 5 present value of future cash flows X implementation cost r risk free rate time to choose between a call or put years time to expiration years volatility cumulative standard normal distribution q continuous dividend payments Computation hg 2 oJf oe Option Value seal 2 ne 2 eso neret 2 5 I Pa m User Manual 138 Real Options Super Lattice Solver Real Options SLS Complex Chooser The holder of the option has the right to choose between a call and a put at different times Tc and Tp with different strike levels and Xp of calls and puts Note that some of these equations cannot be readily solved using Excel spreadsheets Instead due to the recutsive methods used to solve certain bivariate distributions and critical values the use of programming sctipts is required Definitions of Vatiables 5 present value of future cash flows X implementation cost r risk free rate T time to expiration years for call and put Tp volatility cumulative standard normal distribution Q cumulative bivariat
208. un While the software and its models are based on his books the training courses cover the real options subject matter in more depth including the solution to sample business cases and the framing of real options of actual cases It is highly recommended that the user familiarizes him or herself with the fundamental concepts of real options as outlined in Real Options Analysis Tools and Techniques 2nd Edition Wiley 2006 The Real Options SLS software s design and analytics were created by Dr Johnathan Mun and the software s programming was developed by lead developer J C Chin User Manual 3 Real Options Super Lattice Solver Real Options SLS TABLE OF CONTENTS PREFACE ME 2 Welcome to the Real Options Super Lattice Solver S te dia etate pompier C Hd n DRM 2 Whe shot use E SECTION I GETTING STARTED eterne tro eth 7 1 1 Introduction to the Super Lattice cst 8 1 2 Sinple Asset Super aire SOET bl ced acid 11 1 3 Single Ass SES Banpo UR 12 1 4 Muitiple Asset Super Lattice Solver MISTS 18 LS UAI ROME Tatte 21 TESIS Taice pio a ttt sic trian sca E A A bab Dai E E E aetna 22 1 7 SLS Excel Solution SES MSLS and Changing Volatility Models in 23 ARN M 27 1 9 Exotic Financial Options oct tiri
209. uncertainties in the market competitive threats sales and pricing structure the annualized volatility of the cash flows using the logarithmic present value returns approach comes to 45 Suppose the risk free rate is 5 for the 2 year period Using the SLS the analysis results yields 254 95M indicating that the option value to wait and defer is worth over 4 95M after accounting for the 250M NPV if executing now In playing with several scenarios the breakeven point is found when dividend yield is 1 34 This means that if the cost of waiting lost net revenues in sales by pursuing the smaller market rather than the larger market and loss of market share by delaying exceeds 81 34M per year then it is not optimal to wait and the pharmaceutical firm should engage in the inhaled version immediately The loss in returns generated each year does not sufficiently cover the risks incurred An oil and gas company is currently deciding on a deep sea exploration and drilling project The platform provides an expected NPV of 1 000M This project is wrought with risks price of oil and production rate are both uncertain and the annualized volatility is computed to be 55 The firm is thinking of purchasing an expansion option by spending an additional 10M to build a slightly larger platform that it does not currently need but if the price of oil is high or when production rate is low the firm can execute this expansion option and execute additional
210. unning simulations and provides the user sample templates to create such models 8 Real Options Super Lattice Solver Real Options SLS User Manual e The SLS Functions are additional real options and financial options models accessible directly through Excel This module facilitates model building linking and embedding and running simulations e The Option Charts are used to visually analyze the payoff structure of the options under analysis the sensitivity and scenario tables of options to vatious inputs convergence of the lattice results and other valuable analyses The SLS software is created by Dr Johnathan Mun professor consultant and the author of numerous books including Rea Options Analysis Tools and Techniques 2nd Edition Wiley 2005 Modeling Risk Wiley 2006 and Valuing Employee Stock Options Under 2004 FAS 123 Wiley 2004 This software also accompanies the materials presented at different training courses on real options simulation and employee stock options valuation taught by Dr Mun While the software and its models are based on his books the training courses cover the real options subject matter in more depth including the solution of sample business cases and the framing of real options of actual cases It is highly suggested that the user familiarizes him or herself with the fundamental concepts of real options in Real Options Analysis Tools and Techniques 2nd Edition Wiley 2005 prior to attempting an
211. ut 18 62 and 18 76 are more symmetrical in value than with a regular call and put 31 99 and 13 14 e The regular American Call regular European Call because without dividends it is never optimal to execute early However because of the mean reverting tendencies being able to execute early is valuable especially before the asset value decreases So we see that MR American Call gt MR European Call but of course both are less than the regular Call American Call Option with a Mean Reverting Underlying Asset using a Trinomial Lattice V 2 S 100 3 2 2 L 5
212. ver SLS Multiple Super Lattice Solver MSLS Multinomial Lattice Solver MNLS Lattice Maker Advanced Exotic Options Valuator SLS Excel Solution and SLS Functions These modules are highly powerful and customizable binomial and multinomial lattice solvers and can be used to solve many types of options including the three main families of options real options which deals with physical and intangible assets financial options which deals with financial assets and the investments of such assets and employee stock options which deals with financial assets provided to employees within a corporation This text illustrates some sample real options financial options and employee stock options applications that users will most frequently encounter The Single Asset Model is used primarily for solving options with a single underlying asset using binomial lattices Even highly complex options with a single underlying asset can be solved using the SLS The Multiple Asset Model is used for solving options with multiple underlying assets and sequential compound options with multiple phases using binomial lattices Highly complex options with multiple underlying assets and phases can be solved using the MSLS The Multinomial Model uses multinomial lattices trinomial quadranomial pentanomial to solve specific options that cannot be solved using binomial lattices The Lattice Maker is used to create lattices in Excel with visible and l
213. volatility is zero the uncertainty is zero and 15 equal to 100 the value inside the parenthesis is infinity meaning that the standard normal distribution value is 100 alternatively you can state that with zero uncertainties you have a 100 certainty By separating the cash flows you can now use these as inputs into the options model whether it s using the Black Scholes or binomial lattices 2 E Otte IDT xg no G0 ecco cT Continuing with the example in Figure B4 the calculations of interest are on rows 51 to 55 Row 51 shows the present values of the cash flows to Year 0 assume that the base year is 2002 while row 52 shows the present values of the cash flows to Year 1 ignoring the sunk cost of cash flow at Year 0 These two rows are computed in Excel and ate linked formulas You should then copy and paste the values only into row 53 use Excel s Edit Paste Special Values Only to do this Then compute the intermediate variable X 054 using the following Excel formula LN SUM E52 H52 SUM D53 H53 Then simulate this DCF model using Simulator by assigning the relevant input assumptions in the model and set this intermediate variable X as the output forecast The standard deviation from this X is the periodic volatility Annualizing the volatility is required by multiplying this periodic volatility with the square root of the number of periodicities in a year 1
214. where the breakeven points are for different combinations of volatilities and dividends Thus using this information you can make better go or no go decisions for instance breakeven volatility points can be traced back into the discounted cash flow model to estimate the probability of crossing over and that this ability to wait becomes valuable Figure 45 Multiple Asset Super Lattice Solver 2 Simple Two Phased Sequential Compound Option 3 Underlying 100 30 Vi NNNM sanpera Phasel 5 5 0 50 Max Phase2 CostO Max Phase2 Cost OptionOpen Phase2 80 5 0 100 Max Underlying Cost 0 Max Underlying Cost OptionOpen E PHASE1 27 6734 m p
215. will automatically convert it to OpfionOpen before it runs Consequently a potential problem exists because a model that defines OptionOpen as a custom variable will have errors as OptionOpen is now a special variable model that uses advanced worksheet function in the custom equations will not work Functions supported include ABS ACOS ASIN ATAN2 ATAN CEILING COS COSH EXP FLOOR LOG MAX MIN REMAINDER ROUND SIN SINH SQRT TAN TANH TRUNCATE and IF Variables in SLS are case sensitive except for function names Models that mix and match cases will not work in SLS Therefore it is suggested that when using custom variables in SLS and MSLS you are consistent with the use of case for the custom variable names AND and OR functions are missing and are replaced with special characters in SLS The amp and symbols represent the AND and OR operators For example Asset gt 0 Cost lt 0 means OR Asset gt 0 Cost lt 0 while Asset gt 0 amp Cost lt 0 is AND Asset gt 0 Cost lt 0 Blackout Step Specifications To define the blackout steps use the following examples as a guide e 3 Step 3 is a blackout step e 3 5 Steps 3 and 5 are blackout steps 40 Real Options Super Lattice Solver Real Options SLS User Manual e 3 5 7 Steps 3 5 6 7 are blackout steps e 1 3 5 6 Steps 1 3 5 6 blackout steps e 5 7 Steps 5 6 7 are blackout steps
216. ximize your gains speculation with unlimited upside while minimizing your losses hedging against the downside by setting the maximum losses as the premium paid on the option The same idea can be applied to assets A firm s assets might include plants patents projects research and development initiatives and so forth Each of these assets carries a level of uncertainty For example will a firm s multimillion dollar research project develop into revenue generating product Will investing in a successful start up company help a firm expand into new markets Management asks such questions every day The Real Options Super Lattice Solver software collectively the SLS MSLS and MNLS provide analysts and executives the ability to determine the value of investing in an uncertain future 2 Real Options Super Lattice Solver Real Options SLS Who should use this software The SLS MSLS MNLS Lattice Maker Excel Solution and Excel Functions are appropriate for analysts who are comfortable with spreadsheet modeling in Excel and with real options valuation The software accompanies the books Rea Options Analysis Tools and Techniques 2nd Edition Wiley 2005 Modeling Risk Wiley 2006 and Valuing Employee Stock Options Wiley 2004 all by Dr Johnathan Mun who designed the software There several accompanying training courses Certified Risk Analyst CRA The Basics of Real Options and Advanced Real Options also taught by Dr M
217. xpenses positively correlated to revenues but are assumed to be distributed following a Triangular distribution while the effects of market competition are simulated using a Poisson distribution with a small rate times 131 Real Options Super Lattice Solver Real Options SLS User Manual the probability of technical success simulated as Binomial distribution We cannot determine theoretically what a Lognormal minus a Triangular times Poisson and Binomial after accounting for their correlations would be Instead we rely on the Central Limit Theorem and assume the final result is normally distributed especially if a large number of trials are used in the simulations Finally we are interested in the logarithmic relative returns volatility not the standard deviation of the actual cash flows stock prices Stock ptices and cash flows ate usually Lognormally distributed stock prices cannot be below zero but the logs of the relative returns are always normally distributed In fact this can be seen in Figure B11 and B12 where the historical stock prices of Microsoft from March 1986 to December 2004 are tabulated Assumption 2 We assume that the standard deviation is the same as the volatility Again referring to Figure 12 using the expected returns chart the average is computed at 0 58 and the 90th percentile is 8 60 and the implied volatility is found to be 37 Using the data downloaded we compute the empi
218. y the average sum of all and then divided by some variation of n of 120 Real Options Super Lattice Solver 1 0 Real Options SLS the deviations of each point of a dataset from its mean X X adjusted for a degree of freedom for small datasets where a higher standard deviation implies a wider distributional width and thus carries a higher risk The variation of each point around the mean is squared to capture its absolute distances otherwise for a symmetrical distribution the variations to the left of the mean might equal the variations to the right of the mean creating a zero sum and the entire result is taken to the square root to bring the value back to its original unit Finally the denominator 7 7 adjusts for a degree of freedom in small sample sizes illustrate suppose there are three people in a room and we ask all three of them to randomly choose a number of their choice as long as the average is 100 The first person might choose any value and so could the second person However when it comes to the third person he or she can only choose a single unique value such that the average is exactly 100 Thus in a room of 3 people only 2 people 7 7 are truly free to choose So for smaller sample sizes taking the 7 correction makes the computations more conservative This is why we use sample standard deviations in Step 4 volatility Step 5 Compute the annualized volatility The volatil
219. zed Abandonment Options The Abandonment Option looks at the value of a project s or asset s flexibility in being abandoned over the life of the option As an example suppose that a firm owns a project or asset and that based on traditional discounted cash flow DCF models it estimates the present value of the asset Underlying Asset to be 120M for the abandonment option this is the net present value of the project or asset Monte Carlo simulation indicates that the Vo atility of this asset value is significant estimated at 25 Under these conditions there is a lot of uncertainty as to the success or failure of this project the volatility calculated models the different sources of uncertainty and computes the risks in the discounted cash flow DCF model including price uncertainty probability of success competition cannibalization and so forth and the value of the project might be significantly higher or significantly lower than the expected value of 120M Suppose an abandonment option is created whereby a counterparty is found and a contract is signed that lasts 5 years Maturity such that for some monetary consideration now the firm has the ability to sell the asset or project to the counterparty at any time within these 5 years indicative of an American option for a specified Salvage of 90M The counterparty agrees to this 30M discount and signs the contract What has just occurred is that the firm bought itself a 90M in

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