1. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 1 B40.2302 Class #4  BM6 chapters 20, 21  Based on slides created by Matthew Will  Modified 5/24/2015 by Jeffrey Wurgler

2.  Spotting and Valuing Options Principles of Corporate Finance Brealey and Myers Sixth Edition Slides by Matthew Will, Jeffrey Wurgler Chapter 20 © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill

3. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 3 Topics Covered  Calls, Puts and Shares  Financial Alchemy with Options  Option Valuation  Constructing equivalent portfolios  Risk-neutral valuation  Black-Scholes

4. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 4 Option Terminology Put Option Right to sell an asset at a specified exercise price on or before a specified exercise date. Call Option Right to buy an asset at a specified exercise price on or before a specified exercise date.

5. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 5 Option Value  The value of an option at expiration depends on the difference between the stock price and the exercise price. Example - Value at expiration given $85 exercise price

6. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 6 Option Value Payoff on a riskless bond/loan at maturity … is fixed (lender’s perspective). Share Price Bond value 0

7. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 7 Option Value Payoff to a share when you want to sell it … depends on share price (share buyer’s perspective). Share Price Share value 50 0

8. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 8 Option Value Call option value at expiration given a $85 exercise price (call buyer’s perspective). Share Price Call option value 85 105 $20 0

9. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 9 Option Value Put option value at expiration given a $85 exercise price (put buyer’s perspective). Share Price Put option value 80 85 $5 0

10. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 10 Option Obligations

11. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 11 Option Value Call option value at expiration given a $85 exercise price (call seller’s perspective). Share Price Call option $ payoff 85 0

12. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 12 Option Value Put option value at expiration given a $85 exercise price (put seller’s perspective). Share Price Put option $ payoff 85 0

13. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 13 Financial Alchemy Protective Put = Buy stock and buy put Share Price Position Value “Protective Put” Long Put Long Stock

14. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 14 Financial Alchemy Straddle = Long call and long put - Profits from high volatility Share Price Position Value Straddle

15. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 15 Put-Call Parity  The following two strategies give exactly the same payoff (a “protective put” payoff)…  Buy share and buy put  Lend money and buy call  … so they must sell at exactly the same price  This leads to the “put-call parity” formula

16. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 16 Put-Call Parity Value of a call + PV(Exercise price) = Value of put + Current share price  Holds only for European options  Requires put and call with same exercise price  If stock pays dividend, need to make adjustment

17. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 17 Safe versus risky debt  An application of option logic to capital structure:  When a firm borrows, the lender acquires the company and the shareholders obtain the option to buy it back by paying off the debt  Shhs have thus purchased a call option on the firm  The “strike price” is the amount of debt D that must be repaid

18. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 18 Safe versus risky debt Shareholder value at maturity given $D borrowing (shareholder’s perspective). Firm asset value Shareholder payoff D 0

19. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 19 Safe versus risky debt Lender value at maturity given $D lending to a risky firm (lender’s perspective). Firm asset value Debtholder payoff D 0 D

20. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 20 Option Value Upper Limit Stock Price Lower Limit {Stock price - exercise price, 0} whichever is higher

21. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 21 Option Value Option Price Stock Price Upper limit: share price Lower limit: payoff if exercised immediately ACTUAL VALUE Exercise Price Upper and lower limits to call option value

22. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 22 Option Value Option Price Stock Price ACTUAL VALUE Exercise Price Notice the shape of an unexpired option’s value

23. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 23 Option Value Determinants of Call Option Price 1 - Underlying stock price (+) 2 - Exercise (“strike”) price (-) 3 - Standard deviation of stock returns (+) 4 - Time to option expiration (+) 5 - Interest rate (+)

24. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 24 Why can’t do DCF for options?  Can in principle forecast cash flows  But discount rate is changing over time!  Risk of an option changes every time the stock price moves!  E.g. when price goes up, option payoff becomes more certain, option’s risk & beta go down…  A huge nightmare!

25. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 25 Constructing Option Equivalents  Trick to valuing options is to set up an “equivalent” or “replicating” portfolio that we can already value.  Equivalent portfolio involves both buying a certain fraction of a share (called “option delta” or “hedge ratio”) and borrowing.

26. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 26 Constructing Option Equivalents Intel call option Strike = $85, six months to exercise, 2.5% interest for six months Intel is right now at $85 and can either rise to $106.25 or fall to $68 over next six months (keep it simple) Payoffs to call option are therefore: $0 if price falls $21.25 if price rises Notice this is same payoff structure you would get from an equivalent portfolio that is long 5/9 of one share and borrows $36.86 from the bank! So must have same value.

27. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 27 Constructing Option Equivalents  If stock goes down, 5/9 of share is worth 5/9*68=$37.38 And have to repay $36.86*1.025= -$37.78 Total = $0, just like option  If stock goes up, 5/9 of share is worth 5/9*106.25=$59.03 And have to repay $36.86*1.025= -$37.78 Total = $21.25, just like option  Price of option must be the same as price of equivalent portfolio. Equiv. portf. has a value today of 5/9*(85) -36.86 = $10.36. So option is worth $10.36.

28. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 28 Risk-neutral valuation  Value of that option was $10.36, independent of investor risk attitudes It was based on an arbitrage argument Even risk-averse investors like arbitrages!  Suggests another way to value options Pretend people are risk-neutral Work out expected future value of option in that case Discount it back at the risk-free rate to get value today  The option-equivalent and RN methods are two different ways to implement “the binomial method”

29. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 29 Risk-neutral valuation Intel call option redux Risk-neutral investors would set the expected return on the stock equal to interest rate: 2.5% per six months Know that Intel can either rise 25% or fall 20%. We can calculate “RN probabilities” of a price rise: 2.5%=RNProb(rise)*25%+(1-RNProb(rise))*(-20%)  RNProb(rise)=0.50 Value of call if (rise) is $21.25, if not is $0 Take expected value with Rnprobs and discount at r f  (0.50*21.25+0.50*0)/(1.025) = $10.36 Same answer as replicating portfolio technique!

30. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 30 Black-Scholes V Call = N(d 1 )*P- N(d 2 )*PV(S) Our examples have just been simple up-or-down movements In these cases, the binomial method is perfect In reality, there may be a continuum of outcomes Black-Scholes formula uses a replicating portfolio argument to derive option value under these circumstances

31. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 31 V Call - Call option price N(d 1 ) - Cumulative normal density function at (d 1 ) P - Current stock price N(d 2 ) - Cumulative normal density function at (d 2 ) S - Strike price (take PV using risk-free rate) t - time to maturity of option (as fraction of year) - standard deviation of annual returns Black-Scholes V Call = N(d 1 )*P- N(d 2 )*PV(S)

32. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 32 (d 1 ) = -.3070 N(d 1 ) =.3794 Example What is the price of a call option given the following? P = 36r = 10% =.40 S = 40t = 90 days / 365 (d 2 ) = -.5056 N(d 2 ) =.3065 Black-Scholes

33. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 33 Black-Scholes V Call = N(d 1 )*P - N(d 2 )*S*e -rt = [.3794]*36 - [.3065]*40*e - (.10)(.2466) = $ 1.70 Example What is the price of a call option given the following? P = 36r = 10% =.40 S = 40t = 90 days / 365

34.  Real Options Principles of Corporate Finance Brealey and Myers Sixth Edition Slides by Matthew Will, Jeffrey Wurgler Chapter 21 © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill

35. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 35 Topics Covered  Real Options  Follow-on investments  Abandon  Wait (and learn)  Vary output or production methods  Valuation examples mixed in

36. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 36 Real option value Real option value = Value with option - Value without option

37. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 37 Key questions When is there a real option? - Clearly defined underlying asset whose value changes unpredictably over time - Payoffs to asset are contingent on a decision or event When does the real option have significant value? - Usually when only you can take advantage of it - As barriers to competition fall, options often worth less Can that value be estimated using an option pricing model? - If underlying asset is traded, and exercise price is known - Usually not as precise as DCF

38. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 38 Case 1: Follow-on investments  Option to undertake expansion or follow-on investments if tide turns in future  May want to undertake project that is NPV<0 (before considering option value)

39. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 39 Case 1: Follow-on investments Example: Building Mark I computer gives option to build Mark II computer if platform catches on  NPV of Mark I computer (itself) = - $46 million  But gives option to go ahead with Mark II:  Decision arises 3 years from now  Required investment in Mark II is $900 million  Forecasted cash flows of Mark II are $463 (PV as of today)  Mark II cash flows are uncertain: an annual SD of 35 percent  Annual interest rate is 10%  Proceed with Mark I? How valuable is the follow-on option?

40. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 40 Case 1: Follow-on investments Example: Building Mark I computer gives option to build Mark II computer if platform catches on  Option to invest in Mark II is just a 3-year call option on an asset worth $463 million with a $900 million exercise price!  Black-Scholes call value = +$53.59 million  This makes up for the -$46 NPV of the Mark I on its own  Go ahead with Mark I

41. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 41 Case #2: Option to abandon  Opposite of expansion option (a put not a call)  Can bail out (cut your losses) if things look bad

42. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 42 Case #2: Option to abandon Example: Choice between two production technologies. A is specialized: low unit cost, low salvage value. B is general: high unit cost, decent salvage value.  A has cash flows of 18.5 if high demand, 8.5 if low demand  B has cash flows of 18 if high demand, 8 if low demand.  If can’t ever abandon, want A.  But suppose, one year into project know what demand will be. Can abandon and get 10 out of B (0 for A). If low demand, B is better. What is value of the put option associated with B?

43. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 43 Case #2: Option to abandon Example (A vs. B continued) If can’t be abandoned, suppose B is worth $12 million –If high demand, B value rises 50% to $18 million –If low demand, B value falls 33% to $8 million If can be abandoned, B’s put option is worth $0 if demand is high, $2 million if demand is low Say abandonment possible 1 year from now Say 1 year interest rate is 5% Perfect setup for binomial method – implement with RN

44. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 44 Case #2: Option to abandon Example (A vs. B continued) 5%= RNProb(hi. dem.)*(50%)+ (1-RNProb(hi. dem.))*(-33%)  RNProb(high demand) =.46 Expected put option payoff =.46*0+(1-.46)*2 = $1.08 million Discount at 5%  put value is $1.03 million. In total, B is worth $12 + $1.03 = $13.03 million (Compare this to the NPV of A, which has no option)

45. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 45  What if have decent project (NPV>0 today) but may get even better? Not a now-or-never DCF calculation.  When to pull trigger? What is the value of the option to wait? Case #3: Option to wait

46. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 46  Basic option value principle: More time to expiration, more time to gather information = More value (all else equal) Case #3: Option to wait Option Value Underlying asset value

47. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 47 Example: Build factory today (NPV>0 already) or delay a year? If delay, factory may be more or less valuable, depending on demand.  Tradeoff: Building today gets cash flowing. But waiting may help avoid a costly mistake.  What is value of option to wait? Build today or wait a year? Case #3: Option to wait

48. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 48 Example: Build today or delay for 1 year?  Today: If invest $180 million, PV = $200 million  If low demand, CF 1 =$16 and PV going forward = $160 So return would be (16+160)/(200) = -12%  If high demand, CF 1 =$25 and PV going forward = $250 So return (25+250)/(200) = 37.5%  Suppose riskless rate is 5%.  Another binomial problem. Can solve with RN method Case #3: Option to wait

49. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 49 Example: Build today or delay for 1 year? 5%= RNProb(hi. dem.)*(37.5%)+ (1-RNProb(hi. dem.))*(-12%)  RNProb(high demand) =.343 Expected call option payoff =.343*(250-180) + (1-.343)*0 = $24.01 million Discount at 5%  call value is $22.87 million. So “delay for 1 year” value is $22.87 million vs. “build today” value is $200 - $180 = $20 million Case #3: Option to wait

50. © The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 50 Case #4: Flexible production  Flexible production facilities give option to:  Vary product mix as demand changes Computer-controlled knitting machines  Vary production technology as costs change Utilities with “cofiring equipment” that can use coal or natural gas Auto manufacturers with production facilities in different countries