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B40.2302 Class #4

B40.2302 Class #4. BM6 chapters 20, 21 Based on slides created by Matthew Will Modified 8/14/2014 by Jeffrey Wurgler. Principles of Corporate Finance Brealey and Myers Sixth Edition. Spotting and Valuing Options. Slides by Matthew Will, Jeffrey Wurgler. Chapter 20.

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B40.2302 Class #4

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  1. B40.2302 Class #4 • BM6 chapters 20, 21 • Based on slides created by Matthew Will • Modified 8/14/2014 by Jeffrey Wurgler

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

  3. Topics Covered • Calls, Puts and Shares • Financial Alchemy with Options • Option Valuation • Constructing equivalent portfolios • Risk-neutral valuation • Black-Scholes

  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. 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. Option Value Payoff on a riskless bond/loan at maturity … is fixed (lender’s perspective). Bond value 0 Share Price

  7. Option Value Payoff to a share when you want to sell it … depends on share price (share buyer’s perspective). 50 Share value 0 50 Share Price

  8. Option Value Call option value at expiration given a $85 exercise price (call buyer’s perspective). Call option value $20 0 85 105 Share Price

  9. Option Value Put option value at expiration given a $85 exercise price (put buyer’s perspective). Put option value $5 0 80 85 Share Price

  10. Option Obligations

  11. Option Value Call option value at expiration given a $85 exercise price (call seller’s perspective). 0 Call option $ payoff 85 Share Price

  12. Option Value Put option value at expiration given a $85 exercise price (put seller’s perspective). 0 Put option $ payoff 85 Share Price

  13. Financial Alchemy Protective Put = Buy stock and buy put Long Stock “Protective Put” Position Value Long Put Share Price

  14. Financial Alchemy Straddle = Long call and long put - Profits from high volatility Straddle Position Value Share Price

  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. 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. 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. Safe versus risky debt Shareholder value at maturity given $D borrowing (shareholder’s perspective). Shareholder payoff 0 D Firm asset value

  19. Safe versus risky debt Lender value at maturity given $D lending to a risky firm (lender’s perspective). D Debtholder payoff 0 D Firm asset value

  20. Option Value Stock Price Upper Limit Lower Limit {Stock price - exercise price, 0} whichever is higher

  21. Option Value Upper and lower limits to call option value Upper limit: share price Option Price Lower limit: payoff if exercised immediately ACTUAL VALUE Stock Price Exercise Price

  22. Option Value Notice the shape of an unexpired option’s value Option Price ACTUAL VALUE Stock Price Exercise Price

  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. 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. 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. 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 payoffstructure 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. 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. 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. 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 rf (0.50*21.25+0.50*0)/(1.025) = $10.36 • Same answer as replicating portfolio technique!

  30. Black-Scholes • 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 VCall = N(d1)*P- N(d2)*PV(S)

  31. Black-Scholes VCall = N(d1)*P- N(d2)*PV(S) VCall - Call option price N(d1) - Cumulative normal density function at (d1) P - Current stock price N(d2) - Cumulative normal density function at (d2) S - Strike price (take PV using risk-free rate) t - time to maturity of option (as fraction of year) - standard deviation of annual returns

  32. Black-Scholes Example What is the price of a call option given the following? P = 36 r = 10% = .40 S = 40 t = 90 days / 365 (d1) = - .3070 N(d1) = .3794 (d2) = - .5056 N(d2) = .3065

  33. Black-Scholes Example What is the price of a call option given the following? P = 36 r = 10% = .40 S = 40 t = 90 days / 365 VCall = N(d1)*P - N(d2)*S*e-rt = [.3794]*36 - [.3065]*40*e - (.10)(.2466) = $ 1.70

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

  35. Topics Covered • Real Options • Follow-on investments • Abandon • Wait (and learn) • Vary output or production methods • Valuation examples mixed in

  36. Real option value Real option value = Value with option - Value without option

  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. 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. 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. 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. 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. 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. 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. 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. Case #3: Option to wait • 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?

  46. Case #3: Option to wait • Basic option value principle: More time to expiration, more time to gather information = More value (all else equal) Option Value Underlying asset value

  47. Case #3: Option to wait 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?

  48. Case #3: Option to wait Example: Build today or delay for 1 year? • Today: If invest $180 million, PV = $200 million • If low demand, CF1 =$16 and PV going forward = $160 • So return would be (16+160)/(200) = -12% • If high demand, CF1 =$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

  49. Case #3: Option to wait 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

  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

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