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© 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
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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
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© 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
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© 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.
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© 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
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© 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
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© 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
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© 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
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© 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
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© The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 10 Option Obligations
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© 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
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© 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
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© 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
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© 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
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© 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
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© 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
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© 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
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© 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
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© 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
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© 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
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© 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
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© 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
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© 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 (+)
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© 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!
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© 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.
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© 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.
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© 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.
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© 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”
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© 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!
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© 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
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© 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)
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© 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
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© 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
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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
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© 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
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© The McGraw-Hill Companies, Inc., 2000 Irwin/McGraw Hill 20- 36 Real option value Real option value = Value with option - Value without option
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© 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
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© 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)
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© 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?
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© 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
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© 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
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© 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?
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© 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
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© 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)
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© 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
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© 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
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© 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
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© 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
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© 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
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© 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
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