1、Chapter Outline22.1 Options22.2 Call Options22.3 Put Options22.4 Selling Options22.5 Reading The Wall Street Journal22.6 Combinations of Options22.7 Valuing Options22.8 An OptionPricing Formula22.9 Stocks and Bonds as Options22.10 Capital-Structure Policy and Options22.11 Mergers and Options22.12 In

2、vestment in Real Projects and Options22.13 Summary and Conclusions22.1 OptionsMany corporate securities are similar to the stock options that are traded on organized exchanges. Almost every issue of corporate stocks and bonds has option features.In addition, capital structure and capital budgeting d

3、ecisions can be viewed in terms of options.22.1 Options Contracts: PreliminariesAn option gives the holder the right, but not the obligation, to buy or sell a given quantity of an asset on (or perhaps before) a given date, at prices agreed upon today.Calls versus PutsCall options gives the holder th

4、e right, but not the obligation, to buy a given quantity of some asset at some time in the future, at prices agreed upon today. When exercising a call option, you “call in the asset.Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset at some time in t

5、he future, at prices agreed upon today. When exercising a put, you “put the asset to someone.22.1 Options Contracts: PreliminariesExercising the OptionThe act of buying or selling the underlying asset through the option contract.Strike Price or Exercise PriceRefers to the fixed price in the option c

6、ontract at which the holder can buy or sell the underlying asset.ExpiryThe maturity date of the option is referred to as the expiration date, or the expiry. European versus American optionsEuropean options can be exercised only at expiry.American options can be exercised at any time up to expiry.Opt

7、ions Contracts: PreliminariesIn-the-MoneyThe exercise price is less than the spot price of the underlying asset.At-the-MoneyThe exercise price is equal to the spot price of the underlying asset.Out-of-the-MoneyThe exercise price is more than the spot price of the underlying asset.Options Contracts:

8、PreliminariesIntrinsic ValueThe difference between the exercise price of the option and the spot price of the underlying asset.Speculative ValueThe difference between the option premium and the intrinsic value of the option.Option Premium=Intrinsic ValueSpeculative Value+22.2 Call OptionsCall option

9、s gives the holder the right, but not the obligation, to buy a given quantity of some asset on or before some time in the future, at prices agreed upon today. When exercising a call option, you “call in the asset.Basic Call Option Pricing Relationships at ExpiryAt expiry, an American call option is

10、worth the same as a European option with the same characteristics.If the call is in-the-money, it is worth ST - E.If the call is out-of-the-money, it is worthless.CaT = CeT = MaxST - E, 0WhereST is the value of the stock at expiry (time T)E is the exercise price.CaT is the value of an American call

11、at expiryCeT is the value of a European call at expiryCall Option Payoffs-201009080706001020304050-40200-604060Stock price ($)Option payoffs ($)Buy a callExercise price = $50Call Option Payoffs-201009080706001020304050-40200-604060Stock price ($)Option payoffs ($)Write a callExercise price = $50Call

12、 Option Profits-201009080706001020304050-40200-604060Stock price ($)Option profits ($)Write a callBuy a callExercise price = $50; option premium = $1022.3 Put OptionsPut options gives the holder the right, but not the obligation, to sell a given quantity of an asset on or before some time in the fut

13、ure, at prices agreed upon today. When exercising a put, you “put the asset to someone.Basic Put Option Pricing Relationships at ExpiryAt expiry, an American put option is worth the same as a European option with the same characteristics.If the put is in-the-money, it is worth E - ST.If the put is o

14、ut-of-the-money, it is worthless.PaT = PeT = MaxE - ST, 0Put Option Payoffs-201009080706001020304050-40200-604060Stock price ($)Option payoffs ($)Buy a putExercise price = $50Put Option Payoffs-201009080706001020304050-40200-604060Option payoffs ($)write a putExercise price = $50Stock price ($)Put O

15、ption Profits-201009080706001020304050-40200-604060Stock price ($)Option profits ($)Buy a putWrite a putExercise price = $50; option premium = $1010-1022.4 Selling OptionsThe seller (or writer) of an option has an obligation.The purchaser of an option has an option.-201009080706001020304050-40200-60

16、4060Stock price ($)Option profits ($)Buy a putWrite a put10-10-201009080706001020304050-40200-604060Stock price ($)Option profits ($)Write a callBuy a call22.5 Reading The Wall Street Journal22.5 Reading The Wall Street JournalThis option has a strike price of $; a recent price for the stock is $.25

17、 July is the expiration month22.5 Reading The Wall Street JournalThis makes a call option with this exercise price in-the-money by $3.25 = $ $. Puts with this exercise price are out-of-the-money.22.5 Reading The Wall Street JournalOn this day, 2,365 call options with this exercise price were traded.

18、22.5 Reading The Wall Street JournalThe CALL option with a strike price of $ is trading for $4.75.Since the option is on 100 shares of stock, buying this option would cost $475 plus commissions.22.5 Reading The Wall Street JournalOn this day, 2,431 put options with this exercise price were traded.22

19、.5 Reading The Wall Street JournalThe PUT option with a strike price of $ is trading for $.8125.Since the option is on 100 shares of stock, buying this option would cost $81.25 plus commissions.22.6 Combinations of OptionsPuts and calls can serve as the building blocks for more complex option contra

20、cts.If you understand this, you can become a financial engineer, tailoring the risk-return profile to meet your clients needs.Protective Put Strategy: Buy a Put and Buy the Underlying Stock: Payoffs at ExpiryBuy a put with an exercise price of $50Buy the stockProtective Put strategy has downside pro

21、tection and upside potential$50$0$50Value at expiryValue of stock at expiryProtective Put Strategy ProfitsBuy a put with exercise price of $50 for $10Buy the stock at $40$40Protective Put strategy has downside protection and upside potential$40$0-$40$50Value at expiryValue of stock at expiryCovered

22、Call StrategySell a call with exercise price of $50 for $10Buy the stock at $40$40Covered call$40$0-$40$10-$30$30$50Value of stock at expiryValue at expiryLong Straddle: Buy a Call and a PutBuy a put with an exercise price of $50 for $10$40A Long Straddle only makes money if the stock price moves $2

23、0 away from $50.$40$0-$20$50Buy a call with an exercise price of $50 for $10-$10$30$60$30$70Value of stock at expiryValue at expiryShort Straddle: Sell a Call and a PutSell a put with exercise price of$50 for $10$40A Short Straddle only loses money if the stock price moves $20 away from $50.-$40$0-$

24、30$50Sell a call with an exercise price of $50 for $10$10$20$60$30$70Value of stock at expiryValue at expiryLong Call SpreadSell a call with exercise price of $55 for $5$55long call spread$5$0$50Buy a call with an exercise price of $50 for $10-$10-$5$60Value of stock at expiryValue at expiryPut-Call

25、 ParitySell a put with an exercise price of $40Buy the stock at $40 financed with some debt: FV = $XBuy a call option with an exercise price of $40$0-$40$40-P0$40Buy the stock at $40-$40-P0In market equilibrium, it mast be the case that option prices are set such that:Otherwise, riskless portfolios

26、with positive payoffs exist.Value of stock at expiryValue at expiry22.7 Valuing OptionsThe last section concerned itself with the value of an option at expiry.This section considers the value of an option prior to the expiration date.A much more interesting question.Option Value DeterminantsCall Put

27、Stock price+ Exercise price +Interest rate + Volatility in the stock price+ +Expiration date+ +The value of a call option C0 must fall within max (S0 E, 0) C0 MaxST - E, 0ProfitlossESTMarket ValueIntrinsic valueST - ETime valueOut-of-the-moneyIn-the-moneySTThe value of a call option C0 must fall wit

28、hin max (S0 E, 0) C0 S0.22.8 An OptionPricing FormulaWe will start with a binomial option pricing formula to build our intuition.Then we will graduate to the normal approximation to the binomial for some real-world option valuation.Binomial Option Pricing ModelSuppose a stock is worth $25 today and

29、in one period will either be worth 15% more or 15% less. S0= $25 today and in one year S1is either $28.75 or $21.25. The risk-free rate is 5%. What is the value of an at-the-money call option? $25$21.25$28.75S1S0Binomial Option Pricing ModelA call option on this stock with exercise price of $25 will

30、 have the following payoffs. We can replicate the payoffs of the call option. With a levered position in the stock.$25$21.25$28.75S1S0C1$3.75$0Binomial Option Pricing ModelBorrow the present value of $21.25 today and buy 1 share. The net payoff for this levered equity portfolio in one period is eith

31、er $7.50 or $0. The levered equity portfolio has twice the options payoff so the portfolio is worth twice the call option value.$25$21.25$28.75S1S0debt- $21.25portfolio$7.50$0( - ) =C1$3.75$0- $21.25Binomial Option Pricing Model The levered equity portfolio value today is todays value of one share l

32、ess the present value of a $21.25 debt:$25$21.25$28.75S1S0debt- $21.25portfolio$7.50$0( - ) =C1$3.75$0- $21.25Binomial Option Pricing ModelWe can value the option today as half of the value of the levered equity portfolio:$25$21.25$28.75S1S0debt- $21.25portfolio$7.50$0( - ) =C1$3.75$0- $21.25If the

33、interest rate is 5%, the call is worth:The Binomial Option Pricing Model$25$21.25$28.75S1S0debt- $21.25portfolio$7.50$0( - ) =C1$3.75$0- $21.25If the interest rate is 5%, the call is worth:The Binomial Option Pricing Model$25$21.25$28.75S1S0debt- $21.25portfolio$7.50$0( - ) =C1$3.75$0- $21.25$2.38C0

34、Binomial Option Pricing Modelthe replicating portfolio intuition.Many derivative securities can be valued by valuing portfolios of primitive securities when those portfolios have the same payoffs as the derivative securities.The most important lesson (so far) from the binomial option pricing model i

35、s:The Risk-Neutral Approach to ValuationWe could value V(0) as the value of the replicating portfolio. An equivalent method is risk-neutral valuationS(0), V(0)S(U), V(U)S(D), V(D)q1- qThe Risk-Neutral Approach to ValuationS(0) is the value of the underlying asset today.S(0), V(0)S(U), V(U)S(D), V(D)

36、S(U) and S(D) are the values of the asset in the next period following an up move and a down move, respectively.q1- qV(U) and V(D) are the values of the asset in the next period following an up move and a down move, respectively.q is the risk-neutral probability of an “up move.The Risk-Neutral Appro

37、ach to ValuationThe key to finding q is to note that it is already impounded into an observable security price: the value of S(0):S(0), V(0)S(U), V(U)S(D), V(D)q1- qA minor bit of algebra yields:Example of the Risk-Neutral Valuation of a Call:$21.25,C(D)q1- qSuppose a stock is worth $25 today and in

38、 one period will either be worth 15% more or 15% less. The risk-free rate is 5%. What is the value of an at-the-money call option?The binomial tree would look like this:$25,C(0)$28.75,C(D)Example of the Risk-Neutral Valuation of a Call:$21.25,C(D)2/31/3 The next step would be to compute the risk neu

39、tral probabilities$25,C(0)$28.75,C(D)Example of the Risk-Neutral Valuation of a Call:$21.25, $02/31/3After that, find the value of the call in the up state and down state.$25,C(0)$28.75, $3.75Example of the Risk-Neutral Valuation of a Call:Finally, find the value of the call at time 0:$21.25, $02/31

40、/3$25,C(0)$28.75,$3.75$25,$2.38This risk-neutral result is consistent with valuing the call using a replicating portfolio.Risk-Neutral Valuation and the Replicating PortfolioThe Black-Scholes ModelThe Black-Scholes Model isWhereC0 = the value of a European option at time t = 0r = the risk-free inter

41、est rate.N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d.The Black-Scholes Model allows us to value options in the real world just as we have done in the 2-state world.The Black-Scholes ModelFind the value of a six-month call option on th

42、e Microsoft with an exercise price of $150The current value of a share of Microsoft is $160The interest rate available in the U.S. is r = 5%.The option maturity is 6 months (half of a year).The volatility of the underlying asset is 30% per annum.Before we start, note that the intrinsic value of the

43、option is $10our answer must be at least that amount.The Black-Scholes ModelLets try our hand at using the model. If you have a calculator handy, follow along.Then, First calculate d1 and d2The Black-Scholes ModelN(d1) = N(0.52815) = 0.7013N(d2) = N(0.31602) = 0.62401Assume S = $50, X = $45, T = 6 m

44、onths, r = 10%, and = 28%, calculate the value of a call and a put.From a standard normal probability table, look up N(d1) = 0.812 and N(d2) = 0.754 (or use Excels “normsdist function)Another Black-Scholes Example22.9 Stocks and Bonds as OptionsLevered Equity is a Call Option.The underlying asset co

45、mprise the assets of the firm.The strike price is the payoff of the bond.If at the maturity of their debt, the assets of the firm are greater in value than the debt, the shareholders have an in-the-money call, they will pay the bondholders and “call in the assets of the firm.If at the maturity of th

46、e debt the shareholders have an out-of-the-money call, they will not pay the bondholders (i.e. the shareholders will declare bankruptcy) and let the call expire.22.9 Stocks and Bonds as OptionsLevered Equity is a Put Option.The underlying asset comprise the assets of the firm.The strike price is the

47、 payoff of the bond.If at the maturity of their debt, the assets of the firm are less in value than the debt, shareholders have an in-the-money put.They will put the firm to the bondholders.If at the maturity of the debt the shareholders have an out-of-the-money put, they will not exercise the optio

48、n (i.e. NOT declare bankruptcy) and let the put expire.22.9 Stocks and Bonds as OptionsIt all comes down to put-call parity.Value of a call on the firmValue of a put on the firmValue of a risk-free bondValue of the firm=+Stockholders position in terms of call optionsStockholders position in terms of

49、 put options22.10 Capital-Structure Policy and OptionsRecall some of the agency costs of debt: they can all be seen in terms of options.For example, recall the incentive shareholders in a levered firm have to take large risks.Balance Sheet for a Company in DistressAssetsBVMVLiabilitiesBVMVCash$200$2

50、00LT bonds$300?Fixed Asset$400$0Equity$300?Total$600$200Total$600$200What happens if the firm is liquidated today?The bondholders get $200; the shareholders get nothing.Selfish Strategy 1: Take Large Risks (Think of a Call Option)The GambleProbabilityPayoffWin Big10%$1,000Lose Big90%$0Cost of investment is $200 (all the firms cash)Required return is 50%Expected CF from the Gamble = $1000 0.10 + $0 = $100