第二十二章 期权与公司理财：基本概念

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1. Chapter Outline22.1 Options 22.2 Call Options 22.3 Put Options 22.4 Selling Options 22.5 Reading The Wall Street Journal 22.6 Combinations of Options 22.7 Valuing Options 22.8 An Option‑Pricing Formula 22.9 Stocks and Bonds as Options 22.10 Capital-Structure Policy and Options 22.11 Mergers and Options 22.12 Investment in Real Projects and Options 22.13 Summary and Conclusions
2. 22.1 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 decisions can be viewed in terms of options.
3. 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 Puts Call options gives the holder the 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 the future, at prices agreed upon today. When exercising a put, you “put” the asset to someone.
4. 22.1 Options Contracts: PreliminariesExercising the Option The act of buying or selling the underlying asset through the option contract. Strike Price or Exercise Price Refers to the fixed price in the option contract at which the holder can buy or sell the underlying asset. Expiry The maturity date of the option is referred to as the expiration date, or the expiry. European versus American options European options can be exercised only at expiry. American options can be exercised at any time up to expiry.
5. Options Contracts: PreliminariesIn-the-Money The exercise price is less than the spot price of the underlying asset. At-the-Money The exercise price is equal to the spot price of the underlying asset. Out-of-the-Money The exercise price is more than the spot price of the underlying asset.
6. Options Contracts: PreliminariesIntrinsic Value The difference between the exercise price of the option and the spot price of the underlying asset. Speculative Value The difference between the option premium and the intrinsic value of the option.Option Premium=Intrinsic ValueSpeculative Value+
7. 22.2 Call OptionsCall options 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.
8. Basic Call Option Pricing Relationships at ExpiryAt expiry, an American call option is 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 = Max[ST - E, 0] Where ST is the value of the stock at expiry (time T) E is the exercise price. CaT is the value of an American call at expiry CeT is the value of a European call at expiry
9. Call Option Payoffs-201009080706001020304050-40200-604060Stock price ($)Option payoffs ($)Buy a callExercise price = $50
10. Call Option Payoffs-201009080706001020304050-40200-604060Stock price ($)Option payoffs ($)Write a callExercise price = $50
11. Call Option Profits-201009080706001020304050-40200-604060Stock price ($)Option profits ($)Write a callBuy a callExercise price = $50; option premium = $10
12. 22.3 Put OptionsPut options gives the holder the right, but not the obligation, to sell a given quantity of an asset on or before some time in the future, at prices agreed upon today. When exercising a put, you “put” the asset to someone.
13. Basic Put Option Pricing Relationships at ExpiryAt expiry, an American put option is worth the same as a European option with the same characteristics. If the put is in-the-money, it is worth E - ST. If the put is out-of-the-money, it is worthless. PaT = PeT = Max[E - ST, 0]
14. Put Option Payoffs-201009080706001020304050-40200-604060Stock price ($)Option payoffs ($)Buy a putExercise price = $50
15. Put Option Payoffs-201009080706001020304050-40200-604060Option payoffs ($)write a putExercise price = $50Stock price ($)
16. Put Option Profits-201009080706001020304050-40200-604060Stock price ($)Option profits ($)Buy a putWrite a putExercise price = $50; option premium = $1010-10
17. 22.4 Selling OptionsThe seller (or writer) of an option has an obligation.The purchaser of an option has an option.-201009080706001020304050-40200-604060Stock price ($)Option profits ($)Buy a putWrite a put10-10-201009080706001020304050-40200-604060Stock price ($)Option profits ($)Write a callBuy a call
18. 22.5 Reading The Wall Street Journal
19. 22.5 Reading The Wall Street JournalThis option has a strike price of $135; a recent price for the stock is $138.25 July is the expiration month
20. 22.5 Reading The Wall Street JournalThis makes a call option with this exercise price in-the-money by $3.25 = $138¼ – $135. Puts with this exercise price are out-of-the-money.
21. 22.5 Reading The Wall Street JournalOn this day, 2,365 call options with this exercise price were traded.
22. 22.5 Reading The Wall Street JournalThe CALL option with a strike price of $135 is trading for $4.75.Since the option is on 100 shares of stock, buying this option would cost $475 plus commissions.
23. 22.5 Reading The Wall Street JournalOn this day, 2,431 put options with this exercise price were traded.
24. 22.5 Reading The Wall Street JournalThe PUT option with a strike price of $135 is trading for $.8125.Since the option is on 100 shares of stock, buying this option would cost $81.25 plus commissions.
25. 22.6 Combinations of OptionsPuts and calls can serve as the building blocks for more complex option contracts. If you understand this, you can become a financial engineer, tailoring the risk-return profile to meet your client’s needs.
26. 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 protection and upside potential$50$0$50Value at expiryValue of stock at expiry
27. Protective 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 expiry
28. Covered 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 expiry
29. Long 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 $20 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 expiry
30. Short 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-$30$50Sell a call with an exercise price of $50 for $10$10$20$60$30$70Value of stock at expiryValue at expiry
31. Long 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 expiry
32. Put-Call 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-P0]In market equilibrium, it mast be the case that option prices are set such that:Otherwise, riskless portfolios with positive payoffs exist.Value of stock at expiryValue at expiry
33. 22.7 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.
34. Option Value Determinants Call Put 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 < S0. The precise position will depend on these factors.
35. Market Value, Time Value and Intrinsic Value for an American CallCaT > Max[ST - E, 0]ProfitlossESTMarket ValueIntrinsic valueST - ETime valueOut-of-the-moneyIn-the-moneySTThe value of a call option C0 must fall within max (S0 – E, 0) < C0 < S0.
36. 22.8 An Option‑Pricing 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.
37. Binomial Option Pricing ModelSuppose a stock is worth $25 today and 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$28.75S1S0
38. Binomial Option Pricing ModelA call option on this stock with exercise price of $25 will 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$0
39. Binomial 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 either $7.50 or $0. The levered equity portfolio has twice the option’s 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- $21.25
40. Binomial Option Pricing Model The levered equity portfolio value today is today’s value of one share less the present value of a $21.25 debt:$25$21.25$28.75S1S0debt- $21.25portfolio$7.50$0( - ) ===C1$3.75$0- $21.25
41. Binomial 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- $21.25
42. If 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
43. If 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
44. 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 is:
45. The Risk-Neutral Approach to Valuation We 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- q
46. The Risk-Neutral Approach to Valuation S(0) is the value of the underlying asset today.S(0), V(0)S(U), V(U)S(D), V(D) 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- q V(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.
47. The Risk-Neutral Approach 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:
48. Example of the Risk-Neutral Valuation of a Call:$21.25,C(D)q1- qSuppose a stock is worth $25 today and in 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)
49. 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 neutral probabilities$25,C(0)$28.75,C(D)
50. 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, $3.75
51. Example of the Risk-Neutral Valuation of a Call:Finally, find the value of the call at time 0: $21.25, $02/31/3$25,C(0)$28.75,$3.75$25,$2.38
52. This risk-neutral result is consistent with valuing the call using a replicating portfolio.Risk-Neutral Valuation and the Replicating Portfolio
53. The Black-Scholes ModelThe Black-Scholes Model isWhere C0 = the value of a European option at time t = 0r = the risk-free interest 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.
54. The Black-Scholes ModelFind the value of a six-month call option on the Microsoft with an exercise price of $150 The current value of a share of Microsoft is $160 The 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 option is $10—our answer must be at least that amount.
55. The Black-Scholes ModelLet’s try our hand at using the model. If you have a calculator handy, follow along.Then, First calculate d1 and d2
56. The Black-Scholes ModelN(d1) = N(0.52815) = 0.7013 N(d2) = N(0.31602) = 0.62401
57. Assume S = $50, X = $45, T = 6 months, 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 Excel’s “normsdist” function)Another Black-Scholes Example
58. 22.9 Stocks and Bonds as OptionsLevered Equity is a Call Option. The underlying asset comprise 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 the 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.
59. 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 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 option (i.e. NOT declare bankruptcy) and let the put expire.
60. 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=+–Stockholder’s position in terms of call optionsStockholder’s position in terms of put options
61. 22.10 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.
62. Balance Sheet for a Company in DistressAssets BV MV Liabilities BV MV Cash $200 $200 LT bonds $300 ? Fixed Asset $400 $0 Equity $300 ? Total $600 $200 Total $600 $200 What happens if the firm is liquidated today?The bondholders get $200; the shareholders get nothing.
63. Selfish Strategy 1: Take Large Risks (Think of a Call Option)The Gamble Probability Payoff Win Big 10% $1,000 Lose Big 90% $0 Cost of investment is $200 (all the firm’s cash) Required return is 50% Expected CF from the Gamble = $1000 × 0.10 + $0 = $100
64. Selfish Stockholders Accept Negative NPV Project with Large RisksExpected cash flow from the Gamble To Bondholders = $300 × 0.10 + $0 = $30 To Stockholders = ($1000 - $300) × 0.10 + $0 = $70 PV of Bonds Without the Gamble = $200 PV of Stocks Without the Gamble = $0 PV of Bonds With the Gamble = $30 / 1.5 = $20 PV of Stocks With the Gamble = $70 / 1.5 = $47The stocks are worth more with the high risk project because the call option that the shareholders of the levered firm hold is worth more when the volatility is increased.
65. 22.11 Mergers and OptionsThis is an area rich with optionality, both in the structuring of the deals and in their execution.
66. 22.12 Investment in Real Projects & OptionsClassic NPV calculations typically ignore the flexibility that real-world firms typically have. The next chapter will take up this point.
67. 22.13 Summary and ConclusionsThe most familiar options are puts and calls. Put options give the holder the right to sell stock at a set price for a given amount of time. Call options give the holder the right to buy stock at a set price for a given amount of time. Put-Call parity
68. 22.13 Summary and ConclusionsThe value of a stock option depends on six factors: 1. Current price of underlying stock. 2. Dividend yield of the underlying stock. 3. Strike price specified in the option contract. 4. Risk-free interest rate over the life of the contract. 5. Time remaining until the option contract expires. 6. Price volatility of the underlying stock. Much of corporate financial theory can be presented in terms of options. Common stock in a levered firm can be viewed as a call option on the assets of the firm. Real projects often have hidden option that enhance value.