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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-0 CHAPTER 22 Options and Corporate Finance: Basic Concepts

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-1 Chapter Outline 22.1 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 22.1 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

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-2 22.1 Options Many 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. Many 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-3 22.1 Options Contracts: Preliminaries An 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. An 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-4 22.1 Options Contracts: Preliminaries Exercising 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. Exercising 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-5 Options Contracts: Preliminaries In-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. In-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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-6 Options Contracts: Preliminaries Intrinsic 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. Intrinsic 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 Value Speculative Value +

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-7 22.2 Call Options Call 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. Call 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-8 Basic Call Option Pricing Relationships at Expiry At 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 S T – E. If the call is out-of-the-money, it is worthless: C = Max[S T – E, 0] Where S T is the value of the stock at expiry (time T) E is the exercise price. C is the value of the call option at expiry At 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 S T – E. If the call is out-of-the-money, it is worthless: C = Max[S T – E, 0] Where S T is the value of the stock at expiry (time T) E is the exercise price. C is the value of the call option at expiry

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-9 Call Option Payoffs –20 120 204060 80 100 –40 20 40 60 Stock price ($) Option payoffs ($) Buy a call Exercise price = $50 50

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-10 Call Option Payoffs –20 120 204060 80 100 –40 20 40 60 Stock price ($) Option payoffs ($) Sell a call Exercise price = $50 50

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-11 Call Option Profits Exercise price = $50; option premium = $10 Sell a call Buy a call –20 120 204060 80 100 –40 20 40 60 Stock price ($) Option payoffs ($) 50 –10 10

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-12 22.3 Put Options Put 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. Put 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-13 Basic Put Option Pricing Relationships at Expiry At 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 – S T. If the put is out-of-the-money, it is worthless. P = Max[E – S T, 0] At 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 – S T. If the put is out-of-the-money, it is worthless. P = Max[E – S T, 0]

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-14 Put Option Payoffs –20 0 204060 80 100 –40 20 0 40 60 Stock price ($) Option payoffs ($) Buy a put Exercise price = $50 50

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-15 Put Option Payoffs –20 0204060 80 100 –40 20 0 40 –50 Stock price ($) Option payoffs ($) Sell a put Exercise price = $50 50

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-16 Put Option Profits –20 204060 80 100 –40 20 40 60 Stock price ($) Option payoffs ($) Buy a put Exercise price = $50; option premium = $10 –10 10 Sell a put 50

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-17 22.4 Selling Options Exercise price = $50; option premium = $10 Sell a call Buy a call 50 60 40 100 –40 40 Stock price ($) Option payoffs ($) Buy a put Sell a put The seller (or writer) of an option has an obligation. The purchaser of an option has an option. –10 10 Buy a call Sell a put Buy a put Sell a call

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-18 22.5 Reading The Wall Street Journal

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-19 22.5 Reading The Wall Street Journal This option has a strike price of $135; a recent price for the stock is $138.25 July is the expiration month

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-20 22.5 Reading The Wall Street Journal This 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-21 22.5 Reading The Wall Street Journal On this day, 2,365 call options with this exercise price were traded.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-22 22.5 Reading The Wall Street Journal The 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-23 22.5 Reading The Wall Street Journal On this day, 2,431 put options with this exercise price were traded.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-24 22.5 Reading The Wall Street Journal The 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-25 22.6 Combinations of Options Puts 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. Puts 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-26 Protective Put Strategy: Buy a Put and Buy the Underlying Stock: Payoffs at Expiry Buy a put with an exercise price of $50 Buy the stock Protective Put payoffs $50 $0 $50 Value at expiry Value of stock at expiry

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-27 Protective Put Strategy Profits Buy a put with exercise price of $50 for $10 Buy the stock at $40 $40 Protective Put strategy has downside protection and upside potential $40 $0 -$40 $50 Value at expiry Value of stock at expiry -$10

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-28 Covered Call Strategy Sell a call with exercise price of $50 for $10 Buy the stock at $40 $40 Covered Call strategy $0 -$40 $50 Value at expiry Value of stock at expiry -$30 $10

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-29 Long Straddle: Buy a Call and a Put 30 4060 70 30 40 Stock price ($) Option payoffs ($) Buy a put with exercise price of $50 for $10 Buy a call with exercise price of $50 for $10 A Long Straddle only makes money if the stock price moves $20 away from $50. $50 –20

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-30 Long Straddle: Buy a Call and a Put –30 30 4060 70 –40 Stock price ($) Option payoffs ($) $50 This Short Straddle only loses money if the stock price moves $20 away from $50. Sell a put with exercise price of $50 for $10 Sell a call with an exercise price of $50 for $10 20

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-31 bond Put-Call Parity: p 0 + S 0 = c 0 + E/(1+ r) T 25 Stock price ($) Option payoffs ($) Consider the payoffs from holding a portfolio consisting of a call with a strike price of $25 and a bond with a future value of $25. Call Portfolio payoff Portfolio value today = c 0 + (1+ r) T E

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-32 Put-Call Parity: p 0 + S 0 = c 0 + E/(1+ r) T 25 Stock price ($) Option payoffs ($) Consider the payoffs from holding a portfolio consisting of a share of stock and a put with a $25 strike. Portfolio value today = p 0 + S 0 Portfolio payoff

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-33 Put-Call Parity: p 0 + S 0 = c 0 + E/(1+ r) T Since these portfolios have identical payoffs, they must have the same value today: hence Put-Call Parity: c 0 + E/(1+r) T = p 0 + S 0 25 Stock price ($) Option payoffs ($) 25 Stock price ($) Option payoffs ($) Portfolio value today = p 0 + S 0 Portfolio value today (1+ r) T E = c 0 +

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-34 22.7 Valuing Options The 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-35 Option Value Determinants Call Put 1. Stock price+ – 2. Exercise price– + 3. Interest rate + – 4. Volatility in the stock price+ + 5. Expiration date+ + The value of a call option C 0 must fall within max (S 0 – E, 0) < C 0 < S 0. The precise position will depend on these factors.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-36 Market Value, Time Value and Intrinsic Value for an American Call The value of a call option C 0 must fall within max (S 0 – E, 0) < C 0 < S 0. 25 Option payoffs ($) Call STST loss E Profit STST Time value Intrinsic value Market Value In-the-moneyOut-of-the-money

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-37 22.8 An Option ‑ Pricing Formula We 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-38 Binomial Option Pricing Model Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. S 0 = $25 today and in one year S 1 is 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 = $25×(1 –.15) $28.75 = $25×(1.15) S1S1 S0S0

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-39 Binomial Option Pricing Model 1. A call option on this stock with exercise price of $25 will have the following payoffs. 2. We can replicate the payoffs of the call option. With a levered position in the stock. 1. A call option on this stock with exercise price of $25 will have the following payoffs. 2. We can replicate the payoffs of the call option. With a levered position in the stock. $25 $21.25 $28.75 S1S1 S0S0 C1C1 $3.75 $0

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-40 Binomial Option Pricing Model Borrow 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. Borrow 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 $28.75 S1S1 S0S0 debt – $21.25 portfolio $7.50 $0 ( – ) = = = C1C1 $3.75 $0 – $21.25

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-41 Binomial Option Pricing Model The value today of the levered equity portfolio is today’s value of one share less the present value of a $21.25 debt: $25 $21.25 $28.75 S1S1 S0S0 debt – $21.25 portfolio $7.50 $0 ( – ) = = = C1C1 $3.75 $0 – $21.25

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-42 Binomial Option Pricing Model We can value the call option today as half of the value of the levered equity portfolio: $25 $21.25 $28.75 S1S1 S0S0 debt – $21.25 portfolio $7.50 $0 ( – ) = = = C1C1 $3.75 $0 – $21.25

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-43 If the interest rate is 5%, the call is worth: The Binomial Option Pricing Model $25 $21.25 $28.75 S1S1 S0S0 debt – $21.25 portfolio $7.50 $0 ( – ) = = = C1C1 $3.75 $0 – $21.25 $2.38 C0C0

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-44 the replicating portfolio intuition. Binomial Option Pricing Model 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:

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-45 Delta and the Hedge Ratio This practice of the construction of a riskless hedge is called delta hedging. The delta of a call option is positive. Recall from the example: This practice of the construction of a riskless hedge is called delta hedging. The delta of a call option is positive. Recall from the example: The delta of a put option is negative. Swing of call Swing of stock

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-46 Delta Determining the Amount of Borrowing: Value of a call = Stock price × Delta – Amount borrowed $2.38 = $25 × ½ – Amount borrowed Amount borrowed = $10.12 Determining the Amount of Borrowing: Value of a call = Stock price × Delta – Amount borrowed $2.38 = $25 × ½ – Amount borrowed Amount borrowed = $10.12

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-47 The Risk-Neutral Approach to Valuation We could value V(0) as the value of the replicating portfolio. An equivalent method is risk-neutral valuation S(0), V(0) S(U), V(U) S(D), V(D) q 1- q )1( )()1()( )0( f r DVqUVq V

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-48 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. q 1- 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-49 The Risk-Neutral Approach to Valuation The 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) q 1- q A minor bit of algebra yields: )1( )()1()( )0( f r DSqUSq S )1( )()1()( )0( f r DVqUVq V

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-50 Example of the Risk-Neutral Valuation of a Call: Suppose 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: Suppose 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: $21.25,C(D) q 1- q $25,C(0) $28.75,C(D) )15.1(25$75.28$ )15.1(25$.21$

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-51 Example of the Risk-Neutral Valuation of a Call: $21.25,C(D) 2/3 1/3 The next step would be to compute the risk neutral probabilities $25,C(0) $28.75,C(D) )()( )()0()1( DSUS DSSr q f 32 50.7$ 5$ 25.21$75.28$ 25.21$25$)05.1( q

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-52 Example of the Risk-Neutral Valuation of a Call: $21.25, $0 2/3 1/3 After that, find the value of the call in the up state and down state. $25,C(0) $28.75, $3.75 ]0,75.28$25max[$)( DC25$75.28$)( UC

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-53 Example of the Risk-Neutral Valuation of a Call: Finally, find the value of the call at time 0: $21.25, $0 2/3 1/3 $25,C(0) $28.75,$3.75 $25,$2.38 )1( )()1()( )0( f r DCqUCq C )05.1( 0$)31(75.3$32 )0( C 38.2$ )05.1( 50.2$ )0( C

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-54 This risk-neutral result is consistent with valuing the call using a replicating portfolio. Risk-Neutral Valuation and the Replicating Portfolio

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-55 The Black-Scholes Model The Black-Scholes Model is Where C 0 = the value of a European option at time t = 0 r = 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. )N() 210 dEedSC rT T T σ rES d ) 2 ()/ln( 2 1 Tdd 12

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-56 The Black-Scholes Model Find 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. Find 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-57 The Black-Scholes Model Let’s try our hand at using the model. If you have a calculator handy, follow along. Then, First calculate d 1 and d 2 T TσrES d )5.()/ln( 2 1 5282.0 5.30.0 5).)30.0(5.05(.)150/160ln( 2 1 d 31602.05.30.052815.0 12 Tdd

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-58 The Black-Scholes Model N(d 1 ) = N(0.52815) = 0.7013 N(d 2 ) = N(0.31602) = 0.62401 )N() 210 dEedSC rT 5282.0 1 d31602.0 2 d 92.20$ 62401.01507013.0160$ 0 5.05. 0 C eC

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-59 Assume S = $50, X = $45, T = 6 months, r = 10%, and = 28%, calculate the value of a call and a put. From a standard normal probability table, look up N(d 1 ) = 0.812 and N(d 2 ) = 0.754 (or use Excel’s “normsdist” function) Another Black-Scholes Example 884.0 50.028.0 50.0 2 28.0 010.0 45 50 ln 2 1 d 686.050.028.0884.0 2 d 32.8$)754.0(45)812.0(50 ).0(10.0)5.0(0 eeC125.1$45$50$32.8$ )50.0(10.0 eP

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-60 22.9 Stocks and Bonds as Options Levered 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. Levered 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-61 22.9 Stocks and Bonds as Options Levered 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. Levered 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-62 22.9 Stocks and Bonds as Options It all comes down to put-call parity. Value of a call on the firm Value of a put on the firm Value of a risk-free bond Value of the firm = + – Stockholder’s position in terms of call options Stockholder’s position in terms of put options c 0 = S 0 + p 0 – (1+ r) T E

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-63 22.10 Capital-Structure Policy and Options Recall 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. Recall 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.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-64 Balance Sheet for a Company in Distress AssetsBVMVLiabilitiesBVMV Cash$200$200LT bonds$300 Fixed Asset$400$0Equity$300 Total$600$200Total$600$200 What happens if the firm is liquidated today? AssetsBVMVLiabilitiesBVMV Cash$200$200LT bonds$300 Fixed Asset$400$0Equity$300 Total$600$200Total$600$200 What happens if the firm is liquidated today? The bondholders get $200; the shareholders get nothing. $200 $0

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-65 Selfish Strategy 1: Take Large Risks The GambleProbabilityPayoff Win Big10%$1,000 Lose Big90%$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 The GambleProbabilityPayoff Win Big10%$1,000 Lose Big90%$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 NPV = –$200 + $100 (1.10) NPV = –$133

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-66 Selfish Stockholders Accept Negative NPV Project with Large Risks Expected CF 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 Expected CF 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 $20 = $30 (1.50) PV of Bonds With the Gamble: $47 = $70 (1.50) PV of Stocks With the Gamble: The 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 of the firm is increased.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-67 22.11 Mergers and Options This is an area rich with optionality, both in the structuring of the deals and in their execution. In the first half of 2000, General Mills was attempting to acquire the Pillsbury division of Diageo PLC. The structure of the deal was Diageo’s stockholders received 141 million shares of General Mills stock (then valued at $42.55) plus contingent value rights of $4.55 per share. This is an area rich with optionality, both in the structuring of the deals and in their execution. In the first half of 2000, General Mills was attempting to acquire the Pillsbury division of Diageo PLC. The structure of the deal was Diageo’s stockholders received 141 million shares of General Mills stock (then valued at $42.55) plus contingent value rights of $4.55 per share.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-68 22.11 Mergers and Options Cash payment to newly issued shares $0 Value of General Mills in 1 year $42.55 $38 $4.55 The contingent value rights paid the difference between $42.55 and General Mills’ stock price in one year up to a maximum of $4.55.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-69 22.11 Mergers and Options The contingent value plan can be viewed in terms of puts: Each newly issued share of General Mills given to Diageo’s shareholders came with a put option with an exercise price of $42.55. But the shareholders of Diageo sold a put with an exercise price of $38 The contingent value plan can be viewed in terms of puts: Each newly issued share of General Mills given to Diageo’s shareholders came with a put option with an exercise price of $42.55. But the shareholders of Diageo sold a put with an exercise price of $38

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-70 22.11 Mergers and Options $38 $0 Value of General Mills in 1 year $42.55 –$38 Own a put Strike $42.55 Sell a put Strike $38 – $38.00 $4.55 $42.55 Cash payment to newly issued shares

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-71 22.11 Mergers and Options Value of a share $38 $4.55 $0 $42.55 Value of General Mills in 1 year Value of a share plus cash payment $42.55

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-72 22.12 Investment in Real Projects & Options Classic NPV calculations typically ignore the flexibility that real-world firms typically have. The next chapter will take up this point. Classic NPV calculations typically ignore the flexibility that real-world firms typically have. The next chapter will take up this point.

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-73 22.13 Summary and Conclusions The 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 The 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 c0–c0– (1+ r) T E = S 0 + p 0

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McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 22-74 22.13 Summary and Conclusions The 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. 1. Common stock in a levered firm can be viewed as a call option on the assets of the firm. 2. Real projects often have hidden option that enhance value. The 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. 1. Common stock in a levered firm can be viewed as a call option on the assets of the firm. 2. Real projects often have hidden option that enhance value.

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