Options and Corporate Finance: Basic Concepts CHAPTER 22 Options and Corporate Finance: Basic Concepts

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 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.

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.

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.

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.

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. Option Premium = Intrinsic Value Speculative Value +

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.

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 ST – E. If the call is out-of-the-money, it is worthless: C = Max[ST – E, 0] Where ST 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

Call Option Payoffs Exercise price = $50 Buy a call 60 40 20 20 40 60 80 100 120 50 Stock price ($) –20 Exercise price = $50 –40

Call Option Payoffs Exercise price = $50 Sell a call 60 40 20 20 40 60 80 100 120 50 Stock price ($) –20 Exercise price = $50 Sell a call –40

Call Option Profits Exercise price = $50; option premium = $10 –20 120 20 40 60 80 100 –40 Stock price ($) Option payoffs ($) Buy a call 10 50 –10 Exercise price = $50; option premium = $10 Sell a call

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.

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 – ST. If the put is out-of-the-money, it is worthless. P = Max[E – ST, 0]

Put Option Payoffs Exercise price = $50 Buy a put 60 50 40 20 Buy a put 20 40 60 80 100 50 Stock price ($) –20 Exercise price = $50 –40

Put Option Payoffs Exercise price = $50 Sell a put 40 20 Sell a put 20 40 60 80 100 50 Stock price ($) –20 Exercise price = $50 –40 –50

Put Option Profits Exercise price = $50; option premium = $10 60 40 Option payoffs ($) 20 Sell a put 10 Stock price ($) 20 40 50 60 80 100 –10 Buy a put –20 –40 Exercise price = $50; option premium = $10

22.4 Selling Options The seller (or writer) of an option has an obligation. The purchaser of an option has an option. 40 Buy a call Option payoffs ($) Buy a put Sell a call 10 Sell a put Stock price ($) 50 Buy a call 40 60 100 –10 Buy a put Sell a put Exercise price = $50; option premium = $10 Sell a call –40

22.5 Reading The Wall Street Journal

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

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.

22.5 Reading The Wall Street Journal On this day, 2,365 call options with this exercise price were traded.

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.

22.5 Reading The Wall Street Journal On this day, 2,431 put options with this exercise price were traded.

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.

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.

Value of stock at expiry Protective Put Strategy: Buy a Put and Buy the Underlying Stock: Payoffs at Expiry Protective Put payoffs Value at expiry $50 Buy the stock Buy a put with an exercise price of $50 $0 Value of stock at expiry $50

Protective Put Strategy Profits Value at expiry Buy the stock at $40 $40 Protective Put strategy has downside protection and upside potential $0 -$10 $40 $50 Buy a put with exercise price of $50 for $10 Value of stock at expiry -$40

Value of stock at expiry Covered Call Strategy Value at expiry Buy the stock at $40 $10 Covered Call strategy $0 Value of stock at expiry $40 $50 Sell a call with exercise price of $50 for $10 -$30 -$40

Long Straddle: Buy a Call and a Put 40 Buy a call with exercise price of $50 for $10 Option payoffs ($) 30 Stock price ($) 40 60 30 70 Buy a put with exercise price of $50 for $10 –20 $50 A Long Straddle only makes money if the stock price moves $20 away from $50.

Long Straddle: Buy a Call and a Put This Short Straddle only loses money if the stock price moves $20 away from $50. Option payoffs ($) 20 Sell a put with exercise price of $50 for $10 Stock price ($) 30 40 60 70 $50 –30 Sell a call with an exercise price of $50 for $10 –40

Put-Call Parity: p0 + S0 = c0 + E/(1+ r)T Portfolio value today = c0 + (1+ r)T E Portfolio payoff Call Option payoffs ($) bond 25 25 Stock price ($) 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.

Put-Call Parity: p0 + S0 = c0 + E/(1+ r)T Portfolio payoff Portfolio value today = p0 + S0 Option payoffs ($) 25 Stock price ($) 25 Consider the payoffs from holding a portfolio consisting of a share of stock and a put with a $25 strike.

Put-Call Parity: p0 + S0 = c0 + E/(1+ r)T 25 Stock price ($) Option payoffs ($) Portfolio value today = p0 + S0 Portfolio value today (1+ r)T E = c0 + Since these portfolios have identical payoffs, they must have the same value today: hence Put-Call Parity: c0 + E/(1+r)T = p0 + S0

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.

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.

Market Value, Time Value and Intrinsic Value for an American Call Profit ST Call Option payoffs ($) 25 Market Value Time value Intrinsic value ST E Out-of-the-money In-the-money loss The value of a call option C0 must fall within max (S0 – E, 0) < C0 < S0.

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.

Binomial Option Pricing Model Suppose 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? S0 $21.25 = $25×(1 –.15) $28.75 = $25×(1.15) S1 $25

Binomial Option Pricing Model A 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. S0 S1 C1 $28.75 $3.75 $25 $21.25 $0

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. S0 ( – ) = S1 debt portfolio C1 $28.75 – $21.25 = $7.50 $3.75 $25 $21.25 – $21.25 = $0 $0

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: S0 ( – ) = S1 debt portfolio C1 $28.75 – $21.25 = $7.50 $3.75 $25 $21.25 – $21.25 = $0 $0

Binomial Option Pricing Model We can value the call option today as half of the value of the levered equity portfolio: S0 ( – ) = S1 debt portfolio C1 $28.75 – $21.25 = $7.50 $3.75 $25 $21.25 – $21.25 = $0 $0

The Binomial Option Pricing Model If the interest rate is 5%, the call is worth: $2.38 C0 S0 ( – ) = S1 debt portfolio C1 $28.75 – $21.25 = $7.50 $3.75 $25 $21.25 – $21.25 = $0 $0

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

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: D = Swing of call Swing of stock The delta of a put option is negative.

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

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(U), V(U) q S(0), V(0) 1- q S(D), V(D) ) 1 ( f r D V q U + ´ - =

The Risk-Neutral Approach to Valuation S(U), V(U) q q is the risk-neutral probability of an “up” move. S(0), V(0) 1- q S(0) is the value of the underlying asset today. 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. V(U) and V(D) are the values of the asset in the next period following an up move and a down move, respectively.

The Risk-Neutral Approach to Valuation S(0), V(0) S(U), V(U) S(D), V(D) q 1- q ) 1 ( f r D V q U + ´ - = The key to finding q is to note that it is already impounded into an observable security price: the value of S(0): ) 1 ( f r D S q U + ´ - = A minor bit of algebra yields:

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: $21.25,C(D) q 1- q $25,C(0) $28.75,C(D) ) 15 . 1 ( 25 $ 75 28 ´ = 21 -

Example of the Risk-Neutral Valuation of a Call: The next step would be to compute the risk neutral probabilities ) ( 1 D S U r q f - ´ + = 3 2 50 . 7 $ 5 25 21 75 28 ) 05 1 ( = - ´ q $28.75,C(D) 2/3 $25,C(0) 1/3 $21.25,C(D)

Example of the Risk-Neutral Valuation of a Call: After that, find the value of the call in the up state and down state. 25 $ 75 . 28 ) ( - = U C $28.75, $3.75 2/3 $25,C(0) ] , 75 . 28 $ 25 max[$ ) ( - = D C 1/3 $21.25, $0

Example of the Risk-Neutral Valuation of a Call: Finally, find the value of the call at time 0: ) 1 ( f r D C q U + ´ - = ) 05 . 1 ( $ 3 75 2 ´ + = C $21.25, $0 2/3 1/3 $25,C(0) $28.75,$3.75 38 . 2 $ ) 05 1 ( 50 = C $25,$2.38

Risk-Neutral Valuation and the Replicating Portfolio This risk-neutral result is consistent with valuing the call using a replicating portfolio. The replicating portfolio consists of buying one share of stock today and borrowing the present value of $21.25. The payoffs to the portfolio are twice those of the call, therefore the portfolio is worth twice as much as a call. Since we can value the portfolio, we can value the call.

The Black-Scholes Model The Black-Scholes Model is ) N( 2 1 d Ee S C rT ´ - = Where C0 = the value of a European option at time t = 0 r = the risk-free interest rate. T σ r E S d s ) 2 ( / ln( 1 + = N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d. T d s - = 1 2 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 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.

The Black-Scholes Model Let’s try our hand at using the model. If you have a calculator handy, follow along. First calculate d1 and d2 T σ r E S d s ) 5 . ( / ln( 2 1 + = 5282 . 5 30 ). ) ( 05 (. 150 / 160 ln( 2 1 = + d Then, 31602 . 5 30 52815 1 2 = - T d s

The Black-Scholes Model ) N( 2 1 d Ee S C rT ´ - = 5282 . 1 = d N(d1) = N(0.52815) = 0.7013 N(d2) = N(0.31602) = 0.62401 31602 . 2 = d 92 . 20 $ 62401 150 7013 160 5 05 = ´ - C e

Another Black-Scholes Example Assume S = $50, X = $45, T = 6 months, r = 10%, and  = 28%, calculate the value of a call and a put. ( ) 884 . 50 28 2 10 45 ln 1 = ÷ ø ö ç è æ + - d 686 . 50 28 884 2 = - d From a standard normal probability table, look up N(d1) = 0.812 and N(d2) = 0.754 (or use Excel’s “normsdist” function) 32 . 8 $ ) 754 ( 45 812 50 10 5 = - e C 125 . 1 $ 45 50 32 8 ) ( 10 = + - e P

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.

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.

22.9 Stocks and Bonds as Options It all comes down to put-call parity. c0 = S0 + p0 – (1+ r)T E 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

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.

Balance Sheet for a Company in Distress Assets 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? $200 $0 The bondholders get $200; the shareholders get nothing.

Selfish Strategy 1: Take Large Risks 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 NPV = –$200 + $100 (1.10) NPV = –$133

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 $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.

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.

22.11 Mergers and Options 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. Cash payment to newly issued shares $4.55 $38 $0 $42.55 Value of General Mills in 1 year

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

22.11 Mergers and Options Cash payment to newly issued shares Own a put Strike $42.55 $42.55 – $38.00 $4.55 $42.55 $0 Sell a put Strike $38 $42.55 Value of General Mills in 1 year $38 –$38

22.11 Mergers and Options Value of General Mills in 1 year Value of a share Value of a share plus cash payment $42.55 $4.55 $0 Value of General Mills in 1 year $38 $42.55

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.

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 c0– (1+ r)T E = S0 + p0

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. 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.