Options and Corporate Finance Chapter 17 Options and Corporate Finance

Key Concepts and Skills Understand option terminology Be able to determine option payoffs and profits Understand the major determinants of option prices Understand and apply put-call parity Be able to determine option prices using the binomial and Black-Scholes models

Chapter Outline 17.1 Options 17.2 Call Options 17.3 Put Options 17.4 Selling Options 17.5 Option Quotes 17.6 Combinations of Options 17.7 Valuing Options 17.8 An Option Pricing Formula 17.9 Stocks and Bonds as Options 17.10 Options and Corporate Decisions: Some Applications 17.11 Investment in Real Projects and Options

17.1 Options An option gives the holder the right, but not the obligation, to buy or sell a given quantity of an asset on (or before) a given date, at prices agreed upon today. Exercising the Option The act of buying or selling the underlying asset 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 (Expiration Date) The maturity date of the option

Options European versus American options In-the-Money At-the-Money European options can be exercised only at expiry. American options can be exercised at any time up to expiry. In-the-Money Exercising the option would result in a positive payoff. At-the-Money Exercising the option would result in a zero payoff (i.e., exercise price equal to spot price). Out-of-the-Money Exercising the option would result in a negative payoff. Call is in the money if spot price is greater than strike price (opposite for a put).

17.2 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 Option Pricing 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 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

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

Option Value Intrinsic Value Speculative Value Option Premium = Call: Max[ST – E, 0] Put: Max[E – ST , 0] Speculative Value The difference between the option premium and the intrinsic value of the option. Option Premium = Intrinsic Value Speculative Value +

17.4 Selling Options The seller (or writer) of an option has an obligation. The seller receives the option premium in exchange.

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

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

Option Diagrams Revisited 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

17.5 Option Quotes

Option Quotes This option has a strike price of $135; a recent price for the stock is $138.25; July is the expiration month.

Option Quotes 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.

Option Quotes On this day, 2,365 call options with this exercise price were traded.

The CALL option with a strike price of $135 is trading for $4.75. Option Quotes 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.

Option Quotes On this day, 2,431 put options with this exercise price were traded.

Option Quotes 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.

17.6 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.

Protective Put Strategy (Payoffs) 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 with exercise price of $50 for $10 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.

Short Straddle 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 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 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

17.7 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.

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

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.

17.8 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

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 most important lesson (so far) from the binomial option pricing model is: the 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.

Delta 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.

$2.38 = $25 × ½ – Amount borrowed 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 S(U), V(U) q S(0), V(0) 1- q S(D), V(D) We could value the option, V(0), as the value of the replicating portfolio. An equivalent method is risk-neutral valuation:

The Risk-Neutral Approach 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 option in the next period following an up move and a down move, respectively.

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

Example of Risk-Neutral Valuation 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: $28.75,C(U) q $25,C(0) 1- q $21.25,C(D)

Example of Risk-Neutral Valuation The next step would be to compute the risk neutral probabilities $28.75,C(U) 2/3 $25,C(0) 1/3 $21.25,C(D)

Example of Risk-Neutral Valuation After that, find the value of the call in the up state and down state. $28.75, $3.75 2/3 $25,C(0) 1/3 $21.25, $0

Example of Risk-Neutral Valuation 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

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 Where C0 = 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.

The Black-Scholes Model Find the value of a six-month call option on 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 Then,

The Black-Scholes Model N(d1) = N(0.52815) = 0.7013 N(d2) = N(0.31602) = 0.62401

17.9 Stocks and Bonds as Options Levered equity is a call option. The underlying asset comprises 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.

Stocks and Bonds as Options Levered equity is a put option. The underlying asset comprises 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.

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

Mergers and Diversification Diversification is a frequently mentioned reason for mergers. Diversification reduces risk and, therefore, volatility. Decreasing volatility decreases the value of an option. Assume diversification is the only benefit to a merger: Since equity can be viewed as a call option, should the merger increase or decrease the value of the equity? Since risky debt can be viewed as risk-free debt minus a put option, what happens to the value of the risky debt? Overall, what has happened with the merger and is it a good decision in view of the goal of stockholder wealth maximization?

Example Company A Company B Market value of assets $40 million Consider the following two merger candidates. The merger is for diversification purposes only with no synergies involved. Risk-free rate is 4%. Company A Company B Market value of assets $40 million $15 million Face value of zero coupon debt $18 million $7 million Debt maturity 4 years Asset return standard deviation 40% 50%

Example Use the Black and Scholes OPM (or an options calculator) to compute the value of the equity. Value of the debt = value of assets – value of equity Company A Company B Market Value of Equity 25.72 9.88 Market Value of Debt 14.28 5.12 Used the options calculator at Numa to compute the equity value

Example Combined Firm Market value of equity 34.18 The asset return standard deviation for the combined firm is 30% Market value assets (combined) = 40 + 15 = 55 Face value debt (combined) = 18 + 7 = 25 Combined Firm Market value of equity 34.18 Market value of debt 20.82 Total MV of equity of separate firms = 25.72 + 9.88 = 35.60 Wealth transfer from stockholders to bondholders = 35.60 – 34.18 = 1.42 (exact increase in MV of debt)

M&A Conclusions Mergers for diversification only transfer wealth from the stockholders to the bondholders. The standard deviation of returns on the assets is reduced, thereby reducing the option value of the equity. If management’s goal is to maximize stockholder wealth, then mergers for reasons of diversification should not occur.

Options and Capital Budgeting Stockholders may prefer low NPV projects to high NPV projects if the firm is highly leveraged and the low NPV project increases volatility. Consider a company with the following characteristics: MV assets = 40 million Face Value debt = 25 million Debt maturity = 5 years Asset return standard deviation = 40% Risk-free rate = 4%

Example: Low NPV Current market value of equity = $22.706 million Current market value of debt = $17.294 million Project I Project II NPV $3 $1 MV of assets $43 $41 Asset return standard deviation 30% 50% MV of equity $23.831 $25.381 MV of debt $19.169 $15.169 MV of assets = current MV + NPV We assume no increase in debt as a result of the project.

Example: Low NPV Which project should management take? Even though project B has a lower NPV, it is better for stockholders. The firm has a relatively high amount of leverage: With project A, the bondholders share in the NPV because it reduces the risk of bankruptcy. With project B, the stockholders actually appropriate additional wealth from the bondholders for a larger gain in value.

Example: Negative NPV We’ve seen that stockholders might prefer a low NPV to a high one, but would they ever prefer a negative NPV? Under certain circumstances, they might. If the firm is highly leveraged, stockholders have nothing to lose if a project fails, and everything to gain if it succeeds. Consequently, they may prefer a very risky project with a negative NPV but high potential rewards. This discussion is consistent with Selfish Strategy 1 discussed in the chapter on bankruptcy.

Example: Negative NPV Consider the previous firm. They have one additional project they are considering with the following characteristics Project NPV = -$2 million MV of assets = $38 million Asset return standard deviation = 65% Estimate the value of the debt and equity MV equity = $25.453 million MV debt = $12.547 million

Example: Negative NPV In this case, stockholders would actually prefer the negative NPV project to either of the positive NPV projects. The stockholders benefit from the increased volatility associated with the project even if the expected NPV is negative. This happens because of the large levels of leverage.

Options and Capital Budgeting As a general rule, managers should not accept low or negative NPV projects and pass up high NPV projects. Under certain circumstances, however, this may benefit stockholders: The firm is highly leveraged The low or negative NPV project causes a substantial increase in the standard deviation of asset returns

17.12 Investment in Real Projects and Options Classic NPV calculations generally ignore the flexibility that real-world firms typically have.

Quick Quiz What is the difference between call and put options? What are the major determinants of option prices? What is put-call parity? What would happen if it doesn’t hold? What is the Black-Scholes option pricing model? How can equity be viewed as a call option? Should a firm do a merger for diversification purposes only? Why or why not? Should management ever accept a negative NPV project? If yes, under what circumstances?