Chapter 17. Options and Corporate Finance. Key Concepts and Skills

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1 Chapter 17 Options and Corporate Finance Prof. Durham 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 pricing models

2 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 a price 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 specified underlying asset. Expiry (Expiration Date) The maturity date of the option 17-2 Options (continued) European versus American Options European options can be exercised only at expiry. American options can be exercised at any time up to expiry. In-the-Money Option Exercising the option would result in a positive payoff In the case of a call [put] option, because the exercise price is greater [less] than the spot price). At-the-Money and Out-of-the-Money Options At : Exercising the option would result in a zero payoff. Out of : Exercising would result in a negative payoff

3 17.2 Call Options Call options give 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 (that is, at time T), 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 T = Max[ S T E, 0 ] Where S T is the value of the stock at expiry (i.e., at time T), E is the exercise price, and C T is the value of the call option at expiry

4 Call Option Payoffs (at time T) 60 Option payoff at time T ($) Stock price, S T ($) Exercise price = $ Call Option Profits (at time T) 60 Profit at time T ($) Buy a call Stock price, S T ($) Exercise price = $50, and supposing an earlier Option premium / price = $

5 17.3 Put Options Put options give 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 S T. If the put is out of the money, it is worthless. P T = Max[ E S T, 0 ] Where S T is the stock s value at expiry (i.e., at time T), E is the exercise price, and P T is the put option s value at expiry

6 Put Option Payoffs (at time T) 60 Option payoff at time T ($) Buy a put Stock price, S T ($) Exercise price = $ Put Option Profits (at time T) 60 Profit at time T ($) Stock price, S T ($) 100 Buy a put Exercise price = $50, and supposing an Option premium / price = $

7 17.7 Option Value Prior to Expiry Intrinsic Value (at any time t, prior to T) For a Call: Max[ S t E, 0 ] For a Put: Max[ E S t, 0 ] Speculative Value (also called Time Premium) The difference between the option premium (a.k.a., option price) and the intrinsic value of the option. Option Premium = Intrinsic Value + Speculative Value Selling an Option (Call or a Put) The seller (or writer) of an option has an obligation. A seller of a call is obligated to sell the underlying asset if the holder of the call option exercises A seller of a put is obligated to buy the underlying asset if the holder of the put option exercises The seller receives the option premium (a.k.a., the option price) in exchange

8 Call Option Payoffs (from selling!) 60 Payoff & Profit at time T ($) Exercise price = $50 and supposing an option premium of $ Stock price, S T ($) Profit Payoff Put Option Payoffs (from selling!) 40 Payoff & Profit at time T ($) Payoff Sell a put Stock price ($) Exercise price = $50 Profit Sell a put and supposing an option premium of $

9 17.5 Option Quotes The call option with a strike price of $135 is trading for $4.75. Since the option contract is on 100 shares of stock, buying this call option would cost $475 plus commissions 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

10 Put-Call Parity: P 0 + S 0 = C 0 + E/(1+ R) T Payoffs ($) Portfolio payoff: Max( 25, S T ) E Portfolio value = C 0 + today (1+ R) T Call payoff Max( 0, S T $25 ) 25 Bond payoff $25 25 Stock price, S T ($) 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 $ Put-Call Parity, continued Portfolio payoff Max( $25, S T ) Stock payoff, S T Payoffs ($) 25 Portfolio value today = P 0 + S 0 Put payoff Max( 0, $25 S T ) 25 Stock price, S T ($) Consider the payoffs from holding a portfolio consisting of a share of stock and a put with a $25 strike

11 Put-Call Parity, continued Portfolio payoffs ($) Portfolio value today E = C 0 + (1+ R) T Portfolio payoffs ($) Portfolio value today = P 0 + S Stock price, S T ($) Since these portfolios have identical payoffs, they must have the same value today; hence, Put-Call Parity applies: 25 Stock price, S T ($) C 0 + E/(1+R) T = P 0 + S Valuing Options Sections 17.2 and 17.3 taught us how to value call and put options at expiry. This section considers the value of an option prior to the expiration date. A much more interesting question

12 American Call: Value at Any Time t jda Option payoffs ($) Upper bound: S t Market Value C t Speculative value Intrinsic value Lower bound: Max (S t E, 0 ) Out-of-the-money E In-the-money S t C t is bounded: Max( S t E, 0 ) < C t < S t Option Value Determinants 1. Stock price 2. Exercise price 3. Risk-free interest rate 4. Volatility in the stock price 5. Expiration date Call The value of a call option C 0 is bounded: Max( S t E, 0 ) < C t < S t. Put The precise position will depend on these factors

13 17.8 An Option Pricing Formula We ll start with a binomial option pricing formula to build our intuittion. Then we ll 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. S 0 = $25 today and in one year S 1 is either $28.75 or $ The risk-free rate is 5%. What is the value of an at-the-money call option? S 0 S 1 $28.75 = $25 ( ) $25 $21.25 = $25 ( )

14 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. S 0 S 1 C 1 $28.75 $25 Max ( , 0 ) $3.75 $21.25 Max ( , 0 ) $ Binomial Option Pricing Model Today, do two things: (1) borrow the present value of $21.25 and (2) buy 1 share of stock for $ The net payoff for this levered equity portfolio in one period is either $7.50 or $0. At t=1, the levered equity portfolio has twice the call option s payoff, so the portfolio should be worth twice the call option value. S 0 ( ) S 1 Debt pmt. $28.75 $21.25 = $7.50 C 1 $3.75 $25 $21.25 $21.25 = $0.00 $

15 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: S 0 ( debt pmt.) S 1 $28.75 $21.25 = $7.50 C 1 $3.75 $25 $21.25 $21.25 = $0.00 $ Binomial Option Pricing Model Stated differently, the time-1 payoff pattern on two call options is equal to the time-1 payoff pattern on the levered equity portfolio. Thus, today s price of one call option should be equal to onehalf the price of the levered equity portfolio: C 0 = ½ x [ $25 $25 / ( 1 + R f ) ] Supposing a risk-free rate of 5%, we can solve for C 0 : C 0 = ½ x [ $25 $25 / 1.05 ] C 0 = $

16 Binomial Option Pricing Model The most important lesson (so far) from the binomial option pricing model is: the intuition of the replicating portfolio Many derivative securities can be valued by valuing portfolios of primitive securities when those portfolios have the same payoffs as the derivative securities The Black-Scholes Model C 0 = S 0 N(d 1 ) E e ( R T) N(d f 2 ) where C 0 = the value of a European option at t=0, R f = the risk-free rate, S 0 is the current underlying stock s price, T = maturity date, and E is the call s exercise price. N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d. This model allows us to value options in the real world (with a continuum of possible stock prices at expiration) just as we did in the two-state world

17 The Black-Scholes Model Find the value of a six-month call option on a stock with an exercise price (E) of $150. The current value of a share of stock (S 0 ) is $160. The interest rate available in the US is R f = 5%/year. The option s time to maturity (T) is one-half of a year. The volatility (σ) of the underlying asset is 30%/year. 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 d 1 and d 2 : d 1 = [ ln ( S 0 / E ) + ( R f σ 2 ) T ] / [ σ T 1/2 ] Then,

18 The Black-Scholes Model N(d 1 ) = N( ) = N(d 2 ) = N( ) =