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## Options and Corporate Finance

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**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**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 Chapter Outline**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 • 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 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. 17.2 Call Options**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**60 Buy a call 40 Option payoffs ($) 20 80 120 20 40 60 100 50 Stock price ($) –20 Exercise price = $50 –40**60**40 Option payoffs ($) 20 80 120 20 40 60 100 Stock price ($) –20 –40 Call Option Profits Buy a call 10 50 –10 Exercise price = $50; option premium = $10**Put options gives the holder the right, but not the**obligation, to sell a given quantity of an asset on or before some time in the future, at prices agreed upon today. When exercising a put, you “put” the asset to someone. 17.3 Put Options**Put Option Pricing at Expiry**• At expiry, an American put option is worth the same as a European option with the same characteristics. • If the put is in-the-money, it is worth E – ST. • If the put is out-of-the-money, it is worthless. P= Max[E – ST, 0]**Put Option Payoffs**60 50 40 Option payoffs ($) 20 Buy a put 0 80 0 20 40 60 100 50 Stock price ($) –20 Exercise price = $50 –40**Put Option Profits**60 40 Option payoffs ($) 20 10 Stock price ($) 80 50 20 40 60 100 –10 Buy a put –20 Exercise price = $50; option premium = $10 –40**Option Premium**Intrinsic Value Speculative Value + = Option Value • Intrinsic Value • Call: Max[ST –E, 0] • Put: Max[E–ST , 0] • Speculative Value • The difference between the option premium and the intrinsic value of the option.**17.4 Selling Options**• The seller (or writer) of an option has an obligation. • The seller receives the option premium in exchange.**Call Option Payoffs**60 40 Option payoffs ($) 20 80 120 20 40 60 100 50 Stock price ($) –20 Sell a call Exercise price = $50 –40**Put Option Payoffs**40 20 Option payoffs ($) Sell a put 0 80 0 20 40 60 100 50 Stock price ($) –20 Exercise price = $50 –40 –50**Option Diagrams Revisited**Buy a call 40 Option payoffs ($) Buy a put Sell a call Sell a put 10 Stock price ($) 50 40 60 100 Buy a call –10 Buy a put Sell a put Exercise price = $50; option premium = $10 Sell a call –40**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.**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.**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. 17.6 Combinations of Options**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**$10**-$30 Covered Call Strategy Value at expiry Buy the stock at $40 Covered Call strategy $0 Value of stock at expiry $40 $50 Sell a call with exercise price of $50 for $10 -$40**–20**Long Straddle Buy a call with exercise price of $50 for $10 40 Option payoffs ($) 30 Stock price ($) 40 60 30 70 Buy a put with exercise price of $50 for $10 $50 A Long Straddle only makes money if the stock price moves $20 away from $50.**20**Short Straddle This Short Straddle only loses money if the stock price moves $20 away from $50. Option payoffs ($) Sell a put with exercise price of $50 for $10 Stock price ($) 30 70 40 60 $50 –30 Sell a call with an exercise price of $50 for $10 –40**E**Portfolio value today = c0 + (1+ r)T bond Put-Call Parity: p0 + S0 = c0 + E/(1+ r)T Portfolio payoff Call Option payoffs ($) 25 Stock price ($) 25 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 $25 strike.**Portfolio value today**Portfolio value today = p0 + S0 E = c0 + Option payoffs ($) Option payoffs ($) (1+ r)T 25 25 Stock price ($) Stock price ($) 25 25 Put-Call Parity Since these portfolios have identical payoffs, they must have the same value today: hence Put-Call Parity: c0 + E/(1+r)T = p0 + S0**The last section concerned itself with the value of an**option at expiry. This section considers the value of an option prior to the expiration date. A much more interesting question. 17.7 Valuing Options**Market Value**American Call Profit ST Call Option payoffs ($) 25 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.**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. 17.8 An Option Pricing Formula**S1**$28.75 = $25×(1.15) $21.25 = $25×(1 –.15) 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 $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 ( – ) = – $21.25 $7.50 $28.75 = $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 ( – ) = – $21.25 $7.50 $28.75 = $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 ( – ) = – $21.25 $7.50 $28.75 = $3.75 $25 – $21.25 $21.25 $0 = $0**C0**$2.38 Binomial Option Pricing Model If the interest rate is 5%, the call is worth: S0 S1 debt portfolio C1 ( – ) = – $21.25 $7.50 $28.75 = $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.**Swing of call**D = Swing of stock 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: • 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**We could value the option, 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)**The Risk-Neutral Approach**S(0) is the value of the underlying asset today. S(U), V(U) q q is the risk-neutral probability of an “up” move. S(0), V(0) 1- q 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.**S(U), V(U)**q S(0), V(0) 1- q S(D), V(D) The Risk-Neutral Approach • 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: