© Prentice Hall, Corporate Financial Management 3e Emery Finnerty Stowe Derivatives Applications

Some Properties of Options As stock price increases, value of call option increases. value of put option decreases. As exercise price increases, value of call option decreases. value of put option increases. As the time to maturity increases, value of call option increases. value of put option increases.

Option Trading Exchange listed options Chicago Board of Options Exchange (CBOE) NYSE ASE Pacific Stock Exchange Philadelphia Stock Exchange Over-the-counter options Non-standardized contracts

A Simple Model of Option Valuation Consider the call option sold by Carla to Alex. The option has one year to maturity, and an exercise price of $65,000. At the time of the sale, the land value was $60,000. Assume that in one year, the land value can be either $67,870 (up by 13.12%) or $48,000 (down by 20%) in one year. If the risk-free rate is 4% per year, and assuming that the land should yield this rate, what is the value of the call option on this land?

A Simple Model of Option Valuation If the land value goes up to $67,870, the value of the call option would be $67,870 – $65,000 or $2,870. If the land value declines to $48,000, the call option would expire out-of-the-money and be worthless.

A Simple Model of Option Valuation Let p u denote the probability that land will go up in value. Then the probability that the land value will decline is (1– p u ). Since the land is to yield the riskless rate of return of 4%, 4% = p u (13.12%) + (1 – p u ) (–20.0%) Solving this for p u, we get p u = 72.5%. The probability of a decline in land value is thus 27.5%.

A Simple Model of Option Valuation The expected value of the call option at maturity is: p u (Value of call if land goes up) + (1 – p u ) (Value of call if land goes down) = (0.725) ($2,870) + (0.275) ($0) = $2,080 The present value of this is the value of the call option today: $2,080/1.04 = $2,000

Black-Scholes Option Pricing Model Assumptions: The option and the underlying asset trade in perfect markets. The returns on the underlying assets are normally distributed with a constant  over the life of the option. The riskless rate of interest is constant over the option’s life. Option contracts are European (cannot be exercised prior to maturity). Underlying asset does not provide any cash flows over the life of the option.

Black-Scholes Option Pricing Model S = Strike price of the call option. P 0 = current value of the underlying asset. k = riskless APR with continuous compounding.  t = time in years to option expiration.  = standard deviation of the (continuously compounded) returns on the asset. N(d) = Cumulative distribution function for a standard normal random variable d.

Black-Scholes Option Pricing Model The value of the call option, CALL, is given by:

Black-Scholes Option Pricing Model Find the value of a European call option on Hightone Records. The current stock price is $48, and the stock’s volatility is 30%. The risk free rate is 5% per year. The call option matures in 6 months and has a strike price of $50.

Black-Scholes Option Pricing Model

N(d 1 ) = N(d 2 ) =

Put-Call Parity Relationship Find the value of a put option on Hightone Records. The put option also has a strike price of $50 and expires 6 months from today. We will use put-call parity:

Warrants A warrant is a long-term call option issued by the firm. Entitles holder to buy a fixed number of shares from the firm, at a stated price, within a stated time period. When a warrant is exercised, the number of outstanding shares increases.

Convertible Debt At the option of the bondholder, a convertible bond can be converted into a pre-specified number of shares of the firm’s common stock. Each bond can be converted into common stock at a stated conversion price. Conversion price exceeds issuer’s share price at the time of issue by about 10% to 20%. Conversion price is adjusted for stock splits, stock dividends, rights offerings, and other distributions.

Convertible Debt Conversion ratio is the number of shares that can be purchased with one bond. Bondholders who convert do not receive accrued interest. If bonds are called, conversion option expires just before the redemption date.

Convertible Debt The market value of a convertible bond always exceeds its conversion value (unless the conversion option is about to expire). The difference in the market value and the bond’s conversion value is the time premium of the conversion option. Time premium is zero at option expiration.

Convertible Debt As long as the underlying stock does not pay any dividends, bondholders would never convert voluntarily. Sell the bond in the open market since the market value exceeds the conversion value. If the stock does pay dividends, bondholders would not convert as long as the interest on the bond exceeds the total dividends from the stock.

Forced Conversion As long as the call price is less than the market value of the underlying stock, bondholders will not voluntarily convert. To force conversion, the firm should call the bonds when the conversion value reaches the effective call price. The effective call price = optional redemption premium plus accrued interest.

Convertible Preferred Stock Similar to convertible bonds: Convertible preferred stock can be exchanged into shares of common stock, at the option of the preferred stockholder. Convertible exchangeable preferred stock can be converted into convertible debt.

Valuing Warrants Warrants are long-term call options written by the firm. When a warrant is exercised, the number of shares outstanding increases. Let  be the proportionate increase in the number of outstanding shares after all warrants are exercised.

Valuing Warrants The value of the warrant before it is issued is simple C/(1+  ) where C is the value of a call option to buy one share. After the warrant is issued, an efficient market will reflect the dilution in the firm’s stock price, and the warrant’s value is equal to that of the call option.

Hedging with Options Suppose you own 1 share of General Cinema. The current share price is $50. You purchase a put option on this stock, with an exercise price of $50. The put costs $1.00, and expires in 30 days. What will be your net profit (loss) under alternative stock prices 30 days later?

Hedging with Options Stock Price Gain on Stock Value of Put Total Gain Net Gain $48 $49 $50 $51 $52 $53 ($2) ($1) ($0) $1 $2 $3 $2 $1 $0 $1 $2 $3 ($1) $0 $1 $2