Chapter 22 Principles PrinciplesofCorporateFinance Ninth Edition Valuing Options Slides by Matthew Will Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw Hill/Irwin

22- 2 Topics Covered  Simple Option Valuation Model  Binomial Model  Black-Scholes Formula  Black Scholes in Action  Option Values at a Glance  The Option Menagerie

22- 3 Option Valuation Methods  Genentech call options have an exercise price of $80. Case 1 Stock price falls to $60 Option value = $0 Case 2 Stock price rises to $ Option value = $26.67

22- 4 Option Valuation Methods  Assume you borrow 4/7 of the value of the Genentech exercise price ($33.45). Value of Call = 80 x (4/7) – = $12.26

22- 5 Option Valuation Methods  Since the Genentech call option is equal to a leveraged position in 4/7 shares, the option delta can be computed as follows.

22- 6 Option Valuation Methods  If we are risk neutral, the expected return on Genentech call options is 2.5%. Accordingly, we can determine the probability of a rise in the stock price as follows.

22- 7 Option Valuation Method  The Genentech option can then be valued based on the following method.

22- 8 Binomial Pricing The prior example can be generalized as the binomial model and shown as follows.

22- 9 Example Price = 36  =.40 t = 90/365  t = 30/365 Strike = 40r = 10% a = u = d =.8917 Pu =.5075 Pd =.4925 Binomial Pricing

Binomial Pricing

Binomial Pricing

= price Binomial Pricing

= price = intrinsic value Binomial Pricing

= price = intrinsic value The greater of Binomial Pricing

= price = intrinsic value Binomial Pricing

Binomial Model The price of an option, using the Binomial method, is significantly impacted by the time intervals selected. The Genentech example illustrates this fact.

Option Value Components of the Option Price 1 - Underlying stock price 2 - Striking or Exercise price 3 - Volatility of the stock returns (standard deviation of annual returns) 4 - Time to option expiration 5 - Time value of money (discount rate)

Option Value Black-Scholes Option Pricing Model

O C - Call Option Price P - Stock Price N(d 1 ) - Cumulative normal density function of (d 1 ) PV(EX) - Present Value of Strike or Exercise price N(d 2 ) - Cumulative normal density function of (d 2 ) r - discount rate (90 day comm paper rate or risk free rate) t - time to maturity of option (as % of year) v - volatility - annualized standard deviation of daily returns Black-Scholes Option Pricing Model

Black-Scholes Option Pricing Model

N(d 1 )= Black-Scholes Option Pricing Model

Cumulative Normal Density Function

Call Option Example - Genentech What is the price of a call option given the following? P = 80r = 5%v =.4068 EX = 80t = 180 days / 365

Call Option Example - Genentech What is the price of a call option given the following? P = 80r = 5%v =.4068 EX = 80t = 180 days / 365

Call Option Example - Genentech What is the price of a call option given the following? P = 80r = 5%v =.4068 EX = 80t = 180 days / 365

Call Option Example What is the price of a call option given the following? P = 36r = 10%v =.40 EX = 40t = 90 days / 365

Call Option Example What is the price of a call option given the following? P = 36r = 10%v =.40 EX = 40t = 90 days / 365

Call Option Example What is the price of a call option given the following? P = 36r = 10%v =.40 EX = 40t = 90 days / 365

Black Scholes Comparisons

Implied Volatility The unobservable variable in the option price is volatility. This figure can be estimated, forecasted, or derived from the other variables used to calculate the option price, when the option price is known. Implied Volatility (%)

Put - Call Parity Put Price = Oc + EX - P - Carrying Cost + Div. Carrying cost = r x EX x t

Put - Call Parity Example ABC is selling at $41 a share. A six month May 40 Call is selling for $4.00. If a May $.50 dividend is expected and r=10%, what is the put price? O P = O C + EX - P - Carrying Cost + Div. O P = (.10x 40 x.50) +.50 O P = O p = $1.50

Expanding the binomial model to allow more possible price changes 1 step 2 steps 4 steps (2 outcomes) (3 outcomes) (5 outcomes) etc. Binomial vs. Black Scholes

Example What is the price of a call option given the following? P = 36r = 10%v =.40 EX = 40t = 90 days / 365 Binomial price = $1.51 Black Scholes price = $1.70 The limited number of binomial outcomes produces the difference. As the number of binomial outcomes is expanded, the price will approach, but not necessarily equal, the Black Scholes price. Binomial vs. Black Scholes

How estimated call price changes as number of binomial steps increases No. of stepsEstimated value Black-Scholes40.5 Binomial vs. Black Scholes

Dilution

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