Chapter 18 Valuing Options Principles of Corporate Finance Concise Edition Valuing Options Slides by Matthew Will McGraw Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

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

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 $106.67 Option value = $26.67

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) – 33.45 = $12.26

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.

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.

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

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

Binomial Pricing Example Price = 36 s = .40 t = 90/365 D t = 30/365 Strike = 40 r = 10% a = 1.0083 u = 1.1215 d = .8917 Pu = .5075 Pd = .4925

Binomial Pricing 40.37 32.10 36

Binomial Pricing 40.37 32.10 36

Binomial Pricing 50.78 = price 45.28 40.37 36 40.37 32.10 32.10 28.62 25.52 45.28 36 28.62 40.37 32.10 36

Binomial Pricing 50.78 = price 45.28 10.78 = intrinsic value 40.37 .37 32.10 25.52 45.28 36 28.62 40.37 32.10 36

Binomial Pricing The greater of 50.78 = price 45.28 10.78 = intrinsic value 40.37 .37 32.10 25.52 45.28 5.60 36 28.62 The greater of 40.37 32.10 36

Binomial Pricing 1.51 50.78 = price 45.28 10.78 = intrinsic value 5.60 40.37 .37 32.10 25.52 45.28 5.60 36 .19 28.62 40.37 2.91 32.10 .10 36 1.51

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) 22

Black-Scholes Option Pricing Model Option Value Black-Scholes Option Pricing Model 23

Black-Scholes Option Pricing Model OC- Call Option Price P - Stock Price N(d1) - Cumulative normal density function of (d1) PV(EX) - Present Value of Strike or Exercise price N(d2) - Cumulative normal density function of (d2) 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 7

Black-Scholes Option Pricing Model 7

Black-Scholes Option Pricing Model N(d1)= 8

Cumulative Normal Density Function 9

Call Option Example - Genentech What is the price of a call option given the following? P = 80 r = 5% v = .4068 EX = 80 t = 180 days / 365 11

Call Option Example - Genentech What is the price of a call option given the following? P = 80 r = 5% v = .4068 EX = 80 t = 180 days / 365 12

Call Option Example - Genentech What is the price of a call option given the following? P = 80 r = 5% v = .4068 EX = 80 t = 180 days / 365 13

Call Option Example What is the price of a call option given the following? P = 36 r = 10% v = .40 EX = 40 t = 90 days / 365 11

Call Option Example What is the price of a call option given the following? P = 36 r = 10% v = .40 EX = 40 t = 90 days / 365 12

Call Option Example What is the price of a call option given the following? P = 36 r = 10% v = .40 EX = 40 t = 90 days / 365 13

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 Price = Oc + EX - P - Carrying Cost + Div. Put - Call Parity Put Price = Oc + EX - P - Carrying Cost + Div. Carrying cost = r x EX x t 14

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? OP = OC + EX - P - Carrying Cost + Div. OP = 4 + 40 - 41 - (.10x 40 x .50) + .50 OP = 3 - 2 + .5 Op = $1.50 15

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

Binomial vs. Black Scholes Example What is the price of a call option given the following? P = 36 r = 10% v = .40 EX = 40 t = 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.

How estimated call price changes as number of binomial steps increases Binomial vs. Black Scholes How estimated call price changes as number of binomial steps increases No. of steps Estimated value 1 48.1 2 41.0 3 42.1 5 41.8 10 41.4 50 40.3 100 40.6 Black-Scholes 40.5

Dilution

Web Resources Web Links Click to access web sites Internet connection required www.numa.com www.fintools.net/options/optcalc.html www.optionscentral.com www.pcquote.com/options www.pmpublishing.com www.schaffersresearch.com/stock/calculator.asp