Chapter 21 Valuing Options Principles of Corporate Finance Eighth Edition Chapter 21 Valuing Options Slides by Matthew Will McGraw-Hill/Irwin Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved

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

Binomial Pricing

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

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)= 32 34 36 38 40 8

Cumulative Normal Density Function 9

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

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.schaffersresearch.com