Chapter 21 Principles of Corporate Finance Tenth Edition

Chapter 21 Principles of Corporate Finance Tenth Edition Valuing Options Slides by Matthew Will Mc. Graw-Hill/Irwin Copyright © 2011 by the Mc. Graw-Hill Companies, Inc. All rights reserved.

Topics Covered Ø Simple Option Valuation Model Ø A Binomial Model for Valuing Options Ø Black-Scholes Formula Ø Black Scholes in Action Ø Option Values at a Glance Ø The Option Menagerie 21 -2

Option Valuation Methods Ø Google call options have an exercise price of $430 Case 1 Case 2 Stock price falls to $322. 50 Stock price rises to $573. 33 Option value = $0 Option value = $143. 33 21 -3

Option Valuation Methods Ø Assume you buy 4/7 of a Google share and borrow $181. 58 from the bank (@1. 5%). Value of Call = 430 x (4/7) – 181. 58 = $64. 13 21 -4

Option Valuation Methods Ø Since the Google call option is equal to a leveraged position in 4/7 shares, the option delta can be computed as follows. 21 -5

Option Valuation Methods Ø If we are risk neutral, the expected return on Google call options is 1. 5%. Accordingly, we can determine the probability of a rise in the stock price as follows. 21 -6

Option Valuation Method Ø The Google option can then be valued based on the following method. 21 -7

Option Valuation Method Ø The Google PUT option can then be valued based on the following method. Case 1 Case 2 Stock price falls to $322. 50 Stock price rises to $573. 33 Option value = $107. 50 Option value = $0 21 -8

Option Valuation Methods Ø Since the Google PUT option is equal to a leveraged position in 3/7 shares, the option delta can be computed as follows. 21 -9

Option Valuation Methods Ø Assume you SELL 3/7 of a Google share and lend $242. 09 (@1. 5%). Value of PUT = -(3/7) x 430 + 242. 09 = $57. 82 21 -10

Binomial Pricing 21 -11 Present and possible future prices of Google stock assuming that in each three-month period the price will either rise by 22. 6% or fall by 18. 4%. Figures in parentheses show the corresponding values of a six-month call option with an exercise price of $430.

Binomial Pricing 21 -12 Now we can construct a leveraged position in delta shares that would give identical payoffs to the option: We can now find the leveraged position in delta shares that would give identical payoffs to the option:

Binomial Pricing 21 -13 Present and possible future prices of Google stock. Figures in parentheses show the corresponding values of a six-month call option with an exercise price of $430. Option Value: PV option = PV (. 569 shares)- PV($199. 58) =. 569 x $430 - $199. 58/1. 0075 = $46. 49

Binomial Pricing The prior example can be generalized as the binomial model and shown as follows. 21 -14

Binomial Pricing Example Price = 36 Strike = 40 s =. 40 t = 90/365 D t = 30/365 r = 10% a = 1. 0083 u = 1. 1215 d =. 8917 Pu =. 5075 Pd =. 4925 21 -15

Binomial Pricing 40. 37 36 32. 10 21 -16

Binomial Pricing 40. 37 36 32. 10 21 -17

Binomial Pricing 45. 28 50. 78 = price 36 40. 37 36 32. 10 28. 62 25. 52 21 -18

Binomial Pricing 45. 28 50. 78 = price 10. 78 = intrinsic value 40. 37 36 32. 10 28. 62 0 25. 52 0 21 -19

Binomial Pricing The greater of 45. 28 50. 78 = price 5. 60 10. 78 = intrinsic value 40. 37 36 36 . 37 32. 10 28. 62 0 25. 52 0 21 -20

Binomial Pricing 45. 28 50. 78 = price 5. 60 10. 78 = intrinsic value 40. 37 2. 91 36 . 37 . 19 36 1. 51 32. 10 28. 62 0 0 25. 52 0 21 -21

Binomial Model The price of an option, using the Binomial method, is significantly impacted by the time intervals selected. The Google example illustrates this fact. 21 -22

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) 21 -23

Option Value Black-Scholes Option Pricing Model 21 -24

Black-Scholes Option Pricing Model OC- Call Option Price P - Stock Price N(d 1) - Cumulative normal probability density function of (d 1) PV(EX) - Present Value of Strike or Exercise price N(d 2) - Cumulative normal probability 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 21 -25

Black-Scholes Option Pricing Model N(d 1)= 21 -26

Cumulative Normal Density Function 21 -27

Call Option Example - Google What is the price of a call option given the following? P = 430 r = 3% v =. 4068 EX = 430 t = 180 days / 365 21 -28

Call Option Example - Google What is the price of a call option given the following? P = 430 r = 3% v =. 4068 EX = 430 t = 180 days / 365 21 -29

Call Option Example - Google What is the price of a call option given the following? P = 430 r = 3% v =. 4068 EX = 430 t = 180 days / 365 21 -30

Call Option The curved line shows how the value of the Google call option changes as the price of Google stock changes. 21 -31

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 21 -32

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 21 -33

Black-Scholes Option Pricing Model 21 -34

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 21 -35

Black Scholes Comparisons 21 -36

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 (%) VXN 21 -37

Put - Call Parity Put Price = Oc + EX - P - Carrying Cost + Div. Carrying cost = r x EX x t 21 -38

Valuation Variations Ø American Calls with no dividends Ø European Puts with no dividends Ø American Puts with no dividends Ø European Calls and Puts on dividend paying stocks Ø American Calls on dividend paying stocks 21 -39

Binomial vs. Black Scholes Expanding the binomial model to allow more possible price changes 21 -40

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. 21 -41

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 21 -42

Dilution 21 -43

Web Resources Click to access web sites Internet connection required www. numa. com www. math. columbia. edu/~smirnov/options 13. html www. optionscentral. com www. pmpublishing. com www. schaffersresearch. com/streetools/option_tools. aspx? click=jumpto 21 -44