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

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

3. 21-3 Option Valuation Methods  Google call options have an exercise price of $430 Case 1 Stock price falls to $322.50 Option value = $0 Case 2 Stock price rises to $573.33 Option value = $143.33

4. 21-4 Option Valuation Methods  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

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

6. 21-6 Option Valuation Methods  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.

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

8. 21-8 Option Valuation Method PUT  The Google PUT option can then be valued based on the following method. Case 1 Stock price falls to $322.50 Option value = $107.50 Case 2 Stock price rises to $573.33 Option value = $0

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

10. 21-10 Option Valuation Methods  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

11. 21-11 Binomial Pricing 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.

12. 21-12 Binomial Pricing 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:

13. 21-13 Binomial Pricing 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

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

15. 21-15 Example Price = 36  =.40 t = 90/365  t = 30/365 Strike = 40r = 10% a = 1.0083 u = 1.1215 d =.8917 Pu =.5075 Pd =.4925 Binomial Pricing

16. 21-16 40.37 32.10 36 Binomial Pricing

17. 21-17 40.37 32.10 36 Binomial Pricing

18. 21-18 50.78 = price 40.37 32.10 25.52 45.28 36 28.62 40.37 32.10 36 Binomial Pricing

19. 21-19 50.78 = price 10.78 = intrinsic value 40.37.37 32.10 0 25.52 0 45.28 36 28.62 36 40.37 32.10 Binomial Pricing

20. 21-20 50.78 = price 10.78 = intrinsic value 40.37.37 32.10 0 25.52 0 45.28 5.60 36 28.62 40.37 32.10 36 The greater of Binomial Pricing

21. 21-21 50.78 = price 10.78 = intrinsic value 40.37.37 32.10 0 25.52 0 45.28 5.60 36.19 28.62 0 40.37 2.91 32.10.10 36 1.51 Binomial Pricing

22. 21-22 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.

23. 21-23 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)

24. 21-24 Option Value Black-Scholes Option Pricing Model

25. 21-25 O C - 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 Black-Scholes Option Pricing Model

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

27. 21-27 Cumulative Normal Density Function

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

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

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

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

32. 21-32 Call Option Example What is the price of a call option given the following? P = 36r = 10%v =.40 EX = 40t = 90 days / 365

33. 21-33 Call Option Example What is the price of a call option given the following? P = 36r = 10%v =.40 EX = 40t = 90 days / 365

34. 21-34 Black-Scholes Option Pricing Model

35. 21-35 Call Option Example What is the price of a call option given the following? P = 36r = 10%v =.40 EX = 40t = 90 days / 365

36. 21-36 Black Scholes Comparisons

37. 21-37 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

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

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

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

41. 21-41 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

42. 21-42 How estimated call price changes as number of binomial steps increases No. of stepsEstimated value 148.1 241.0 342.1 541.8 1041.4 5040.3 10040.6 Black-Scholes40.5 Binomial vs. Black Scholes

43. 21-43 Dilution

44. 21-44 Web Resources Click to access web sites Internet connection required www.numa.com www.math.columbia.edu/~smirnov/options13.html www.optionscentral.com www.pmpublishing.com www.schaffersresearch.com/streetools/options/option_tools.aspx?click=jumpto