Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.1 Introduction to Binomial Trees Chapter 9

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.2 A Simple Binomial Model A stock price is currently $20 In three months it will be either $22 or $18 Stock Price = $22 Stock Price = $18 Stock price = $20

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.3 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price=? A Call Option ( Figure 9.1, page 202) A 3-month call option on the stock has a strike price of 21.

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.4 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price = $.633 The Answer A 3-month call option on the stock has a strike price of 21. T = 3 months r = 12%

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.5 Stock Price = $22 Stock Price = $18 Stock price = $20 Aside About Probability, #1 If time value of money were not involved, then the stock price would make you think about a probability:

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.6 Stock Price = $22 Stock Price = $18 Stock price = $20 Aside About Probability, #2 If time value of money were not involved, then the stock price would make you think about a probability:.5

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.7 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price = $.50 Aside About Probability, #3 Because of the time value of money, this is wrong. Instead of $.50, the option is worth $

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.8 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price = $.633 The Answer Again A 3-month call option on the stock has a strike price of 21. T = 3 months r = 12% The answer is $.633, not $.50. This answer has nothing to do with probability. It’s an arbitrage price!!!

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.9 Consider the Portfolio:long  shares short 1 call option Portfolio is riskless when 22  – 1 = 18  or  =  – 1 18  Setting Up a Riskless Portfolio

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.10 Riskless portfolio: long.25 shares short 1 call option FINAL VALUE By holding 0.25 shares, you are hedging the risk of the option. 22x.25  – 1 = x.25 = 4.5 Riskless Portfolio

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.11 Riskless portfolio: long.25 shares short 1 call option The difference for the share is = 4. The difference for the option is 1-0 = 1.  = (change in option)/(change in share) =.25 22x.25  – 1 = x.25 = 4.5 How Much to Hedge?

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.12 Try 23 and 18 instead of 22 and 18. What is delta now? Riskless portfolio: long.2 shares short 1 call option  = (change in option)/(change in share) =.2 23x.2  – 1 = x.2 = 3.6 Another Value for Delta

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.13 Valuing the Portfolio ( Risk-Free Rate is 12% ) The riskless portfolio is: long 0.25 shares short 1 call option The value of the portfolio in 3 months is 22  0.25 – 1 = 4.50 The value of the portfolio today is 4.5e – 0.12  0.25 =

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.14 Riskless portfolio: long.25 shares short 1 call option FINAL VALUE PRESENT VALUE The present value of the portfolio is $ What is the option worth? 22x.25  – 1 = x.25 = 4.5 Present Value of Riskless Portfolio

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.15 Valuing the Portfolio ( Risk-Free Rate is 12% ) The riskless portfolio is: long 0.25 shares short 1 call option The value of the portfolio in 3 months is 22  0.25 – 1 = 4.50 The value of the portfolio today is 4.5e – 0.12  0.25 =

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.16 Valuing the Option The portfolio that is long 0.25 shares short 1 option is worth The value of the shares is (= 0.25  20 ) The value of the option is therefore (= – )

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.17 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price = $.633 The Answer Again A 3-month call option on the stock has a strike price of 21. T = 3 months r = 12% The answer is $.633, not $.50. This answer has nothing to do with probability. It’s an arbitrage price!!!

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.18 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price = $.633 But can we pretend there is a probability? The answer is $.633, not $.50. It’s an arbitrage price!!! But is there some probability that would give the same answer?

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.19 Yes! Try p =.6523 The value of the option is e – 0.12  0.25 [   0] = Su = 22 ƒ u = 1 Sd = 18 ƒ d = 0 SƒSƒ

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.20 Stock Price = $22 Stock Price = $18 Stock price = $20 Aside About Probability, #4 What are the probabilities now? Could be.4,.2,.4. Could be.3,.4,.3. Stock Price = $20

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.21 Stock Price = $22 Stock Price = $18 Stock price = $20 Aside About Probability, #5 What are the probabilities now? Could be.4,.2,.4. Could be.3,.4,.3. Stock Price = $20.4.2

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.22 Generalization (Figure 9.2, page 203) A derivative lasts for time T and is dependent on a stock Su ƒ u Sd ƒ d SƒSƒ

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.23 Generalization (continued) Consider the portfolio that is long  shares and short 1 derivative The portfolio is riskless when Su  – ƒ u = Sd  – ƒ d or Su  – ƒ u Sd  – ƒ d

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.24 Generalization (continued) Value of the portfolio at time T is Su  – ƒ u Value of the portfolio today is (Su  – ƒ u )e –rT Another expression for the portfolio value today is S  – f ƒ Hence ƒ = S  – ( Su  – ƒ u )e –rT

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.25 Generalization (continued) Substituting for  we obtain ƒ = [ p ƒ u + (1 – p )ƒ d ]e –rT where

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.26 Risk-Neutral Valuation ƒ = [ p ƒ u + (1 – p )ƒ d ]e -rT The variables p and (1  – p ) can be interpreted as the risk-neutral probabilities of up and down movements The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate Su ƒ u Sd ƒ d SƒSƒ p (1  – p )

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.27 Irrelevance of Stock’s Expected Return When we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevant

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.28 Original Example Revisited Since p is a risk-neutral probability 20e 0.12  0.25 = 22p + 18(1 – p ); p = Alternatively, we can use the formula Su = 22 ƒ u = 1 Sd = 18 ƒ d = 0 S ƒS ƒ p (1  – p )

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.29 Valuing the Option The value of the option is e – 0.12  0.25 [   0] = Su = 22 ƒ u = 1 Sd = 18 ƒ d = 0 SƒSƒ

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.30 A Two-Step Example Figure 9.3, page 207 Each time step is 3 months

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.31 Valuing a Call Option Figure 9.4, page 207 Value at node B = e –0.12  0.25 (   0) = Value at node A = e –0.12  0.25 (   0) = A B C D E F

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.32 A Put Option Example; X=52 Figure 9.7, page A B C D E F

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.33 What Happens When an Option is American (Figure 9.8, page 211) A B C D E F

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.34 Delta Delta (  ) is the ratio of the change in the price of a stock option to the change in the price of the underlying stock The value of  varies from node to node

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 9.35 Choosing u and d One way of matching the volatility is to set where  is the volatility and  t is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein