McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance Chapter Ten Introduction to Binomial Trees

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Chapter Outline 10.1 A one-step binomial model 10.2 Risk Neutral Valuation 10.3 Two-step binomial trees 10.4 A put example 10.5 American Options 10.6 Delta 10.7 Matching volatilities with u and d 10.8 Binomial Trees in Practice

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Prospectus: The last chapter concerned itself with the value of an option at expiry. This section considers the value of an option prior to the expiration date. A much more interesting question.

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 An Option ‑ Pricing Formula We will start with a binomial option pricing formula to build our intuition. Then we will graduate to the normal approximation to the binomial for some real-world option valuation.

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Binomial Option Pricing Model Suppose a stock is worth $25 today and in one period will either be worth $28.75 or $ The risk-free rate is 5%. What is the value of an at-the- money call option? $25 $21.25 $28.75 S1S1 S0S0

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Binomial Option Pricing Model 1.A call option on this stock with exercise price of $25 will have the following payoffs. 2.We can replicate the payoffs of the call option. With a levered position in the stock. $25 $21.25 $28.75 S1S1 S0S0 c1c1 $3.75 $0

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Binomial Option Pricing Model Borrow the present value of $21.25 today and buy 1 share. The net payoff for this levered equity portfolio in one period is either $7.50 or $0. The levered equity portfolio has twice the option’s payoff so the portfolio is worth twice the call option value. $25 $21.25 $28.75 S1S1 S0S0 debt - $21.25 portfolio $7.50 $0 ( - ) = = = c1c1 $3.75 $0 - $21.25

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Binomial Option Pricing Model The levered equity portfolio value today is today’s value of one share less the present value of a $21.25 debt: $25 $21.25 $28.75 S1S1 S0S0 debt - $21.25 portfolio $7.50 $0 ( - ) = = = c1c1 $3.75 $0 - $21.25

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Binomial Option Pricing Model We can value the option today as half of the value of the levered equity portfolio: $25 $21.25 $28.75 S1S1 S0S0 debt - $21.25 portfolio $7.50 $0 ( - ) = = = c1c1 $3.75 $0 - $21.25

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 If the interest rate is 5%, the call is worth: The Binomial Option Pricing Model $25 $21.25 $28.75 S1S1 S0S0 debt - $21.25 portfolio $7.50 $0 ( - ) = = = c1c1 $3.75 $0 - $21.25 $2.39 c0c0

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Binomial Option Pricing Model the replicating portfolio intuition. Many derivative securities can be valued by valuing portfolios of primitive securities when those portfolios have the same payoffs as the derivative securities. The most important lesson (so far) from the binomial option pricing model is:

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Delta and the Hedge Ratio In the example just previous, we replicated the payoffs of the call option with a levered equity portfolio. This has everything to do with anything for the rest of the semester, so let’s take a minute to wrap our brains around it now rather than later. The delta of a stock option is the ratio of change in the price of the option to the change in the price of the underlying asset: The delta is the number of units of stock we should hold for each option shorted in order to create a riskless hedge.

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Delta and the Hedge Ratio This practice of the construction of a riskless hedge is called delta hedging. The delta of a call option is positive. –Recall from the example: The delta of a put option is negative. Deltas change through time. -This is a feature of options that we will return to in chapter 14

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 The Risk-Neutral Approach to Valuation We could value f as the value of the replicating portfolio. An equivalent method is risk-neutral valuation S0fS0f p 1- p S0ufuS0ufu S0dfdS0dfd

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 The Risk-Neutral Approach to Valuation S 0 is the value of the underlying asset today. S 0 u and S 0 d are the values of the asset in the next period following an up move and a down move, respectively. f u and f d are the values of the derivative asset in the next period following an up move and a down move, respectively. p is the risk-neutral probability of an “up” move. S0fS0f p 1- p S0ufuS0ufu S0dfdS0dfd

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 The Risk-Neutral Approach to Valuation The key to finding p is to note that it is already impounded into an observable security price: the value of S 0 : A minor bit of algebra yields: S0fS0f p 1- p S0ufuS0ufu S0dfdS0dfd

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Example of the Risk-Neutral Valuation of a Call: Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. (u = 1.15; d = 0.85) The risk-free rate is 5%. What is the value of an at-the-money call option? The binomial tree would look like this: $21.25 C d p 1- p $25.00 c 0 $28.75 C u

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Example of the Risk-Neutral Valuation of a Call: The next step would be to compute the risk neutral probabilities $21.25 C d $25.00 c 0 $28.75 C u

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Example of the Risk-Neutral Valuation of a Call: After that, find the value of the call in the up state and down state. $21.25 $0 $25.00 c 0 $28.75 $3.75

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 This risk-neutral result is consistent with valuing the call using a replicating portfolio. Risk-Neutral Valuation and the Replicating Portfolio

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 More on the Binomial Model The binomial option pricing model is an alternative to the Black-Scholes option pricing model— especially given the computational efficiency of spreadsheets such as Excel. In some situations, it is a superior alternative. For example if you have path dependency in your option payoff, you must use the binomial option pricing model. –Path dependency occurs when how you arrive at a price (the path you follow) for the underlying asset is important. –One example of a path dependent security is a “no regret” call option where the exercise price is the lowest price of the stock during the option life.

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance Period Binomial Option Pricing Example There is no reason to stop with just two periods. Find the value of a three-period at-the-money call option written on a $25 stock that can go up or down 15 percent each period when the risk-free rate is 5 percent.

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Three Period Binomial Process: Stock Prices $ /3 1/ /3 1/ /3 1/ /3 1/ /3 1/ /3 1/

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 2/3 $ /3 1/ /3 1/ /3 1/ /3 1/ /3 1/ / Three Period Binomial Process: Call Option Prices

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Valuation of a Lookback Option When the stock price falls due to the stock market as a whole falling, the board of directors tends to reset the exercise price of executive stock options. To see how this reset provision adds value, let’s price that same three-period call option (exercise price initially $25) with a reset provision. Notice that the exercise price of the call will be the smallest value of the stock price depending upon the path followed by the stock price to get there.

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Three Period Binomial Process: Lookback Call Option Prices $

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Three Period Binomial Process: Lookback Call Option Prices $ $15.35 $38.02 $20.77 $ $ $20.77 $13.02 $3.10 $6.85 $3.66 $0 $2.71 $0

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Three Period Binomial Process: Lookback Call Option Prices $ $15.35 $38.02 $20.77 $ $28.10 $20.77 $13.02 $3.10 $6.85 $3.66 $0 $2.71 $0

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Three Period Binomial Process: Lookback Call Option Prices $ $15.35 $0 $38.02 $13.02 $20.77 $0 $28.10 $3.10 $28.10 $3.66 $28.10 $6.85 $20.77 $2.71 $20.77 $

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance A put example At the money. Before we start, we expect value less than $5.25 $ /3 1/ /3 1/ /3 1/ /3 1/ /3 1/ /3 1/

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 2/3 $ /3 1/ /3 1/ /3 1/ /3 1/ /3 1/ / A put example

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance A put example We can check our work with put-call parity:

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance American Options At each node prior to expiry, compare immediate exercise with the option’s value. If the proceeds of immediate exercise are higher than the value of the option, exercise. Use the exercise value at that node to work backward through the tree to find the value of an American option at time 0.

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Optimal Early Exercise: American Put 2/3 $ /3 1/ /3 1/ /3 1/ /3 1/ /3 1/ /

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Optimal Exercise of American Calls There are two cases to consider: –A stock paying a known dividend yield –The dollar amount of the dividend is known.

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Known Dividend Yield Ex-dividend date S0S0 S 0 u S 0 d S 0 u 2 S0S0 S 0 d 2 S 0 u 3 (1-  ) S 0 u(1-  ) S 0 d(1-  ) S 0 d 3 (1-  ) Ex-dividend date S 0 u 2 (1-  ) S 0 (1-  ) S 0 d 2 (1-  )

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance 457 Known Dollar Dividend S0S0 S 0 u S 0 d S 0 u 2 S0S0 S 0 d 2 Ex-dividend date S 0 u 2 – D S 0 – D S 0 d 2 – D (S 0 u 2 – D) u (S 0 – D) u (S 0 d 2 – D)u (S 0 d 2 – D)d (S 0 u 2 – D) d (S 0 – D) d

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance Matching Volatility with u and d In practice, we choose the parameters u and d to match the volatility of the stock price.

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved Finance Binomial Trees in Practice The BOPM is easily incorporated into Excel spreadsheets After 30 or so steps, the results are excellent.