An Introduction to Binomial Trees Chapter 11

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 = $20 Stock Price = $18

A Call Option A 3-month call option on the stock has a strike price of 21. Figure 11.1 (p. 244) Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=? Stock Price = $18 Option Price = $0

Setting Up a Riskless Portfolio Consider the Portfolio: long  shares short 1 call option Portfolio is riskless when 22– 1 = 18 or  = 0.25 22– 1x1 18

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 = 4.3670

Valuing the Option The portfolio that is long 0.25 shares short 1 option is worth 4.367 The value of the shares is 5.000 (= 0.2520 ) The value of the option is therefore 0.633 (= 5.000 – 4.367 )

Generalization Stock goes up by u-1 percent or down by d-1 percent. Figure 11.2 (p. 245) Su ƒu Sd ƒd S ƒ

Generalization (continued) Consider the portfolio that is long  shares and short 1 derivative The portfolio is riskless when Su – ƒu = Sd – ƒd or Su – ƒu Sd – ƒd

Generalization (continued) Value of the portfolio at time T is Su – ƒu Value of the portfolio today is (Su – ƒu )e–rT Another expression for the portfolio value today is S – ƒ Hence ƒ = S – (Su – ƒu )e–rT

Generalization (continued) Substituting for  we obtain ƒ = [ p ƒu + (1 – p )ƒd ]e–rT Eqn (11.2) p. 246 where

Generalization – Continued Note: u > 1 and d < 1 Note: d < erT < u (why?) So 0 < p < 1

Risk-Neutral Valuation ƒ = [ p ƒu + (1– p )ƒd ]e-rT Eqn (11.2) The variables p & (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 ƒu Sd ƒd S ƒ p (1– p )

Risk-Neutral Valuation In risk-neutral world (RNW) the expected return on all assets equal the risk-free rate Determine risk-neutral probabilities: p and 1-p Then c = e-rT E[max{0, S-X}] = e-rT [p(S-X) + (1-p)(0)], where expectation is wrt to risk- neutral probabilities

Risk Neutral Valuation Suppose u = 1.1, d = .95, and erT = 1.05 Let S = $1 (it does not matter). Then in RNW E[ST] = S erT: 1.1p + .95(1-p) = 1.05 Or p = (1.05 - .95)/(1.1 - .95) = 2/3 same as eq (10.3) Suppose X = 100, S = 100, and payoff = max{0, ST – 100} Then c = [(2/3)(110-100) + (1/3)(0)](1/1.05) = $6.35

Risk Neutral Valuation: Basic Idea Option value based upon absence of arbitrage profit Arbitrage profit desirable regardless of investors’ risk preferences All investors regardless of risk preferences assign same value to options So in a world of risk neutral investors options would have the same value as in risk-averse world Equal to the present value of expected payoff

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 Why? Because in a risk-neutral world options have the same value as in our risk-averse world. But all assets have an expected return equal to the risk free rate. So expected return does not matter.

Interesting Question Suppose stock A has an expected return of 10% and a risk of 20% Stock B has an expected return of 15% and a risk of 20% How can the call option values be the same? Wouldn’t you rather own option on stock B? In a risk-neutral world both would have an expected return equal to the risk-free rate and a risk of 20%. Therefore, both would have the same value.

Original Example Revisited Su = 22 ƒu = 1 Since p is a risk-neutral probability 20e0.12 0.25 = 22p + 18(1– p ); p = 0.6523 Alternatively, we can use the formula: p S ƒ Sd = 18 ƒd = 0 (1– p )

Valuing the Option Su = 22 ƒu = 1 Sd = 18 ƒd = 0 S ƒ 0.6523 0.3477 The value of the option is e–0.120.25 [0.65231 + 0.34770] = 0.633

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