1 Introduction to Binomial Trees Chapter 11

2 A Simple Binomial Model A stock price is currently $20 A stock price is currently $20 In three months it will be either $22 or $18 In three months it will be either $22 or $18 Stock Price = $22 Stock Price = $18 Stock price = $20

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

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

5 Valuing the Portfolio (Risk-Free Rate is 12%) The riskless portfolio is: 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 in 3 months is 22  0.25 – 1 = 4.50 The value of the portfolio today is 4.5 e – 0.12  0.25 = The value of the portfolio today is 4.5 e – 0.12  0.25 =

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

7 Generalization A derivative lasts for time T and is dependent on a stock A derivative lasts for time T and is dependent on a stock S 0 u ƒ u S 0 d ƒ d S0ƒS0ƒ

8 Generalization (continued) Consider the portfolio that is long  shares and short 1 derivative Consider the portfolio that is long  shares and short 1 derivative The portfolio is riskless when S 0 u  – ƒ u = S 0 d  – ƒ d or The portfolio is riskless when S 0 u  – ƒ u = S 0 d  – ƒ d or S 0 u  – ƒ u S 0 d  – ƒ d  S 0 – f

9 Generalization (continued) Value of the portfolio at time T is S 0 u  – ƒ u Value of the portfolio at time T is S 0 u  – ƒ u Value of the portfolio today is (S 0 u  – ƒ u )e –rT Value of the portfolio today is (S 0 u  – ƒ u )e –rT Another expression for the portfolio value today is S 0  – f Another expression for the portfolio value today is S 0  – f Hence ƒ = S 0  – ( S 0 u  – ƒ u )e –rT Hence ƒ = S 0  – ( S 0 u  – ƒ u )e –rT

10 Generalization (continued) Substituting for  we obtain Substituting for  we obtain ƒ = [ p ƒ u + (1 – p )ƒ d ]e –rT where

11 Risk-Neutral Valuation ƒ = [ p ƒ u + (1 – p )ƒ d ]e -rT 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 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 The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate S0u ƒuS0u ƒu S0d ƒdS0d ƒd S0ƒS0ƒ p (1  – p )

12 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

13 Original Example Revisited Since p is a risk-neutral probability: Since p is a risk-neutral probability: 20 = [ 22p + 18(1 – p )] e  0.25 ; Solve for p: p = Alternatively, we can use the formula Alternatively, we can use the formula S 0 u = 22 ƒ u = 1 S 0 d = 18 ƒ d = 0 S0 ƒS0 ƒ p (1  – p )

14 Valuing the Option The value of the option is e –0.12  0.25 [   0] = = S 0 u = 22 ƒ u = 1 S 0 d = 18 ƒ d = 0 S0ƒS0ƒ

15 Risk-Neutral vs Real World In risk-neutral world anything earns the risk-free rate of return and the probabilities of price movements are different from those in the real world. In risk-neutral world anything earns the risk-free rate of return and the probabilities of price movements are different from those in the real world. Assume the stock from previous slide earns an expected return µ = 16%. Then, by definition: Assume the stock from previous slide earns an expected return µ = 16%. Then, by definition: Real world probability

16 Risk-Neutral vs Real World In the case of the call option it means that the expected option payoff is $ and, hence, the expected option return (implied from that of the stock) is 42.58%. Here is the algebra: In the case of the call option it means that the expected option payoff is $ and, hence, the expected option return (implied from that of the stock) is 42.58%. Here is the algebra:

17 A Two-Step Example Each time step is 3 months Each time step is 3 months

18 Valuing a Call Option Value at node B = e –0.12  0.25 (   0) = Value at node B = e –0.12  0.25 (   0) = Value at node A = e –0.12  0.25 (   0) Value at node A = e –0.12  0.25 (   0) = = A B C D E F

19 A Put Option Example; K=52 r =.05, T=2 years A B C D E F

20 What Happens When the Put from Previous Slide is American A B C D E F At each node choose either the continuation value or the exercise value, whichever is higher.

21 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 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 The value of  varies from node to node Think of  as the number of shares that one needs to sell short to hedge one long call option. The change in the value of this portfolio from node to node is zero: Think of  as the number of shares that one needs to sell short to hedge one long call option. The change in the value of this portfolio from node to node is zero:  C -  S = (1-0) – 0.25(22-18) = 0  C -  S = (1-0) – 0.25(22-18) = 0 Delta changes over time which gives rise to dynamic hedging strategies Delta changes over time which gives rise to dynamic hedging strategies

22 Computation of Delta In practice, compute delta as follows: In practice, compute delta as follows: Delta is then: Delta is then: Call Delta is positive. What about puts? Call Delta is positive. What about puts? S u =25, f u =5 S d =15, f d =0

23 Choosing u and d in practice 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

24 Example A stock price is currently $25. The standard deviation of the stock return is 20% per year and the risk-free rate is 10% per year (continuous). What is the value of the derivative that pays $ in  t=T=2 months? (We’ll use one period tree for simplicity) A stock price is currently $25. The standard deviation of the stock return is 20% per year and the risk-free rate is 10% per year (continuous). What is the value of the derivative that pays $ in  t=T=2 months? (We’ll use one period tree for simplicity)

25 Value a Forward Given the data on the previous slide value a two-month forward on the stock. Delivery price K is $20. Given the data on the previous slide value a two-month forward on the stock. Delivery price K is $20. From before you remember that From before you remember that f 0 = S 0 – Ke -rT = 25 – 20e -0.1x2/12 = 5.33 f 0 = S 0 – Ke -rT = 25 – 20e -0.1x2/12 = 5.33 Using risk neutral valuation: Using risk neutral valuation: f u = S u – K = – 20 = f u = S u – K = – 20 = f d = S d – K = – 20 = f d = S d – K = – 20 = f 0 = [p f u + (1-p) f d )] e -rT = f 0 = [p f u + (1-p) f d )] e -rT = = [0.5824x x3.040]e -0.1x2/12 = 5.33