Chapter 11 Binomial Trees Outline A OneStep Binomial

Chapter 11 Binomial Trees

Outline A One-Step Binomial Model Risk-Neutral Valuation Two-Step Binomial Model American Options Delta Matching Volatility With u and d Options On Other Assets

Binomial Trees Binomial Tree representing different possible paths that might be followed by the stock price over the life of an option In each time step, it has a certain probability of moving up by a certain percentage amount and a certain probability of moving down by a certain percentage amount

A one-step binomial model A Simple Binomial Model A 3 -month call option on the stock has a strike price of 21. A stock price is currently $20 In three months it will be either $22 or $18 Stock Price = $22 Option Price = $1 Stock price = $20 Stock Price = $18 Option Price = $0

Setting Up a Riskless Portfolio Consider the Portfolio: long D shares short 1 call option 22 D – 1 18 D Portfolio is riskless when 22 D – 1 = 18 D => D = 0. 25 A riskless portfolio is therefore=> Long : 025 shares Short : 1 call option

Valuing the Portfolio 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. 5 or 18 × 0. 25=4. 5 The value of the portfolio today is (if Rf=12%) 4. 5 e – 0. 12× 0. 25 = 4. 367

Valuing the Option Stock price today = $20 Suppose the option price = f the portfolio today is 0. 25 × 20 – f = 5 – f It follows that 5 – f =4. 367 So f=0. 633 ---- the current value of option

Generalization S 0 = stock price S 0 u u= percentage increase in ƒu S 0 the stock price S 0 d d= percentage decrease in ƒ ƒd the stock price ƒ= option on stock price whose current price ƒu = payoff from the option(when price moves up) ƒd= payoff from the option(when price moves down) T= the duration of the option

Generalization (continued) Consider the portfolio that is long D shares and short 1 call option S 0 u. D – ƒu S 0 d. D – ƒd The portfolio is riskless when S 0 u. D – ƒu = S 0 d. D – ƒd or

Generalization (continued) Value of the portfolio at time T is (S 0 u. D – ƒu)e–r. T The cost of setting up the portfolio is S 0 D – f Hence S 0 D – ƒ = (S 0 u. D – ƒu )e–r. T ƒ = S 0 D – (S 0 u. D – ƒu )e–r. T Substituting for we obtain

Generalization (continued) Ex. (see Figure 11. 1) u=1. 1, d=0. 9, r=0. 12, T=0. 25, fu=1, ƒd=0 ƒ = [ pƒu + (1 – p)ƒd ]e–r. T = [ 0. 6523× 1 + 0. 3477× 0 ]e– 0. 12× 0. 25 = 0. 633

The option pricing formula in equation(11. 2) does not involve the probabilities of stock price moving up or down. The key reason is that we are not valuing the option in absolute terms. We are calculating its value in terms of the price of the underlying stock. The probabilities of future up or down movements are already incorporated into the stock price.

Risk-Neutral Valuation We assume p and 1 -p as probabilities of up and down movements. Expected option payoff = p × ƒu + (1 – p ) × ƒd The expected stock price at time T is E(ST) = p. S 0 u + (1 -p) S 0 d = p. S 0 (u-d) + S 0 d substituting => E(ST)=S 0 er. T From this equation, we can see that the stock price grows on average at the risk-free rate. Because setting the probability of the up

Risk-Neutral Valuation (continued) In a risk-neutral world all individuals are indifferent to risk. In such a world , investors require no compensation for risk, and the expected return on all securities is the risk-free interest rate. Risk-neutral valuation states that we can with complete impunity assume the world is risk neutral when pricing options.

Original Example Revisited p * European 3 -month call option *Rf=12% S 0=20 ƒ (1 – p) S 0 u = 22 ƒu = 1 S 0 d = 18 ƒd = 0 Since p is the probability that gives a return on the stock equal to the risk-free rate. We can find it from E(ST)=S 0 er. T => 22 p + 18(1 – p ) = 20 e 0. 12 × 0. 25 => p = 0. 6523 At the end of the three months, the call option has a 0. 6523 probability of being worth 1 and a 0. 3477 probability of being worth zero. So the expect

Real world compare with Risk-Neutral world It is not easy to know the correct discount rate to apply to the expected payoff in the real world. Using risk-neutral valuation can solve this problem because we know that in a risk-neutral world the expected return on all assets is the riskfree rate.

Two-Step Binomial Model Stock price=$20 , u=10% , d=10% Each time step is 3 months r=12%, K=21 (Figure 11. 3 Stock prices in a two-step tree) 22 24. 2 19. 8 20 18 16. 2

Valuing a Call Option (Figure 11. 4) D 22 20 1. 2823 A p= B 2. 0257 18 24. 2 3. 2=max{24. 2 -21, 0} E 19. 8 0=max{19. 8 -21, 0} F 16. 2 0=max{16. 2 -21, 0} C 0 Value at node B = e– 0. 12× 0. 25(0. 6523× 3. 2 + 0. 3477× 0) = 2. 0257 Value at node A = e– 0. 12× 0. 25(0. 6523× 2. 0257 + 0. 3477× 0) = 1. 2823

Generalization Figure 11. 6 Stock and option prices in general two-step tree S 0 ƒ S 0 u ƒu S 0 d ƒd S 0 u 2 ƒuu S 0 ud ƒud S 0 d 2 ƒdd

Generalization (continued) *The length of time step is Dt years ƒ = e–r Dt[ pƒu + (1 – p)ƒd ]-------(1) (11. 2) (11. 3) ƒu = e–r Dt[ pƒuu + (1 – p)ƒud ]------(2) ƒd= e–r Dt[ pƒud + (1 – p)ƒdd ]------(3) ƒ = e– 2 r. Dt[ p 2ƒuu +2 p (1 – p)ƒud + (1 – p)2ƒ ]

A Put Example (Figure 11. 7) K = 52, duration = 2 yr, current price = $50 72 u=20%, d=20%, r = 5% D 60 50 4. 1923 A B 1. 4147 40 0=max{52 -72, 0} 48 E 4=max{52 -48, 0} C 9. 4636 F 32 20=max{52 -32, 0} ƒ = e– 2 r. Dt[ p 2ƒuu +2 p (1 – p)ƒud + (1 – p)2ƒdd ] = e– 2*0. 05*1 [ 0. 62822 × 0 + 2× 0. 6282(1 – 0. 6282) × 4 + (1 – 0. 6282)2× 20] = 4. 1923

American Options American options can be valued using a binomial tree The procedure is to work back through the tree from the end to the beginning, testing at each node to see whether early exercise is optimal

American Options (Figure 11. 8) American Put option K = 52, duration = 2 yr, current price = $50, u=20%, 72 d=20%, r = 5% D 0=max{52 -72, 0} 60 B 50 5. 0894 max{5. 0894, 52 -50} A 1. 4147 max{1. 4147, 52 -60} 40 48 4=max{52 -48, 0} E C 12. 0 max{9. 4636, 52 -40} F 32 20=max{52 -32, 0} Value at node B = e–-0. 05× 1(0. 6282× 0 +0. 3718× 4)=1. 4147 Value at node C = e–-0. 05× 1(0. 6282× 4+ 0. 3718× 20)=9. 4636 Value at node A = e–-0. 05× 1(0. 6282× 1. 4147 +0. 3718× 12)=5. 0894

Delta (D) is an important parameter in the pricing and hedging of option. The delta (D) of stock option =

Delta (Figure 11. 7) Delta At the end of the first time step is At the end of the second time step is either The two-step examples show that

Matching Volatility With u and d In practice, when constructing a binomial tree to represent the movements in a stock price. We choose the parameters u and d to match the volatility of the stock price. s = volatility Dt = the length of the time step This is the approach used by Cox, Ross, and Rubinstein

Options On Other Assets Option on stocks paying a continuous dividend yield Dividend yield at rate = q Total return from dividends and capital gains in a risk-neutral world = r. => Capital gains return = r-q The stock expected value after one time step of length Dt is S 0 e(r-q) Dt p. S 0 u+(1 -p)S 0 d=S 0 e(r-q) Dt =>

Options On Other Assets

Options On Other Assets ) Option on stock indices ( a= e(r-q) Dt European 6 -month call option on an index level when index level is 810, K=800, rf=5%, σ=20%, q=2% Node time: 0 810 53. 39 0. 25 895. 19 100. 66 732. 92 5. 06 0. 5 989. 34 189. 34 810. 00 10 663. 17 0. 00

Options On Other Assets Option on currencies ( a= e(r-rf) Dt ) Three-step tree: American 3 -month call. when the value of the currency is 0. 61, K=0. 6, rf=5%, σ=20%, foreign rf=7% 0 0. 61 0. 019 0. 0833 0. 632 0. 033 0. 589 0. 007 0. 1667 0. 654 0. 054 0. 61 0. 015 0. 569 0. 00 0. 25 0. 677 0. 077 0. 632 0. 032 0. 589 0. 00 0. 550 0. 00

Options On Other Assets Option on futures ) ( a= 1 Three-step tree: American 9 -month put. when the futures price is 31, K=30, rf=5%, σ=30% 0 31 2. 84 0. 25 36. 02 0. 93 26. 68 4. 54 0. 5 41. 85 0 31 1. 76 22. 97 7. 03 0. 75 48. 62 0 36. 02 0 26. 68 3. 32 19. 77 10. 23