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 Portfolio is riskless when 22D – 1 = 18D => D = 0.25 A riskless portfolio is therefore=> Long : 025 shares Short : 1 call option 22D – 1 18D

The riskless portfolio is: long 0.25 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%) 4.5e – 0.12×0.25 = 4.367

the portfolio today is Stock price today = $20 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 S0 = stock price u= percentage increase in the stock price d= percentage decrease in 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 S0u ƒu S0d ƒd S0 ƒ

Generalization (continued) Consider the portfolio that is long D shares and short 1 call option The portfolio is riskless when S0uD – ƒu = S0dD – ƒd or S0uD – ƒu S0dD – ƒd

Generalization (continued) Value of the portfolio at time T is (S0uD – ƒu)e–rT The cost of setting up the portfolio is S0D – f Hence S0D – ƒ = (S0uD – ƒu )e–rT ƒ = S0D – (S0uD – ƒu )e–rT Substituting for we obtain ƒ = [ pƒu + (1 – p)ƒd ]e–rT ---(11.2) where Delta值算出來後，我們就來看看投組在第T期的現值為(S0uD – ƒu)e–rT ，投組成本為S0D – f， 這兩式相等可得 ƒ = S0D – (S0uD – ƒu )e–rT，接下來再把

Generalization (continued) Ex. (see Figure11.1) u=1.1, d=0.9,r=0.12,T=0.25,fu=1, ƒd=0 ƒ = [ pƒu + (1 – p)ƒd ]e–rT = [ 0.6523×1 + 0.3477×0 ]e–0.12×0.25 = 0.633 接下來有個例題大家可以看一下，我們可以先把P算出來，再把所有值代入一般式中，就可以求得答案。

The option pricing formula in equation(11 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. 在11.2式的選擇權公式中並沒有牽涉到股價上漲或下跌的機率 主要原因是因為我們用的不是”獨立“的項目去評價選擇權，而是用”標的股票的價格”去計算選擇權價值， 未來股價上漲下跌的機率已經包含在”股價”裡面

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) = pS0u + (1-p) S0d = pS0 (u-d) + S0d substituting => E(ST)=S0erT 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 movement equal to p is therefore equivalent to assuming that the return on the stock equals the risk-free rate. 雖然推導(11.2)式並不需對股價上漲或下跌的機率做任何假設，但一般會將(11.2)式中的變數 p 解釋為股價上漲的機率，1-p 為股價下跌的機率。 根據P及1-P我們可以算出選擇權的預期報酬 然後在T期的時候，股票預期報酬的推導如下 由上述式子我們可以看出股價平均而言是以無風險利率成長，因為設定股價上漲的機率為P就相當於是假設股票的報酬率為無風險利率

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 * European 3-month call option *Rf=12% 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)=S0erT => 22p + 18(1 – p ) = 20e0.12 ×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 value is Expected option payoff = p × ƒu + (1 – p ) × ƒd 0.6523×1 + 0.3477×0 = 0.6523 In a risk-neutral world this should be discounted at the risk-free rate. The value of the option today is 0.6523e–0.12×0.25= 0.633 S0u = 22 ƒu = 1 S0d = 18 ƒd = 0 S0=20 ƒ p (1 – p )

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 risk-free 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) 20 22 18 24.2 19.8 16.2

Valuing a Call Option (Figure 11.4) p= 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 *ƒ = [ pƒu + (1 – p)ƒd ]e–rT ---(11.2) where 24.2 D 3.2=max{24.2-21,0} 22 B 20 1.2823 2.0257 19.8 A E 0=max{19.8-21,0} 18 C 16.2 F 0=max{16.2-21,0}

Generalization S0u2 ƒuu S0u ƒu S0ud S0 ƒud ƒ S0d ƒd S0d2 ƒdd Figure11.6 Stock and option prices in general two-step tree S0u2 ƒuu S0u ƒu S0ud ƒud S0 ƒ S0d ƒd S0d2 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–2rDt[ p2ƒuu +2p (1 – p)ƒud + (1 – p)2ƒdd ]

A Put Example (Figure 11.7) K = 52, duration = 2yr, current price = $50 u=20%, d=20%, r = 5% 50 4.1923 60 40 72 48 32 1.4147 9.4636 A B C D E F 0=max{52-72,0} 4=max{52-48,0} 20=max{52-32,0} ƒ = e–2rDt[ p2ƒuu +2p (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 = 2yr, current price = $50,u=20%, d=20%, r = 5% 50 5.0894 60 40 72 48 32 1.4147 12.0 A B C D E F 0=max{52-72,0} max{1.4147,52-60} 4=max{52-48,0} max{5.0894,52-50} max{9.4636,52-40} 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 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 delta changes over time

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 S0e(r-q) Dt pS0u+(1-p)S0d=S0e(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 0.25 0.5 Node time: 0 0.25 0.5 Node time: 0 0.25 0.5 810 53.39 895.19 100.66 732.92 5.06 989.34 189.34 810.00 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 0.0833 0.1667 0.25 0.61 0.019 0.632 0.033 0.589 0.007 0.654 0.054 0.015 0.569 0.00 0.032 0.677 0.077 0.550

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 0.25 0.5 0.75 31 2.84 36.02 0.93 26.68 4.54 41.85 1.76 22.97 7.03 3.32 48.62 19.77 10.23