Lecture 10 The Multistep Binomial Model Reminder: Mid Term Test Friday 9th March - 12pm Examples Sheet 1 4 (not qu 3 or qu 5 on sheet 4) Lectures 1-9 10.1 A Discrete Model for Stock Price Reminder: The continuous model of Stock Price Earlier in Lecture 2 we proposed the continuous random model for the stock price: ds = µsdt + σsdw Example 10.1. Given current stock price S 0 at t = 0, what possible values of the stock price at time t in this model? Solution 10.1. 88
The discrete model of Stock Price The binomial model for the stock price we have described in the last 2 lectures is a discrete time model: The stock price S changes only at discrete times t, 2 t, 3 t,... The price either moves up S Su or down S Sd with d < e r t < u. The probability of up movement is q. Example 10.2. If there are n steps in the tree, how many possible stock prices can we observe (at all times)? Solution 10.2. 89
10.2 Binomial Stock Price Tree So let us build up a tree of possible stock prices. The tree is called a binomial tree, because the stock price will either move up or down at the end of each time period. Each node represents a possible future stock price. We divide the time to expiration T into several time steps of duration t = T/N, where N is the number of time steps in the tree. What we want to do is have the ability to increase N to a large enough number so that the binomial tree approximates the continuous model. Example 10.3. Sketch the binomial tree for a stock price with N = 4. Solution 10.3. See figure 10.1. We introduce the following notations: Sn m is the n-th possible value of stock price at time-step m t. Sn m = u n d m n S0, 0 where n = 0, 1, 2,..., m. S0 0 is the stock price at the time t = 0. Note that u and d are the same at every node in the tree. For example, at the third time-step 3 t, there are four possible stock prices: S0 3 = d 3 S0, 0 S1 3 = ud 2 S0, 0 S2 3 = u 2 ds0 0 and S3 3 = u 3 S0. 0 At the final time-step N t, there are N + 1 possible values of stock price. 90
MATH20912 Lecture 10 Figure 10.1: A sketch of the 4 step stock binomial tree 91
MATH20912 Lecture 10 10.3 Binomial Call Option Tree Example 10.4. Sketch the binomial tree for a call option price with N = 4. Solution 10.4. See figure 10.2. We denote by C m n the n-th possible value of call option at time-step m t. In order to calculate the price of the call option at S = S 0 and t = 0, we must solve recursively just as we did with the two step tree. At each substep in the tree we apply a Risk Neutral Valuation according to the formula C m n = e r t ( pc m+1 n+1 ) + (1 p)cm+1 n. Here 0 n m, 0 m < N and p = er t d u d. Now of course, before we can move recursively through the tree, we need a final condition to apply at t = T. For a call option we have C N n = max ( S N n E, 0 ), where n = 0, 1, 2,..., N and E is the strike price. The current option price C 0 0 is again the expected payoff in a risk-neutral world, discounted at risk-free rate r: C 0 0 = e rt E p [C T ]. Example 10.5. Why not use arbitrage arguments? Solution 10.5. 92
MATH20912 Lecture 10 Figure 10.2: A sketch of the 4 step call option binomial tree 93
10.4 Approximating the Continuous Model If we wish to come up with a binomial model that approximates the continuous model, we need to choose the parameters of the binomial model u, d and p so that they can match the properties of the continuous model, in particular the mean and variance of the model. We assume that the stock price starts at the value S 0 and the time step is t. Let us find the expected stock price, E [S], and the variance of the return, var [ ] S S, for continuous and discrete models. Expected stock price For the continuous model we have E [S] = S 0 e µ t. On the binomial tree: E [S] = qs 0 u + (1 q)s 0 d. Example 10.6. Combine these two results for the first equation needed to match the models. Solution 10.6. 94
Variance of the stock price For the Continuous model we have: var [ ] S = σ 2 t S Example 10.7. Derive the variance for the binomial tree and hence the second equation. Solution 10.7. 95
MATH20912 Lecture 10 What is Left? This gives us two equation for three unknowns, so what to do? In fact we have a free choice, one of the most popular models is the CRR model which imposes u = d 1 so that an up movement followed by a down movement takes you back to where you started. How to solve these equations? The solution to these equation is rather tedious to derive but you can have a go at it in examples sheet 4! 96