The Multistep Binomial Model

Transcription

1 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 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 Given current stock price S 0 at t = 0, what possible values of the stock price at time t in this model? Solution

2 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 If there are n steps in the tree, how many possible stock prices can we observe (at all times)? Solution

3 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 Sketch the binomial tree for a stock price with N = 4. Solution See figure 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

4 MATH20912 Lecture 10 Figure 10.1: A sketch of the 4 step stock binomial tree 91

5 MATH20912 Lecture Binomial Call Option Tree Example Sketch the binomial tree for a call option price with N = 4. Solution See figure 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 Why not use arbitrage arguments? Solution

6 MATH20912 Lecture 10 Figure 10.2: A sketch of the 4 step call option binomial tree 93

7 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 Combine these two results for the first equation needed to match the models. Solution

8 Variance of the stock price For the Continuous model we have: var [ ] S = σ 2 t S Example Derive the variance for the binomial tree and hence the second equation. Solution

9 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