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Unformatted text preview: Pricing on binary trees – first steps MF lec 28/10 Suppose that a stock price changes in discrete time steps along a binary tree up to time T . That is, for any price S at time t ≤ T- 1 there are two possible time t + 1 prices which we denote by uS and dS , where d < 1 + r < u (to prevent an arbitrage using the stock alone). The parameters u and d can be different at every node of the tree. There are two equivalent approaches to pricing European options on this stock that look quite different. Risk-neutral model At each node before time T the risk neutral probability ¯ p for the share price to increase satisfies- S + ( u ¯ p + d (1- ¯ p ) ) S 1 + r = 0 ⇐⇒ ¯ p = 1 + r- d u- d ∈ (0 , 1) Given a stock price at each node of the tree we can find the relevant u and d and hence evaluate the risk neutral probabilities. Denote the option values at the subsequent nodes by C u and C d . The option value at the current node must also create a fair bet under the risk neutral probability for the stock price i.e.- C + ¯ pC u + (1- ¯ p ) C d 1 + r = 0 ⇐⇒ C = ¯ pC u + (1- ¯ p ) C d 1 + r At time T we know a ( K,T ) call option has value [ S ( T )- K ] + so using the time T values and the risk neutral probabilities we can calculate the option price back through the tree from time...
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This note was uploaded on 02/04/2011 for the course MATH 3301 taught by Professor Dri.m.macphee during the Spring '10 term at Durham.
- Spring '10