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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. Riskneutral 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
 DrI.M.MacPhee
 Math

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