lec_smry3 - Math Finance summary Mich ’10 10 5...

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Unformatted text preview: Math Finance summary Mich ’10 10 5 Approximation methods Stock brokers need to update prices for many millions of options as stock prices change. In practice they use approximation techniques of three basic types: calculations on finite binomial trees; simulation (Monte Carlo) methods; numerical methods for pde. These have various strengths and weaknesses. We have considered binomial tree models in some detail H.16 – they date back to the original idea in Cox, Ross and Rubinstein in 1979 (J of Fin’l Econ). Binomial tree for American put options We first consider a common method for pricing an American put. This is to compute its value on a tree model with a short enough time step to provide an acceptable approximation to the price on the continuous time model. H.18 Consider an American put with parameters ( K, T ) on a stock with initial price S and volatility σ . The interest rate is r per period (continuously compounded). Pick some large integer M and set up the risk-neutral binomial tree of depth M i.e. set δt = T/M , α = ( r- σ 2 / 2) δt , p = 1 / 2 , u = exp( σ √ δt + α ) , d = exp(- σ √ δt + α ) The tree has stock price nodes S m n = u n d m- n S for n = 0 , . . . , m ; m = 0 , 1 , . . . , M so at time t = mδt the stock price S ( t ) = S m n when there have been n upward and m- n downward price jumps, 0 ≤ n ≤ m . Let Λ( S ) = ( K- S ) + denote the return when the put is exercised at price S and let V ( S, m ) denote the risk-neutral expected value of the put at stage m with stock price S if it has not yet been exercised and V ( S, m ; w ) denote the corresponding value if the...
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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.

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lec_smry3 - Math Finance summary Mich ’10 10 5...

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