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# lec_smry2 - Math Finance summary Mich 10 7 Asset price...

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Math Finance summary Mich ’10 7 H.6 Asset price behaviour The cornerstone of a great many models in mathematical finance is the assumption that asset prices S ( t ), t 0 (at least for non-dividend paying stocks) are distributed as a geometric Brownian motion with drift μ and volatility σ . Specifically, log S ( u + t ) S ( u ) N ( ( μ σ 2 / 2) t, σ 2 t ) for all u 0, t > 0 and additionally S ( u + t ) /S ( u ) is independent of S ( v ), v u . There is considerable empirical evidence that this model is often reasonable (Taylor’s theorem for the log function links this model to our earlier observation about relative price changes when they are small). There is also much research into improving it e.g. by making the volatility time dependent. Tuning the binary tree model It is possible to choose u , d and p to make the binary tree model emulate the geometric BM model of stock prices. There are various (essentially equivalent) ways to do this. Divide time period T into a large number n timesteps of length Δ = T/n (with interest rate per time step equal to r Δ) and set u = exp( σ Δ + ( μ σ 2 / 2)Δ) , d = exp( σ Δ + ( μ σ 2 / 2)Δ) , p = 1 2 + o (Δ) where μ is the drift and σ 2 the volatility of the geometric BM stock price model. The parameters can be made risk neutral by setting (1+ r Δ d ) / ( u d ) = p = 1 / 2+ o (Δ) which leads in the limit to a geometric BM model with volatility σ but with risk neutral drift H.12 r in place of μ . Assuming the limit can be passed through the expectation (correct but we do not prove it in this course) this leads to the call option price formula C ( K, T, σ, S, r ) = e rT E r ( [ Se W K ] + ) where S (0) = S and E r indicates that we use W N (( r σ 2 / 2) T, σ 2 T ). 3 The Black-Scholes formula The expectation in the formula for the European call price can be evaluated in the form H.8.5 C = S Φ( d 1 ) Ke rT Φ( d 2 ) (3 . 1) where d 1 = ( r + σ 2 / 2) T + log( S/K ) σ T , d 2 = d 1 σ T = ( r σ 2 / 2) T + log( S/K ) σ T and Φ denotes the standard Normal cdf. This is established by standard integration methods.

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lec_smry2 - Math Finance summary Mich 10 7 Asset price...

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