Stochastic

E the option with payo s2 80 find the stock holdings 0

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Unformatted text preview: ) + m=1 ! Bt + (r t/h pX h h EXm m=1 µ+ 2 /2)t so under the risk neutral measure, P ⇤ , St = S0 · exp((r 2 /2)t + Bt ) The value of the option g (ST ) in the discrete approximation is given by the expected value under its risk neutral measure. Ignoring the detail of proving that the limit of expected values is the expected value of the limit, we have proved the desired result. The Black-Scholes partial di↵erential equation We continue to suppose that the option payo↵ at time T is g (ST ). Let V (t, s) be the value of the option at time t < T when the stock price is s. Reasoning with the discrete time approximation and ignoring the fact that the value in this case depends on h, V (t h, s) = 1 [p⇤ V (t, su) + (1 1 + rh p⇤ )V (t, sd)] 201 6.7. CALLS AND PUTS Doing some algebra we have V (t, s) (1 + rh)V (t s, h) = p⇤ [V (t, s) + (1 V (t, su)] p )[V (t, s) ⇤ V (t, sd)] Dividing by h we have V (t, s) V (t h, s) rV (t h, s) h V (t, su) V (t, s) V (t, sd) ⇤ V (t, s) ⇤ =p + (1 p ) h h (6.30) Letting h ! 0 the left-hand side...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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