stochastic.221 - 220 21.1 Feynman-Kac Theorem The function...

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Unformatted text preview: 220 21.1 Feynman-Kac Theorem The function u satisfies the PDE ut + rxux + 1 2x2 uxx + xuy = 0; 0  t  T; x  0; y 2 IR; 2 the terminal condition uT; x; y  = hy ; x  0; y 2 IR; and the boundary condition ut; 0; y = hy ; 0  t  T; y 2 IR: One can solve this equation. Then v t; S t; Zt is the option value at time t, where 0 The PDE for v is S u du vt; x; y  = e,rT ,t ut; x; y : ,rv + vt + rxvx + 1 2x2vxx + xvy = 0; 2 v T; x; y  = hy ; vt; 0; y = e,rT ,thy : One can solve this equation rather than the equation for u. 21.2 Constructing the hedge Start with the stock price S 0. The differential of the value X t of a portfolio t is dX =  dS + rX , S  dt = S r dt + dB  + rX dt , rS dt =  S dB + rX dt: We want to have X t = v t; S t; so that  Zt X T  = v T; S 0; =h Z T 0 0 S u du ; ZT 0 ! S u du ; ! S u du : (1.1) ...
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