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Black-Scholes equation

Assume that we have a market consisting of a risky asset $S$ and a bank-account $B$, where the corresponding $P$-dynamics are given as \begin{eqnarray*} dS_t&=&S_t\mu(t,S_t) dt+S_t\sigma(t,S_t) dW_t,\\ S_0&=&s_0\\ dB_t&=&rB_t dt,\\ B_0&=&1. \end{eqnarray*} Assume that $F(t,s)$ is the price at time $t$ of a simple claim with maturity $T$ of the form $F(T,s)=\Phi(s)$. Then the price $F$ is a solution to the boundary value problem \begin{eqnarray*} \frac{\partial F(t,s)}{\partial t}&=&-rs\frac{\partial F(t,s)}{\partial s}-\frac{1}{2}s^2\sigma(t,s)^2\frac{\partial^2 F(t,s)}{\partial s^2}+rF(t,s)\\ F(T,s)&=&\Phi(s). \end{eqnarray*} This equation can be solved using the Feynman-Kac representation theorem which gives us the risk neutral valuation formula. \[ F(t,s)=\exp(-(T-t) r)\text{E}\left[ \Phi(S_T)\mid S_t=s\right],\] where \begin{eqnarray*} dS_u&=&rS_u du+\sigma(u,S_u)dW_u,~t\leq u\leq T,\\ S_t&=&s \end{eqnarray*}

 

Questions: Magnus Wiktorsson
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