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Let stock price $S$ satisfy

$$S(t)=S(0)e^{(\int_0^t\sigma(s)dB_s-\frac{1}{2}\int_0^t\sigma(s)^2ds)}$$

I want to calculate the Martingale representation $V(t)=E(F|F_t)$ of European option with strike price $M$ and maturity $T$ which is given by

$$F=(S(T)-M)^+$$

How to find the solution and the Black-Scholes PDE?

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