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I want to show whether the following price at t is of a european option in Black-Scholes Framework. $$S_tlog_e (S_t^3) $$ Is it just trying to substitute the function (and partial derivates) in the Black-Scholes PDE?

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Yes it is actually just substituting it into the Black Scholes PDE. If the PDE is satisfied, $V(t,S(t)),t\ge 0$ is a martingale and hence $V(t,S(t)) = E_t (V(T,S(T))$ so that $V(t,S(t))$ is the expected value, at time t, of an option that pays $V(T,S(T))$ at time $T$. Here I assumed $r=0$ for simplicity

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If it was a solution to a European option PDE pricing problem, at time $T$ you have $V(T,S) = S \log{S^3}$.

Based on this terminal function, you can then solve the BS PDE using a semi-analytical approach to find $V(t,S)$. My guess is it will not be the candidate you posted (haven't checked).

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