= Exponential payoff transform PDE
{title2=$V_t+(A+\rho\theta\sigma B)V_\sigma+\tfrac12B^2V_{\sigma\sigma}+\tfrac12\theta(\theta-1)\sigma^2V=0$}
For log-price dynamics $dX=-\sigma^2dt/2+\sigma dW^X$ and volatility dynamics $d\sigma=A(\sigma)dt+B(\sigma)dW^\sigma$, with correlation $\rho$, the exponential payoff ansatz $U=e^{\theta X}V$ reduces the backward <partial differential equation> to the displayed one-dimensional equation. The terminal value is $V(T,\sigma)=1$. Correlation changes the volatility drift, while the log-price drift supplies the negative $\theta$ term.
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