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Let $X_t$ be a continuous local-martingale modeling the stock price given by $$ X_t = \int_0^t \sigma_t(T,K)dW_t , $$ and $\sigma_t(T,K)$ is an $L^2$-measurable process not adapted to $W_t$'s filtration (like a stochastic volatility process).

I want to compute some Greeks, in particular I'm looking at $K$ be a strike price and $T$ be a future time of maturity. I'm interested in computing the quantity $$ \mathbb{E}\left[ D_t(X_t - K)_+ \right] , $$ where $D_t$ is the Malliavin Derivative operator.

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