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I am implementing a monte carlo to price various options using a local volatility model.

The implied volatility surface from which the local volatility is derived is a function of logmoneyness and expiry : $ \sigma_{implied} (log(F/K), T) $.

When using only the implied volatility to price a european option for example, I read the implied volatility with $log(F_T/K)$, with $K$ the strike of the option and $F_T$ the forward price of the underlying corresponding to the expiry of the option $T$.

With local volatility however, generally speaking, $K$ is replaced by $S$. So in my implementation, should I use $log(F_T/S_t)$ to read the local volatility for each step $t$ in the simulation?

In doing so, I compared the average accumulated local volatility until the expiry and compared it with the implied volatility corresponding to the expiry. These quantities are close together : 3.13873% implied volatility versus 3.23428% accumulated local volatility. Does this difference mean that my implementation needs to be improved to get a more accurate local volatility?

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