# How to use log moneyness in a local volatility context

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?