Like in the title, I am working on running Monte Carlo simulations to price options with the Local Volatility model as a project. I just want to make sure that I am understanding the process, especially the discretization correctly.

The risk neutral dynamics under the Local Volatility model is:

$$ frac{d S_t }{S_t } = mu_t dt + sigma(t,S_t) dW_t $$

Applying Itô’s lemma gives:

$$ d ln(S_t) = (mu_t-frac{1}{2}sigma^2(t,S_t)) dt + sigma(t,S_t) dW_t $$

Using Euler-Maruyama discretization scheme for simplicity:

begin{align}
ln(S_{t+delta t}) &= ln(S_{t}) + int_t^{t+delta t}(mu_t-frac{1}{2} sigma^2(u,S_u)) du + int_t^{t+delta t} sigma(u, S_u) dW_u \
&approx ln(S_{t}) + (mu_t – frac{1}{2} sigma^2(t,S_t)) delta t + z sqrt{sigma^2(t, S_t)delta t} tag{1}
end{align}

Then I can incorporate the local volatility model (and the skew/smile) into my simulations by splitting the time interval between 0 and T into smaller intervals and use the volatility given by the local volatility surface and time step, plug these two into (1) (assuming that I can build a smooth LV surface).

I have two questions.

1/ Would it be correct to use the drift rate equal to the risk free rate for pricing options ?

2/ If I want to use Monte Carlo simulations to get an idea on the probability of the underlying asset ending up between an interval after a defined time period, then I would have to use the "expected return" of the underlying asset instead of the risk free rate ?

Thanks!