# Option price calculation using Local Volatility and Monte Carlo

The below formula is used to convert the implied vol into the local volatility, my question is, once I have converted it into the LV ( and have built the full surface), what models do I use to calculate the option price? Do I simulate the underlying price using Monte Carlo and just calculate the average PV of the payoff, using:

$$dS_{t} = (r_{t}-d_{t})S_{t}\,dt + \sigma (S_{t},t)S_{t}\,dW_{t}$$

Using the LV in the BS formula would give me a different result (ie. ATM IV = 20%, transformed ATM LV = 17%, so I can't use the LV in the BS model)?

$$\sigma^2 \left(T,y\right)=\frac{\frac{\partial w}{\partial T}}{1 -\frac{ y}{w} \frac{\partial w}{\partial y}+\frac{1}{2}\frac{\partial^2 w}{\partial y^2}+\frac{1}{4}\left(\frac{ y^2}{w^2}-\frac{1}{w}-\frac{1}{4}\right)\left( \frac{\partial w}{\partial y}\right)^2}$$

Where y is the money-ness, defined as $$y=\ln \left(\frac{ K}{F} \right)$$, and w is the transformation of Black Scholes implied vol $$w=\sigma_{BS}^2\,T$$