# How to price the american options using local volatility

I have given with a surface of american option prices $$C_{am}(T, K)$$. From these american option prices the implied volatility surface is deduced.

Now I want to find the local volatility $$\sigma(s,t)$$ for my model, such that the price with respect to the price process $$dS_t=rS_tdt+\sigma(S_t,t)S_tdW_t$$ replicate all prices on the price surface.

There is the Dupire Formula for vanilla European option. Clearly, I can't use the Dupire formula to convert my implied volatility surface to the local volatility surface, as Dupire Formula bases heavily on the assumption of European Option(especially, the price should only depend on the stock price at final time). As it turns out, if we use the Dupire Formula, the price obtained by my local volatility model is significant different than the one on the price surface $$C_{am}(T, K)$$.

Is there another way to do that? I know there exists some brute force method using a least square optimization to calibrate the local volatility directly to the price surface $$C_{am}(T, K)$$. However, this method is very time consuming. here for the paper Also I have to adapt the model to discrete dividend. Can anybody provide any helpful materials or method used in the industry?

• Hi @quallenjäger , could you please share how to compute "implied volatility" from American Option prices? I guess, we could set up a backward propagation numerical scheme (binomial tree or finite difference) and use a root-finder to compute the "implied volatility". If so, I see this "implied vol" to have the same meaning as the implied vol backed out from Black Scholes formula from European Option Prices. – bhutes Jun 11 '19 at 7:07
• @bhutes You are right. Only the implied vol backed out from american price surface obeys a different equation than the usual dupire equation, which is used to back out implied vol from european option prices. – quallenjäger Sep 16 '19 at 15:42