# How to calculate Delta of an option in the Local Volatility model?

Let $dS/S = \sigma(S,t) dW$.

Let the local volatility be known, i.e, we know the formula $\sigma(S,t)$.

How do I derive $\Delta$ of a regular call in this model?

Let BS be the Black Scholes price, and let $\sigma_{imp}$ be the implied volatility induced by above local volatility. Then, does $\sigma_{imp}$ depend on $S$ as well?

• No closed form formula for the price, and a fortiori the Greeks. You should resort to compute the delta numerically. Indeed the LV model embeds a spot/IV dynamics. But it is not clear what you are asking IMHO (what is your precise definition of $\sigma_{imp}$) – Quantuple Mar 31 '17 at 7:10
• @quantuple - if you have constructed a parametric form for the call prices, and then convert it to a parametric form for local via via dupire, and then fit that, then you can have a closed form for the price. – will Apr 2 '17 at 11:48
• Sure but that's a special case – Quantuple Apr 2 '17 at 11:55