Numerical example of how to calculate local vol surface from IV surface

Question

I'm looking for an excel example (not a copy of Dupire's eqn) of how to convert an IV surface to a local vol surface. If unsuccessful I'll work through Dupire's eqn but would be helpful to look at an example first.

Answer 1

I know one article (download) that explaining how to calculate local vol surface from IV surface and also chapter 18 of this book is very good In this context. However you know that Dupire’s (1994) formula for local volatility is \begin{align} \sigma_L(k,T)=\sqrt\frac{\frac{\partial C}{\partial T}}{\frac{1}{2}K^2\frac{\partial^2 C}{\partial K^2}} \end{align} where $C = C(K,T)$ is the time-$t$ call price with strike $K$ and maturity $T$ when the spot price is $S_t$.The analytic expressions for the derivatives required of the Dupire (1994) local volatility formula require extensive coding From a coding point of view, it is simpler to approximate the derivatives using finite differences. Write $C(K) = C(K,T)$ to emphasize the dependence of the European call price on the maturity T. We use a small time increment $\Delta t$ and approximate the time derivative as the central difference

\begin{align} \frac{\partial C}{\partial T}\approx\frac{C(K,T+\Delta T)-C(K,T-\Delta T)}{2\Delta t} \end{align} Similarly, we can use a small strike increment $\Delta K$ and approximate the second order strike derivative as the central difference \begin{align} \frac{\partial^2 C}{\partial K^2}\approx\frac{C(K-\Delta K,T)-2C(K,T)+C(K+\Delta K,T)}{(\Delta K )^2} \end{align}