What is Local Volatility? The Dupire Equation and Perfect Calibration

Local volatility models allow for perfect calibration to the current market smile. Discover how the Dupire equation derives volatility from option prices.

I spent 10 hours implementing a Local Volatility surface to understand how quants achieve perfect calibration to market prices. What is Local Volatility? It is a deterministic function of both the underlying asset price and time, denoted as $\sigma_{loc}(S, t)$, that allows the Black-Scholes-Merton model framework to fit any observed Volatility Smile.

Bruno Dupire proved in 1994 that if you have a continuous surface of European call prices $C(K, T)$, there is a unique volatility function that is consistent with those prices: $\sigma_{loc}^2(K, T) = \frac{\frac{\partial C}{\partial T} + (r-d)K\frac{\partial C}{\partial K} + dC}{\frac{1}{2}K^2\frac{\partial^2 C}{\partial K^2}}$. This is the foundation of 'model-independent' pricing for exotic options.

Local volatility is great for 'fitting' the today's market, but it often fails to predict how the smile will move tomorrow. It implies a 'sticky-strike' behavior that is often contradicted by actual market dynamics. A quant using only local volatility to hedge exotic options will find that their 'Greeks' are inconsistent when the volatility surface undergoes a parallel shift.

Use the Dupire formula to calculate the local volatility for an at-the-money strike and compare it to the market's implied volatility; the 'Local Vol' is usually more extreme to account for the curvature of the smile.