Dupire local volatility theory provides a mathematical framework for constructing a local volatility function that is consistent with the observed prices of options across all strike prices and maturities. The model assumes that volatility is a deterministic function of both time and the underlying asset’s price level. This framework resolves the inconsistency of the Black-Scholes model, which assumes constant volatility, by calibrating to the market’s volatility surface. The local volatility function essentially represents the instantaneous volatility at any given point in time and price.
Calibration
The calibration process involves inverting the Dupire equation to derive the local volatility surface from market data, typically using a large set of liquid option prices. Accurate calibration ensures that the model can correctly price exotic options that depend on the asset’s path, while maintaining consistency with vanilla option prices. In crypto markets, this calibration is complicated by fragmented liquidity and high-frequency price changes.
Application
The application of local volatility models extends beyond pricing to calculating accurate risk sensitivities, known as the Greeks, for complex derivative portfolios. By incorporating the local volatility surface, traders can better understand how changes in the underlying price will affect their portfolio’s delta and gamma exposure. This provides a more precise method for dynamic hedging than models relying on constant or stochastic volatility assumptions.
