Local Volatility Models

Local volatility models extend the Black-Scholes framework by allowing volatility to be a deterministic function of both the asset price and time. This approach ensures that the model can perfectly replicate the observed market prices of all vanilla options.

Unlike constant volatility models, local volatility captures the dynamics of the volatility surface across different strikes and maturities. It is particularly useful for pricing path-dependent exotic derivatives where the specific trajectory of the asset price matters.

However, these models can be computationally intensive and may produce unrealistic forward volatility dynamics. They are a significant step up in complexity from simple models and require robust numerical methods.

Practitioners often use them as a calibration tool to ensure internal consistency in pricing.