On Dupire local volatility

The Dupire equation for local volatility is surely one of the biggest technological discoveries in the pricing of equity derivatives. It has significantly changed how the market analyses and manages the risk of structured products. Despite having many deficiencies, the model is being used in an industrial way. By industrial, I really mean like a factory. Although it doesn’t account for many of the associated risks, the local volatility model is the benchmark for pricing any structured product. The local volatility model is also the most widely used approach for risk management. But why is the local volatility model so widely used? The main reasons are that the model is very easy to implement and the simulations using the model are so precise. Nevertheless the local volatility model has many deficiencies and does not accurately price products with non-European options. Furthermore the standard Dupire formula is not the best choice for actually implementing the model. Instead Jim Gatheral’s implementation of local volatility is surely more practical for the simple reason that we tend to see prices as a function of implied volatilities rather than of actual vanilla option prices. Gatheral suggests the following formulation:

Where:

is the local variance

is the strike

is the maturity

is the log moneyness strike

is the forward

is the implied total variance

As seen in the above equation, the local volatility is a function of the implied volatility rather than call/put prices. This approach is much more useful in practice.