Dupire Equation: Uses

The equation Dupire 2   can be used in the following 2 ways: 

 

1. If the local volatility Sigmaais known, the PDE can be used to compute the price today of all call options in a single sweep, starting from the boundary condition C(S, t, K, T) = (S - K)+

In contrast, the BS backward equation requires one PDE for each strike and maturity. 

In the case of calibrating a parametric form of Sigmaato a set of market option prices, one needs to compute the model price of all these options and the forward equation can accelerate the computation to a factor 100. 

 

 

2. If the call prices are known today, one can compute their derivatives and extract the local volatility by the following stripping formula: 

Dupiree

 

Starting from a finite set of listed option prices, a good interpolation in strike and maturities provides a continuum of option prices and we can apply the stripping formula to get the local volatilities. 

 

Once the local volatilities are obtained, one can price exotic instruments with this calibrated local volatility model. 

Properly accounting for the market skew can have a massive impact on the price of exotics --> example: call up-and-out. 

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