Local Volatility Calibration on the Foreign Currency Option Market

Abstract: In this thesis we develop and test a new method for interpolating and extrapolating prices of European options. The theoretical base originates from the local variance gamma model developed by Carr (2008), in which the local volatility model by Dupire (1994) is combined with the variance gamma model by Madan and Seneta (1990). By solving a simplied version of the Dupire equation under the assumption of a continuous ve parameter di usion term, we derive a parameterization dened for strikes in an interval of arbitrary size. The parameterization produces positive option prices which satisfy both conditions for absence of arbitrage in a one maturity setting, i.e. all adjacent vertical spreads and buttery spreads are priced non-negatively. The method is implemented and tested in the FX-option market. We suggest two sub-models, one with three and one with ve degrees of freedom. By using a least-square approach, we calibrate the two sub-models against 416 Reuters quoted volatility smiles. Both sub-models succeeds in generating prices within the bid-ask spread for all options in the sample. Compared to the three parameter model, the model with ve parameters calibrates more exactly to market quoted mids but has a longer calibration time. The three parameter model calibrates remarkably quickly; in a MATLAB implementation using a Levenberg-Marquardt algorithm the average calibration time is approximately 1 ms. Both sub-models produce volatility smiles which are C2 and well-behaving. Further, we suggest a technique allowing for arbitrage-free interpolation of calibrated option price functions in the maturity dimension. The interpolation is performed in parameter space, where every set of parameters uniquely determines an option price function. Furthermore, we produce sucient conditions to ensure absence of calendar spread arbitrage when calibrating the proposed model to several maturities. We use this technique to produce implied volatility surfaces which are suciently smooth, satisfy all conditions for absence of arbitrage and fit market quoted volatility surfaces within the bid-ask spread. In the final chapter we use the results for producing Dupire local volatility surfaces and for pricing variance swaps.