Calibration Of Local Volatility Surfaces Using Tikhonov Regularization

When pricing a financial derivative product, a geometric Brownian motion is often used to describe the behavior of underlying's price. Based on this model, the volatility part is always an important parameter which affects the prices of derivatives. So how to use the observed market prices to get the volatility value which can fairly reflect the underlying's fluctuation, which can be used to get rational prices for related derivatives and look for the hedging strategies is often an important subject in mathematical finance research.This paper aims to calibrate the local volatility surface with help of some observed market European call option prices and use Tikhonov method to regularize the calibration problem. A natural spline functional approach and a linear interpolation are used to get the local volatility surface when given some knots whose values are determined by solving a nonlinear reg-ularization problem. The regularized strategy is minimizing the difference between the European call option prices calculated from the Dupire equa-tion and the known relevant market prices, the L-BFGS algorithms is used when doing the minimization. In this paper, real market European call op-tion data on the SPX index is used to compute the local volatility surface and some back test are demonstrated too.During the implementation, this paper mainly uses C++ and VBA to realize the above mathematical methods, applying the interface program between C++ and VBA the obtained results are more visible in Excel, all the programme can provide a good software interface for getting a local volatility function numerically.