Calibration Of Stochastic Local Volatility Models With Jumps

The calibration of volatility models is an inverse problem to find the parameters of models which could match the model price with the market price. Since the constant volatility assumption in Black-Scholes[5] model can not explain the phenomenon of volatility smile and volatility term structure observed in market, researchers tried to use more flexible models to calibrate the market in the past decades. These models contain stochastic local volatility models and jump models. For stochastic local volatility models, the volatility of the underlying asset is represented as the product of a local volatility function and a stochastic process. That is, the model combines the local volatility model with the stochastic volatility model, which aims to incorporate the advantages and eliminate the disadvantages of both models. For this kind of models, researchers have proposed several calibration methods. A two-stage calibration procedure is most common. The first stage is to calibrate the purely stochastic process part. The past work always focused on the parametric form of stochastic process and used nonlinear least-squares methods to find the optimal parameters. The second step is to calibrate the local function. Based on the result in Derman and Kani[20],the local function can be represented as a special function and then can be calculated using Monte Carlo or PDE method. However in order to calculate the special function, it is necessary to calibrate the local volatility model first. To determine the jump parameters in the jump models, the nonlinear least-squares method is always used to calibrate. Others use some statistical techniques to estimate the parameters from historical underlying price. The successful application of the Tikhonov regularization method in local volatility model has encouraged us to calibrate the general volatility models using the same method.In this thesis, we use the Tikhonov regularization method to calibrate the general stochastic local volatility model with jumps. The general form means that we do dot assume the special structure of three terms in this model, that is the local volatility function, the drift term and di?usion term of stochastic process. Dai et al.[15] discussed the calibration of general stochastic volatility models. But the general stochastic local volatility models with jumps is seldom discussed before. We calibrate the three terms simultaneously. Since the calibration does not base on the result about local volatility function in Derman and Kani[20], we also omit the procedure of calibrating the local volatility model. First, a modified Dupire’s equation associated with stochastic local volatility models with jumps is proposed. Based on the equation, we formulate the calibration problem of each term as an optimal control problem. The necessary condition of the optimal solution can be derived, which satisfies a variational problem.We further simplify these necessary conditions. Then we propose a gradient descent algorithm to numerically find the optimal solution. Since the three terms are calibrated at the same time, an overall iteration algorithm is proposed. An extensive numerical analysis is presented to demonstrate the e?ciency of our numerical algorithm. In the numerical experiments, we test each term separately and then test the three terms as a whole. In the market test, we use the SPX option data to calibrate the model. The test results show the e?ectiveness of our calibration methods.