Regularization Methods For Implied Volatility In Option Pricing

Volatility is an important risk measurement in financial economics and critical for derivative pricing, portfolio selection, risk management and monetary policy etc. It is assumed to be constant in the classic Black-Scholes model. But empiri-cal evidences demonstrate that implied volatility derived from option transaction prices shows time-varying, smile, smirk and term structure characterizations. Im-plied volatility reflects the expectation and judgement of the market. In traditional models of forecasting, implied volatility contains the information of future market, even the most one. So it’s suitable for volatility forecasting in mid-long term. The estimation of it has become an important topic in financial engineering. Financial innovation would be quickly developed in order to promote the internationaliza-tion. Innovation certainly may bring the opportunity, but it also may introduce the risk. So researches on volatility, which is a quantitative tool of risk, would give scientific guidance for financial regulator.Total variation regularization method is an important approach for solving discontinuous inverse problem and has been broadly adopted in image restoration, image reconstruction and other inverse problems. This dissertation is aiming at investigating the inverse problem of option pricing. Nonlinear total variation regu-larization and adjoint models are proposed based on Tikhonov regularization and the solutions are derived. The properties of solutions are analyzed and numeri-cal algorithms are presented. As volatility exhibits jump, overnight and weekend effect, it’s valuable to investigate whether total variation regularization could be applied to implied volatility estimation. The main work is described as follows:Total variation regularization approach to estimate univariate implied volatil-ity is proposed. Optimal control problem is derived to estimate implied volatility with total variation regularization method under the framework of Black-Scholes theory. Properties of the solution are rigorously analyzed, including necessary con-ditions, existence, stability and convergence. Particularly under the condition that maturity is short enough, the uniqueness of the local solution is proved through a measure transformation technique. The rigorous theoretical analysis shows the effectiveness of the presented method.Total variation regularization method to estimate bivariate implied volatility is proposed. Bivariate implied volatility is more general and meets the demand of the market better. Necessary condition is obtained by variation method. And the properties of the solution, including existence, stability, and convergence, are ana-lyzed. Gauss-Jacobi iterate algorithm is presented to solve the model numerically based on the discrete Euler-Lagrange equation. The results of numerical exper-iments show that total variation regularization method could not only maintain nonsmoothness of volatility, but also has smaller error.Total variation regularization TV-L1model is proposed to estimate implied volatility. The minimum total variation TV-L1model is established by combing total variation and L1fidelity. In fact, L1fidelity based regularization model has already been broadly adopted in data driven parameter selection and multiscale image decomposition. Adjoint equation is derived by semi-discretization of ad-justed Black-Scholes equation with adjoint variable. Adjoint equation provides the precise value of gradient which is needed in the procedure of minimization. This avoids the approximate computation of Vega. L-BFGS algorithm is proposed to calculate the implied volatility based on Crank-Nicoloson finite difference method. Numerical experiments show validity and effectiveness of the model and algorithm.This dissertation is aiming at making up for the blank in the inverse problem of option pricing of total variation regularization method and is an improvement of existent models and methods. The properties of the solutions are rigorously analyzed. Fast and effective algorithms are given. Numerical experiments show that total variation regularization methods maintain the nonsmoothness of the implied volatility.