Monte Carlo calibration to implied volatility surface under volatility models

The calibration problem of implied volatility surface under complex financial models can be formulated as a nonlinear high-dimensional optimization problem. To resolve this problem for genuine volatility models, we develop a sequential methodology termed two-stage Monte Carlo calibration method. It consists of the first stage-dimension separation for splitting parametric set into two subsets, and the second stage-standard error reduction for efficient evaluation of option prices. The first stage dimension separation aims to reduce dimensionality of the optimization problem by estimating some volatility model parameters a priori under the historical probability measure such that the total number of model parameters under an option pricing measure is significantly reduced. The second stage standard error reduction aims simultaneously to reduce variance of option payoffs by the martingale control variate algorithm, and to increase the total number of Monte Carlo simulation by the hardware graphics processing unit (GPU) for parallel computing. This two-stage Monte Carlo calibration method is capable of solving a variety of complex volatility models, including hybrid models and multifactor stochastic volatility models. Essentially, it provides a general framework to analyze backward information from the historical spot prices and the forward information from option prices.