Abstract
Accurately modeling the implied volatility surface is of great importance to option pricing, trading and hedging. In this paper, we investigate the use of a Bayesian nonparametric approach to fit and forecast the implied volatility surface with observed market data. More specifically, we explore Gaussian Processes with different kernel functions characterizing general covariance functions. We also obtain posterior distributions of the implied volatility and build confidence intervals for the predictions to assess potential model uncertainty. We apply our approach to market data on the S&P 500 index option market in 2018, analyzing 322,983 options. Our results suggest that the Bayesian approach is a powerful alternative to existing parametric pricing models
Highlights
IntroductionModeling and forecasting implied volatility is of great importance in both finance theory and practical decision making
Implied volatility is a measure of the future expected risk of the underlying price
We show that the Gaussian Processes (GP) models outperform the parametric model in terms of out-of-sample pricing errors, but they provide posterior distributions for option prices that, when translated into confidence intervals, are informative about the accuracy of the estimated prices
Summary
Modeling and forecasting implied volatility is of great importance in both finance theory and practical decision making. In this project, we propose a method to approach the option pricing problem in a Bayesian framework. Implied volatility tends to increase for in-the-money (ITM) and out-ofthe-money (OTM) options. This pattern is usually described as the “volatility smile” as the implied volatility plotted against strike or moneyness looks like the shape of a smile. Other stochastic volatility models include HullWhite (Hull and White; 1987), SABR (Hagan et al.; 2003), etc Despite their popularity in the industry, these models are very complicated and sometimes without closed-form solution. Solving the corresponding partial differential equation with appropriate boundary condi-
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