Functional Stochastic Volatility in Financial Option Surfaces

Phillip A. Jang, Michael Jauch, David S. Matteson

Abstract: Stochastic volatility, or variability that is well approximated as a random process, is widespread in modern finance. While understanding that volatility is essential for sound decision making, the structural and data constraints associated with complex financial instruments limit the applicability of classical volatility modeling. This article investigates stochastic volatility in functional time series with the goal of accurately modeling option surfaces. We begin by introducing a functional analogue of the familiar stochastic volatility models employed in univariate and multivariate time series analysis. We then describe how that functional specification can be reduced to a finite dimensional vector time series model and discuss a strategy for Bayesian inference. Finally, we present a detailed application of the functional stochastic volatility model to daily SPX option surfaces. We find that the functional stochastic volatility model, by accounting for the heteroscedasticity endemic to option surface data, leads to improved quantile estimates. More specifically, we demonstrate through backtesting that Value-at-Risk estimates from the proposed functional stochastic volatility model exhibit correct coverage more consistently than those of a constant volatility model.