Parameter Averaging of Quadratic SDEs With Stochastic Volatility

Abstract

In an influential series of papers, V. Piterbarg demonstrates how to perform time-averaging of parameters in a class of diffusion models with linear local volatility and orthogonal stochastic volatility. In this paper, we consider how to extend the applicability of parameter-averaging techniques to a setting where i) the local volatility function has non-zero convexity; and ii) the correlation between the stochastic volatility process and the underlying asset is non-zero and deterministic. These extension are based on classical small-noise SDE expansions and are of practical use in a number of markets -- foreign exchange being a good example -- where empirical observations of volatility smile moves indicate the presence of non-linear local volatility. For efficient calibration of the time-averaged model, we also derive accurate call option pricing approximations for assets with constant-parameter quadratic local volatility overlaid with (correlated) Heston-type stochastic volatility. Several numerical tests probe the accuracy of the parameter-averaging techniques and the various option pricing approximations.

Keywords: Time-averaging, local-stochastic volatility, quadratic local volatility, Heston process

JEL Classification: G12, G13