An infinite-dimensional affine stochastic volatility model

TL;DR

This paper introduces a novel infinite-dimensional stochastic volatility model using Hilbert space valued Ornstein-Uhlenbeck processes, enabling explicit characteristic functions and flexible covariance modeling, applicable to forward rate dynamics.

Contribution

It develops a new affine infinite-dimensional volatility model with explicit characteristic functions and multiple covariance options, extending existing finite-dimensional models.

Findings

Explicit solutions via generalized Riccati equations

Flexible covariance modeling including jump intensities

Application to forward rate dynamics in finance

Abstract

We introduce a flexible and tractable infinite-dimensional stochastic volatility model. More specifically, we consider a Hilbert space valued Ornstein-Uhlenbeck-type process, whose instantaneous covariance is given by a pure-jump stochastic process taking values in the cone of positive self-adjoint Hilbert-Schmidt operators. The tractability of our model lies in the fact that the two processes involved are jointly affine, i.e., we show that their characteristic function can be given explicitly in terms of the solutions to a set of generalised Riccati equations. The flexibility lies in the fact that we allow multiple modeling options for the instantaneous covariance process, including state-dependent jump intensity. Infinite dimensional volatility models arise e.g. when considering the dynamics of forward rate functions in the Heath-Jarrow-Morton-Musiela modeling framework using the Filipovi\`c space. In this setting we discuss various examples: an infinite-dimensional version of the Barndorff-Nielsen-Shephard stochastic volatility model, as well as a model involving self-exciting volatility.