An infinite‐dimensional affine stochastic volatility model

Sonja Cox, Sven Karbach and Asma Khedher

Mathematical Finance, 2022, vol. 32, issue 3, 878-906

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, that is, we show that their characteristic function can be given explicitly in terms of the solutions to a set of generalized 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, for example, when considering the dynamics of forward rate functions in the Heath–Jarrow–Morton–Musiela (HJMM) modeling framework using the Filipović space. In this setting, we discuss various examples: an infinite‐dimensional version of the Barndorf–Nielsen–Shephard stochastic volatility model, as well as covariance processes with a state dependent intensity.

Date: 2022
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