The Laplace transform of the integrated Volterra Wishart process

TL;DR

This paper derives explicit formulas for the Laplace transform of the integrated Volterra Wishart process, enabling efficient pricing in complex financial models with autocorrelation, long-range dependence, and covariance risk.

Contribution

It introduces new explicit expressions for the Laplace transform of the process using Fredholm determinants and Riccati equations, linking Gaussian process transforms to practical financial applications.

Findings

Explicit Laplace transform formulas derived

Connections made to Riccati equations for special cases

Practical approximation methods for pricing models

Abstract

We establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and resolvent. Furthermore , we link the characteristic exponents to a system of non-standard infinite dimensional matrix Riccati equations. This leads to a second representation of the Laplace transform for a special case of convolution kernel. In practice, we show that both representations can be approximated by either closed form solutions of conventional Wishart distributions or finite dimensional matrix Riccati equations stemming from conventional linear-quadratic models. This allows fast pricing in a variety of highly flexible models, ranging from bond pricing in quadratic short rate models with rich autocorrelation structures, long range dependence and possible default risk, to pricing basket options with covariance risk in multivariate rough volatility models.