A new class of multidimensional Wishart-based hybrid models

In this article, we present a new class of pricing models that extend the application of Wishart processes to the so-called stochastic local volatility (or hybrid) pricing paradigm. This approach combines the advantages of local and stochastic volatility models. Despite the growing interest on the topic, however, it seems that no particular attention has been paid to the use of multidimensional specifications for the stochastic volatility component. Our work tries to fill the gap: we introduce two hybrid models in which the stochastic volatility dynamics is described by means of a Wishart process. The proposed parametrizations not only preserve the desirable features of existing Wishart-based models but significantly enhance the ability of reproducing market prices of vanilla options.

On the application of Wishart process to the pricing of equity derivatives: the multi-asset case

Given the inherent complexity of financial markets, a wide area of research in the field of mathematical finance is devoted to develop accurate models for the pricing of contingent claims. Focusing on the stochastic volatility approach (i.e. we assume to describe asset volatility as an additional stochastic process), it appears desirable to introduce reliable dynamics in order to take into account the presence of several assets involved in the definition of multi-asset payoffs. In this article we deal with the multi asset Wishart Affine Stochastic Correlation model, that makes use of Wishart process to describe the stochastic variance covariance matrix of assets return. The resulting parametrization turns out to be a genuine multi-asset extension of the Heston model: each asset is exactly described by a single instance of the Heston dynamics while the joint behaviour is enriched by cross-assets and cross-variances stochastic correlation, all wrapped in an affine modeling. In this framework, we propose a fast and accurate calibration procedure, and two Monte Carlo simulation schemes.

The Explicit Laplace Transform for the Wishart Process

We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral, which extends the original approach of Bru (1991). We compare our methodology with the alternative results given by the variation-of-constants method, the linearization of the matrix Riccati ordinary differential equation, and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.

The Explicit Laplace Transform for the Wishart Process

We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral, which extends the original approach of Bru (1991). We compare our methodology with the alternative results given by the variation-of-constants method, the linearization of the matrix Riccati ordinary differential equation, and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.