Abstract
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.
Highlights
A Wishart process is a matrix-valued continuous time stochastic process with a marginal Wishart distribution, i.e., a generalization to multiple dimensions of the chisquared distribution, or, in the case of non-integer degrees of freedom, of the gamma distribution
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
Summary
A Wishart process is a matrix-valued continuous time stochastic process with a marginal Wishart distribution, i.e., a generalization to multiple dimensions of the chisquared distribution, or, in the case of non-integer degrees of freedom, of the gamma distribution. We deal with the Wishart Affine Stochastic Correlation model (WASC), introduced in Da Fonseca et al (2007) with the purpose of reproducing well-known multi-asset stylized facts in a tractable way. WASC model makes use of Wishart process to describe the stochastic variance covariance matrix of asset returns. In this article we extend the results presented in La Bua and Marazzina (2019) for the single asset Wishart Multidimensional Stochastic Volatility model (WMSV) to the WASC one. For the multi-asset case, we extend this approximating technique making use of the distributional law of diagonal elements of Wishart process to connect the WASC calibration problem to the Heston one. We extend the Gaussian variable approximation scheme presented in La Bua and Marazzina (2019) to the WASC model.
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