Multivariate Heston–Wishart Models: Portfolio Risk, Correlation, and Optimal Allocation

Multivariate Heston-Wishart models describe asset returns whose variances and correlations evolve jointly through a positive definite covariance process. We develop a consistent framework for using these models to measure portfolio risk and choose dynamic allocations, linking the stochastic equations to analytical formulas and reproducible numerical experiments. The objective is to establish what the specified model can support mathematically and computationally before evaluating its usefulness on market data. Using the established affine leverage construction, we derive the joint transform of asset log returns and covariance, characterize admissible leverage, and obtain Fourier formulas for value at risk and expected shortfall. We distinguish weighted asset log returns, continuously rebalanced wealth, and portfolios held without rebalancing. We also derive the matrix Riccati allocation rule for an investor with constant relative risk aversion and verify its optimality under the stated assumptions. The analysis identifies structural restrictions with practical consequences. Volatility and correlation need not rise together; their local association depends on model parameters. Total variance follows a scalar square root process only under specific restrictions. Stationary portfolio projections share a short horizon kurtosis bound determined by dimension and covariance positivity. Experiments compare Fourier risk calculations with 65,536 simulated covariance paths and analytical portfolio values with 32,768 paths. At the one percent tail level, expected shortfall agrees across methods within Monte Carlo uncertainty. The optimal covariance hedge yields an estimated annual certainty equivalent gain of 19.4 basis points, compared with 18.1 analytically, in an unconstrained example allowing borrowing. A controlled correlation comparison reverses the sign of the apparent risk increment when the fixed correlation benchmark matches average portfolio variance. These results provide a checked foundation for risk measurement and allocation within the model. They establish numerical consistency and clarify interpretation; parameter estimation, latent covariance filtering, and forecasting or investment advantages on financial data remain empirical questions.