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. We compare our methodology with the alternative results given by the variation of constants method, the linearization of the Matrix Riccati ODE's and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.Comment: Accepted on: Journal of Applied Probability 51(3), 201

Matrix-State Particle Filter for Wishart Stochastic Volatility Processes

This work deals with multivariate stochastic volatility models, which account for a time-varying variance-covariance structure of the observable variables. We focus on a special class of models recently proposed in the literature and assume that the covariance matrix is a latent variable which follows an autoregressive Wishart process. We review two alternative stochastic representations of the Wishart process and propose Markov-Switching Wishart processes to capture different regimes in the volatility level. We apply a full Bayesian inference approach, which relies upon Sequential Monte Carlo (SMC) for matrix-valued distributions and allows us to sequentially estimate both the parameters and the latent variables.Multivariate Stochastic Volatility; Matrix-State Particle Filters; Sequential Monte Carlo; Wishart Processes, Markov Switching.

High-dimensional limits of eigenvalue distributions for general Wishart process

In this article, we obtain an equation for the high-dimensional limit measure of eigenvalues of generalized Wishart processes, and the results is extended to random particle systems that generalize SDEs of eigenvalues. We also introduce a new set of conditions on the coefficient matrices for the existence and uniqueness of a strong solution for the SDEs of eigenvalues. The equation of the limit measure is further discussed assuming self-similarity on the eigenvalues.Comment: 28 page

Student-t Processes as Alternatives to Gaussian Processes

We investigate the Student-t process as an alternative to the Gaussian process as a nonparametric prior over functions. We derive closed form expressions for the marginal likelihood and predictive distribution of a Student-t process, by integrating away an inverse Wishart process prior over the covariance kernel of a Gaussian process model. We show surprising equivalences between different hierarchical Gaussian process models leading to Student-t processes, and derive a new sampling scheme for the inverse Wishart process, which helps elucidate these equivalences. Overall, we show that a Student-t process can retain the attractive properties of a Gaussian process -- a nonparametric representation, analytic marginal and predictive distributions, and easy model selection through covariance kernels -- but has enhanced flexibility, and predictive covariances that, unlike a Gaussian process, explicitly depend on the values of training observations. We verify empirically that a Student-t process is especially useful in situations where there are changes in covariance structure, or in applications like Bayesian optimization, where accurate predictive covariances are critical for good performance. These advantages come at no additional computational cost over Gaussian processes.Comment: 13 pages, 6 figures, 1 table. To appear in "The Seventeenth International Conference on Artificial Intelligence and Statistics (AISTATS), 2014.

Currency option pricing with Wishart process

AbstractIt has been well-documented that foreign exchange rates exhibit both mean reversion and stochastic volatility. In addition to these, recent empirical evidence shows a stochastic skew of implied volatility surface from currency option data, which means that the slope of implied volatility curve of a given maturity is stochastically time varying. This paper develops a currency option pricing model which accommodates for this phenomena. The proposed model postulates that the log-currency value follows a mean reverting process with stochastic volatility driven by Wishart process under risk-neutral measure. Pricing formula for European currency option is derived in terms of Fourier Transform. Benchmarking against the Monte Carlo simulation, our numerical examples reveal that the pricing formula is accurate and remarkably efficient. The proposed model is also generalized to include jumps. The ability of the our model on capturing stochastic skew is illustrated through a numerical example