Multivariate Lèvy models: estimation and asset allocation

Multidimensional asset models based on Lévy processes have been introduced to meet the necessity of capturing market shocks using more refined distribution assumptions compared to the standard Gaussian framework. In particular, along with accurately modeling marginal distributions of asset returns, capturing the dependence structure among them is of paramount importance, for example, to correctly price derivatives written on more than one underlying asset. Most of the literature on multivariate Lévy models focuses in fact on pricing multi-asset products, which is also the case of the model introduced in Ballotta and Bonfiglioli (2014). Believing that risk and portfolio management applications may benefit from a better description of the joint distribution of the returns as well, we choose to adopt Ballotta and Bonfiglioli (2014) model for asset allocation purposes and we empirically test its performances. We choose this model since, besides its flexibility and the ability to properly capture the dependence among assets, it is simple, relatively parsimonious and it has an immediate and intuitive interpretation, retaining a high degree of mathematical tractability. In particular we test two specifications of the general model, assuming respectively a pure jump process, more precisely the normal inverse Gaussian process, or a jump-diffusion process, precisely Merton’s jump-diffusion process, for all the components involved in the model construction. To estimate the model we propose a simple and easy-to-implement three-step procedure, which we assess via simulations, comparing the results with those obtained through a more computationally intensive one-step maximum likelihood estimation. We empirically test portfolio construction based on multivariate Lévy models assuming a standard utility maximization framework; for the exponential utility function we get a closed form expression for the expected utility, while for other utility functions (we choose to test the power one) we resort to numerical approximations. Among the benchmark strategies, we consider in our study what we call a ‘non-parametric optimization approach’, based on Gaussian kernel estimation of the portfolio return distribution, which to our knowledge has never been used. A different approach to allocation decisions aims at minimizing portfolio riskiness requiring a minimum expected return. Following Rockafellar and Uryasev (2000), we describe how to solve this optimization problem in our multivariate Lévy framework, when risk is measured by CVaR. Moreover we present formulas and methods to compute, as efficiently as possible, some downside risk measures for portfolios made of assets following the multivariate Lévy model by Ballotta and Bonfiglioli (2014). More precisely, we consider traditional risk measures (VaR and CVaR), the corresponding marginal measures, which evaluate their sensibility to portfolio weights alterations, and intra-horizon risk measures, which take into account the magnitude of losses that can incur before the end of the investment horizon. Formulas for CVaR in monetary terms and marginal measures, together with our approach to evaluate intra-horizon risk, are among the original contributions of this work.