PORTFOLIO OPTIMIZATION AND STOCHASTIC VOLATILITY ASYMPTOTICS |
| |
Authors: | Jean‐Pierre Fouque Ronnie Sircar Thaleia Zariphopoulou |
| |
Institution: | 1. Department of Statistics & Applied ProbabilityUniversity of California Santa Barbara;2. ORFE DepartmentPrinceton University;3. Departments of Mathematics, and Information, Risk and Operations ManagementThe University of Texas at Austin |
| |
Abstract: | We study the Merton portfolio optimization problem in the presence of stochastic volatility using asymptotic approximations when the volatility process is characterized by its timescales of fluctuation. This approach is tractable because it treats the incomplete markets problem as a perturbation around the complete market constant volatility problem for the value function, which is well understood. When volatility is fast mean‐reverting, this is a singular perturbation problem for a nonlinear Hamilton–Jacobi–Bellman partial differential equation, while when volatility is slowly varying, it is a regular perturbation. These analyses can be combined for multifactor multiscale stochastic volatility models. The asymptotics shares remarkable similarities with the linear option pricing problem, which follows from some new properties of the Merton risk tolerance function. We give examples in the family of mixture of power utilities and also use our asymptotic analysis to suggest a “practical” strategy that does not require tracking the fast‐moving volatility. In this paper, we present formal derivations of asymptotic approximations, and we provide a convergence proof in the case of power utility and single‐factor stochastic volatility. We assess our approximation in a particular case where there is an explicit solution. |
| |
Keywords: | portfolio optimization asymptotic analysis stochastic volatility |
|
|