The tool of stochastic control has been widely used in modern finance. Optimal solutions are usually governed by nonlinear partial differential equations known as Hamilton-Jacobi-Bellman (HJB) equations. Typically, close-form solutions of HJB equations are not available. We study HJB equation for a practical and important topic in finance: determination of optimal asset allocation and optimal consumption strategies under stock price movement with stochastic volatility. The solution to this stochastic control problem is determined by an inhomogeneous HJB equation. Even obtaining accurate numerical solution for this equation is extremely difficult. In this talk, we present closed-form approximate solutions for this problem and provide analytical predictions for the optimal asset allocation strategy and the optimal consumption strategy under stochastic volatility for the class of HARA utility functions. The theoretical predictions are surprisingly accurate. In all financial models, the parameters in the models are not directly observable quantities, and they need to be estimated. The relative error in our theoretical approximation is even smaller than the relative errors in model-parameter estimations. Therefore, our closed-form solution can be treated as “exact” for practical purpose.
Biography
Qiang Zhang is a professor in the Department of Mathematics at City University of Hong Kong. He receives his B.Sc. from Fudan University, MA and Ph.D. from New York University. Prior to joining City University of Hong Kong, Professor Zhang had been with New York University and State University at Stony Brook. His research interests include financial mathematics, risk management, optional portfolio selection, fluid dynamics, granular materials and mathematical physics.