Optimal Portfolio Policies and Consumption for Important Jump Events: Approximate Canonical Model Approach

At important events or announcements, there may be large changes in the value of market portfolios due to large fluctuations in the underlying assets. Events and their corresponding jumps can occur at random times or at scheduled times. However, the amplitude of the response to an event in either case can be unpredictable and thus can be considered as random. While the volatility of portfolios are often modeled by continuous Brownian motion processes, discontinuous jump processes are more appropriate for modeling important external events that significantly affect the prices of financial assets. Here the discontinuous jump processes are modeled by state and control dependent compound Poisson processes, such that the random jumps come at the times of a pure Poisson process with jump amplitudes that are randomly distributed. The maximal, expected total discounted utility of terminal wealth and instantaneous consumption formulation is in terms of stochastic differential equations with optimal discounted utility objectives. This paper was motivated by a recent paper of Rishel (1999) concerning portfolio optimization when prices are dependent on external events. However, the model has been significantly generalized for more realistic computational considerations with constraints. Although the solution is illustrated with computation for an canonical risk-adverse power utility, the usual canonical solution is not strictly valid, but perturbations about the canonical solution leads to approximations that have feasible computations demands compared to a full stochastic dynamic programming solution.