The effect of the jumps in price and volatility on dynamic portfolio choice

Based on the double-jump diffusion model with the affine structure in the state variable, this paper applies stochastic control method to obtain the analytical solution of dynamic portfolio choice, which maximizes the expected power utility of portfolio terminal wealth, and studies the effect of jumps in security price and volatility on dynamic portfolio choice. Analysis shows that the state variables in portfolio model both have no “market timing” for power utility investor; the jumps in price and volatility make the dynamic portfolio problem to blend a standard dynamic problem with a static buy-and-hold problem; jumps in price make the investor not only never take a leverage or short position on the risky asset but also never invest all of his wealth on the risky asset in dynamic portfolio; the jumps result in positive or negative jump hedging demand, which depends on that the direction of jumps in price is upward or downward, at the same time, the jumps in volatility will in further enhance or weaken the magnitude of jump hedging demand. This study demonstrates that jumps in price and volatility change the standard dynamic portfolio problem in several important ways, and should be considered in the context of dynamic portfolio choice.