Optimal Consumption And Portfolio In A Jump Diffusion Market

Optimal portfolio and consumption is one of the most interesting prob-lems in financial mathematics. The classical Merton's model assumes that the price of the stock follows a geometric Brownian motion, which couldn't display the jumps of the stock price in the practical market, also the distribution daily yield rate has fatter tails, skewness and be leptokuritic than the normal distri-bution. Exponential levy model is very suitable to solve these troubles. If the jump range of the stock price is assumed to follow the logarithmic normal dis-tribution, the density function of the daily yield rate has the shape of weighted sum of many normality density functions with different means and variances. So it is necessary to study the optimal portfolio-consumption problem under exponential levy model of the stock price. N. C. Framstad, B.Φksendal and A. Sulem[4] have studied the optimal portfolio-consumption problem under the utility function for CRRA in a jumps diffusion market.The aim of this paper is to consider an investor in a jump diffusion market who wants to maximize his utility from consumption, allocating his money between a stock and a risk-free bond in chapter 3 and between two different stocks in chapter 4. We discuss the effects of jumps in the model upon the investor's optimal portfolio and consumption. In fact in order to guarantee the total wealth of the investor won't become negative, it does not allow short selling and borrowing in the jump diffusion market. The main method is stochastic dynamic programming, we got a implicit solution of the problem under constant Relative risk aversion(CRRA) utility function, then compared it with that of the Merton's model, and analysed the sensitivity for the solution with respect to some parameters.