Capital Asset Pricing for Markets with Intensity Based Jumps

Abstract

This paper proposes a unified framework for portfolio optimization, derivative pricing, modeling and risk measurement in financial markets with security price processes that exhibit intensity based jumps. It is based on the natural assumption that investors prefer more for less, in the sense that for two given portfolios with the same variance of its increments, the one with the higher expected increment is preferred. If one additionally assumes that the market together with its monetary authority acts to maximize the long term growth of the market portfolio, then this portfolio exhibits a very particular dynamics. In a market without jumps the resulting dynamics equals that of the growth optimal portfolio (GOP). Conditions are formulated under which the well-known capital asset pricing model is generalized for markets with intensity based jumps. Furthermore, the Markowitz efficient frontier and the Sharpe ratio are recovered in this continuous time setting. In this paper the numeraire for derivative pricing is chosen to be the GOP. Primary security account prices, when expressed in units of the GOP, turn out to be supermartingales. In the proposed framework an equivalent risk neutral martingale measure need not exist. Fair derivative prices are obtained as conditional expectations of future payoff structures under the real world probability measure. The concept of fair pricing is shown to generalize the classical risk neutral and the actuarial net present value pricing methodologies.