Continuous-Time Models in Portfolio Theory

We proceed from the classical Markowitz portfolio theory in the static framework. A Markowitz efficient portfolio gives the highest expected return for a given amount of risk. The extension of this model leads to the fund separation theorem of CAPM. The first solution to the dynamic portfolio problem in continuous time was introduced by Merton. He assumed that prices of assets follow an Ito process with constant expected return and covariance of returns, and also market equilibrium. We present the Olson-Rosenberg paradox, which demonstrates a contradiction between the assumption of stationary distribution of returns and the separation theorem on the one hand and the assumption of market equilibrium on the other hand. One consistent approach in the dynamic setting is given by Stochastic Portfolio Theory of R. Femholz. The rates of return and volatilities follow stochastic processes and equilibrium is not assumed. We will also discuss another approach to the Olson-Rosenberg paradox, which aleds to nonlinear differential equations.