Optimal Portfolios with Stochastic Interest Rates

Abstract

The continuous-time portfolio problem has its origin in the pioneering work of Merton (1969, 1971), which deals with finding the optimal investment strategy of an investor. More precisely, the investor looks for an optimal decision on how many shares of which security he should hold at any time between now and a time horizon Tto maximize expected utility from terminal wealth. In the classical Merton problem the investor can allocate money to a riskless money market account (syn. savings account) and to ddifferent stocks of varying degrees of risk. By describing the actions of the investor via the portfolio process (i.e. the percentages of wealth invested in the different securities) Merton was able to reduce the portfolio problem to a control problem which could be solved by using standard stochastic control methodology.1 A drawback of Merton’s model, however, is the assumption of deterministic interest rates. Our main objective in the current chapter is to overcome this restriction. Therefore, we analyze portfolio problems in which the interest rate dynamics of the economy can be described by the Vasicek model, the Dothan model, the Black-Karasinski model, or the Cox-Ingersoll-Ross model. Our presentation of the problem with Ho-Lee or Vasicek term structure goes back to Korn/Kraft (2001), who applied stochastic control theory to arrive at the solution.2 By contrast, the results concerning the Dothan and Black-Karasinski models are new. The portfolio problem with Cox-Ingersoll-Ross term structure was partly solved by Deelstra/Oraselli/Koehl (2000) applying the martingale approach under the assumption of a complete market. In contrast to their approach we apply stochastic control theory so that we can drop the assumption of completeness. Besides, we are able to solve the problem for a set of market coefficients not covered by the results of Deelstra/Oraselli/Koehl (2000).