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Optimal Diversification under Budget and No-Short-Selling Constraints

Srdjan Stojanovic
University of Cincinnati

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A large number of stocks are considered for trading. Their prices evolve according to the known system of stochastic differential equations. An index (or more generally, a modest number of indices), such as the Dow Jones or S&P 500, is tracked for the purpose of gaining information about the market performance. The index is correlated with the traded stocks and also evolves according to the known stochastic differential equation. The portfolio consists of the dollar amount invested in each particular stock. The portfolio is constrained in a natural way: budget constraint (the sum of all the investments is equal to the wealth of the economic agent, or more generally, equal to an arbitrary function of wealth, index, and time) and No-Short-Selling constraint (or more generally, short-selling is bounded by an arbitrary function of wealth, index, and time). Arbitrary utility functions (possibly discontinuous for the probability maximization problems) are considered. The problem is solved completely by deducing the associated backward parabolic Hamilton-Jacobi-Bellman partial differential equation, which characterizes the value function of the above stochastic control problem, and by deducing the associated backward parabolic (possibly degenerate) Monge-Ampere type partial differential equation, which also characterizes the same value function, and furthermore, by designing a numerical algorithms for solving such fully nonlinear partial differential equations. All this is done as an efficient pure Mathematica code.

The main trust of the paper is the ability to handle constraints on the investment portfolio, and in particular, the most difficult constraint among them, the No-Short-Selling constraint. Very much different portfolio strategies are computed depending on the type of the constraints imposed. Some examples are available in the cases: no constraints, budget constraint only, budget and No-Short-Selling constraint. For more information contact the author.