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Optimal Diversification under Budget and No-Short-Selling Constraints
Srdjan Stojanovic
University of Cincinnati
Presenter's website:
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.
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