21-Bond Portfolio Man Page 674 Wednesday, February 4, 2004 1:12 PM
674 The Mathematics of Financial Modeling and Investment Managementmaximization of a functional, with a functional being a real-valued func-
tion defined over other functions. Most classical physics can be expressed
equivalently through differential equations or variational principles.
Variational methodologies also have important applications in engi-
neering, where they are used to select a path that maximizes or mini-
mizes a functional given some exogenous dynamics. For example, one
might want to find the optimal path that an airplane must follow in
order to minimize fuel consumption or flying time. The given dynamics
are the laws of motion and eventually specific laws that describe the
atmosphere and the behavior of the airplane.
Economics and finance theory have inherited this general scheme.
General equilibrium theories can be expressed as variational principles.
However, financial applications generally assume that some dynamics
are given. In the case of bond portfolios, for example, the dynamics of
interest rates are assumed to be exogenously given. The problem is to
find the optimal trading strategy that satisfies some specific objective. In
the case of immunization an objective might be to match liabilities at
the minimum cost with zero exposure to interest rates fluctuations. The
solution is a path of the portfolio’s weights. In continuous time, it
would be a continuous trading strategy.
Such problems are rarely solvable analytically; numerical techniques,
and in particular multistage stochastic optimization, are typically
required. The key advantage of stochastic programming is its ability to
optimize on the entire path followed by exogenously given quantities. In
applications such as bond portfolio optimization, this is an advantage
over myopic strategies which optimize looking ahead only one period.
However, because stochastic programming works by creating a set of sce-
narios and choosing the scenario that optimizes a given objective, it
involves huge computational costs. Only recently have advances in IT
technology made it feasible to create the large number of scenarios
required for stochastic optimization. Hence there is a renewed interest in
these techniques both at academia and inside financial firms.^18Scenario Generation
The generation of scenarios (i.e., joint paths of the stochastic variables) is
key to stochastic programming. Until recently, it was imperative to create
a parsimonious system of scenarios. Complex problems could be solved
only on supercomputers or massively parallel computers at costs prohibi-
tive for most organizations. While parsimony is still a requirement, sys-(^18) A presentation of stochastic programming in finance can be found in Zenios, Prac-
tical Financial Optimization, forthcoming.