23-RiskManagement Page 739 Wednesday, February 4, 2004 1:13 PM
Risk Management 739be replicated by engineering appropriate portfolios. In other words,
there is a market, and therefore a price, for every contingency.
Markets in which this hedging is not possible are called incomplete
markets. In incomplete markets there are contingencies that are not
traded and cannot be priced and replicated. An investor who “owns”
one of these contingencies is stuck with them and has no assurance that
a buyer will be found. An incomplete market might be completed by
adding appropriate assets provided that they are tradable. If the market
is completed, every contingency becomes tradable. However, there is no
guarantee that an arbitrary market can be completed.
The question of market completeness is fairly complicated. There
are two key aspects in the notion of market completeness: (1) the math-
ematics of market completeness and (2) the economic rationale as to
why markets are complete or can be completed. We discuss each below.The Mathematics of Market Completeness
The purely mathematical aspect of the completeness of a given market
model is a widely studied subject. Some market models are complete
while others are not. For instance, a market where stock prices evolve as
geometric random walks and a risk-free asset is available is complete.
On the other hand, a market represented by a stochastic volatility model
is incomplete.
A market is complete if any cash flow stochastic process can be rep-
licated by an appropriate self-financing trading strategy with some ini-
tial investment. Replication means that the self-financing trading
strategy and the original cash flow process are equal processes. Recall
that in Chapter 6 on probability theory we defined four notions of
equality between stochastic processes. The weakest condition of equal-
ity requires that two processes have the same finite-dimensional distri-
butions. This concept of equality is insufficient to define replication.
The strongest condition of equality requires that two processes have the
same paths except for a set of measure zero. Replication requires that
the original cash flow process and the replicating self-financing trading
strategy are equal processes in this strongest sense.
Recall also from Chapter 10 that there are two types of solutions of
stochastic differential equations: strong solutions and weak solutions.
Strong solutions are solutions built on given Brownian motions while
weak solutions include their own Brownian motion. This notion, which
might look abstract and remote, is however important from the point of
view of a replicating strategy. If a replicating process is defined by a sto-
chastic differential equation, the difference between strong and weak
solutions is important.