23-RiskManagement Page 740 Wednesday, February 4, 2004 1:13 PM
740 The Mathematics of Financial Modeling and Investment ManagementMarket completeness entails that there is a core of price processes
such that any cash flow stream can be engineered as a time-varying, but
self-financing, portfolio made up of the core price processes. For example,
in a complete market a complex derivative instrument can be replicated
by a portfolio of simpler instruments. A bank that creates a credit deriva-
tive can always hedge its positions.
As we have seen in Chapter 14 on arbitrage, in the finite-state, one-
step case, market completeness means that the number of linearly inde-
pendent price processes is equal to the number of states. In other words,
a market is complete if there are as many linearly independent price pro-
cesses as states of the world. This notion can be easily expressed in
terms of linear algebra. In the finite-state, discrete-time case the above
conditions must be replaced by the notion of dynamically complete mar-
kets as assets can be traded at intermediate dates. In fact, the number of
linearly independent price processes can be smaller than the number of
states provided that assets can be traded repeatedly. As shown by Dar-
rell Duffie and Chi-Fu Huang^1 and Hua He,^2 what is needed, in this
case, is that there are as many linearly independent price processes as
there are branches leaving a node in the market information structure.
Based on this, it can be demonstrated that the binomial model and its
extension to multiple variables are complete.
When we proceed to the continuous-state, continuous-time case this
notion looses meaning. In this case there is a continuum of states and a
continuum of instants. The infinite number of trading instants allows
markets to be complete even if they are formed by a finite number of
securities. There are restrictions to ensure that a market model is com-
plete. A fundamental theorem assures that, in the absence of arbitrage,
market completeness is associated with the uniqueness of the equivalent
martingale measure. In a complete market the equivalent martingale
measure is unique, while an incomplete market is characterized by infi-
nite martingale measures. This happens because there are contingencies
that cannot be priced by arbitrage.
The condition of market completeness is violated in many important
models. Two, in particular, have attracted attention: jump-diffusion mod-
els and stochastic volatility models. Jump-diffusion models are models
formed by diffusions plus processes where finite jumps occur at random
times, such as at those times represented by a Poisson process. Stochastic(^1) Darrell Duffie and Chi-Fu Huang, “Implementing Arrow-Debreu Equilibria by
Continuous Trading of Few Long-Lived Securities,” Econometrica 53 (1985), pp.
1337–1356
(^2) Hua He, “Convergence from Discrete to Continuous Time Contingent Claims Pric-
es,” Review of Financial Studies 3, no. 4 (1990), pp. 523–546.