The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 463 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Continuous-State, Continuous-Time Models 463

It can be demonstrated by direct computation that the above for-

mula is equal to the Black-Scholes option pricing formula presented ear-

lier in this chapter.

EQUIVALENT MARTINGALE MEASURES AND

COMPLETE MARKETS

In the continuous-state, continuous-time setting, a market is said to be

complete if any finite-variance random variable Y can be obtained as the

terminal value at time T of a self-financing trading strategy θ: Y = θTXT.

A fundamental theorem of arbitrage pricing states that, in the absence

of arbitrage, a market is complete if and only if there is a unique equiv-

alent martingale measure. This is condition can be made more specific

given that the market is populated with assets that follow Itô processes.

Suppose that the price process is X = (V,S^1 ,...,SN–1) where, as in the pre-

vious section:

dSt = μtdt + σtdBt

dVt = rVtdt

and B is a standard Brownian motion B = (B^1 ,...,BD) in RD.

It can be demonstrated that markets are complete if and only if

rank(σ) = d almost everywhere. This condition should be compared with

the conditions for completeness we established in the discrete-state set-

ting in the previous chapter. In that setting, we demonstrated that mar-

kets are complete if and only if the number of linearly independent price

processes is equal to the maximum number of branches leaving a node.

In fact, market completeness is equivalent to the possibility of solving a

linear system with as many equations as branches leaving each node.

In the present continuous-state setting, there are infinite states and

so we need different types of considerations. Roughly speaking, each

price process (which is an Itô process) depends on D independent

sources of uncertainty as we assume that the standard Brownian motion

is D-dimensional. In a finite-state setting this means that, if processes

are Markovian, at each time step any process can jump to D different

values. The market is complete if there are D independent price pro-

cesses. Note that the number D is arbitrary.