The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 455 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Continuous-State, Continuous-Time Models 455

A multivariate standard Brownian motion B = (B 1 ,...,BD) in RD adapted

to the filtration ℑt is defined over this probability space. From Chapter

10 we know that there are mathematical subtleties that we will not take

into consideration, as regards whether (1) the filtration is given and the

Brownian motion is adapted to the filtration or (2) the filtration is gen-

erated by the Brownian motion.

Suppose that there are N price processes X = (X^1 ,...,XN) that form a

multivariate Itô process in RN. Trading strategies are adapted processes θ

= (θ^1 ,...,θΝ) that represent the quantity of each asset held at each instant.

In order to ensure the existence of stochastic integrals, we require the

processes (X^1 ,...,XN) and any trading strategy to be of bounded varia-

tion. Let’s first suppose that there is no payoff-rate process. This assump-

tion will be relaxed in a later section. Suppose also that one of these

processes, say X^1

t , is defined by a short-rate process r, so that

t

∫ruud

X^10

t = e

or

dX^11

t = rtXt dt

where rt is a deterministic function of t called the short-rate process.

Note that X^1

t could be replaced by a trading strategy. We can think of rt

as the risk-free short-term continuously compounding interest rate and

of X^1

t as a risk-free continuously compounding bank account.

The concept of arbitrage and of trading strategy was defined in the

previous section. We now introduce the concept of deflators in a contin-

uous-time continuous-state setting. Any strictly positive Itô process is

called a deflator. Given a deflator Y we can deflate any process X,

obtaining a new deflated process

Y

Xt = XtYt

For example, any stock price process of a nondefaulting firm or the risk-

free bank account is a deflator. For technical reasons it is necessary to intro-

duce the concept of regular deflators. A regular deflator is a deflator that,

after deflation, leaves unchanged the set of admissible bounded-variation

trading strategies.

We can make the first step towards defining a theory of pricing

based on equivalent martingale measures. It can be demonstrated that if