The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 443 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Continuous-State, Continuous-Time Models 443

type: {A: ω ∈ A ⇔ Xt(ω) = x} determined by specific values of the vari-

able Xt. These sets have probability zero as there is an infinite number

of values Xt. As a consequence, we cannot define conditional expecta-

tion as expectation under the usual definition of conditional probabili-

ties the same way we did in the case of finite-state setting.

Trading Strategies and Trading Gains

We have to define the meaning of trading strategies in the continuous-

state, continuous-time setting; this requires the notion of continuous

trading. Mathematically, continuous trading means that the composi-

tion of portfolios changes continuously at every instant and that these

changes are associated with trading gains or losses. A trading strategy is

a (vector-valued) process θ = {θi} such that θθθθt = {} θti is the portfolio

held at time t. To ensure that there is no anticipation of information,

each trading strategy θ must be an adapted process.

Given a trading strategy, we have to define the gains or losses asso-

ciated with it. In discrete time, the trading gains equal the sum of pay-

offs plus the change of a portfolio’s value

T

 i i

∑ ∑dt

i θi

t



 + ∑S

i

TθT – ∑S 0

i θ

0

t = 0 i i i

over a finite interval [0,T].

We must define trading gains when time is a continuous variable.

Recall from Chapter 8 that it is not possible to replace finite sums of

stochastic increments with pathwise Riemann-Stieltjes integrals after

letting the time interval go to zero. The reason is that, though we can

assume that paths are continuous, we cannot assume that they have

bounded variation. As a consequence, pathwise Riemann-Stieltjes inte-

grals generally do not exist. However, we can assume that paths are of

bounded quadratic variation. Under this latter assumption, using Itô

isometry, we can define pathwise Itô integrals and stochastic integrals.

Let’s first assume that the payoff-rate process is zero, so that there

are only price processes. Under this assumption, the trading gain Tt of a

trading strategy can be represented by a stochastic integral:

t t

T i

t = ∫θθθθsdS = ∑∫θ

idS

s s s

0 i 0

In the rest of this section, we will not strictly adhere to the vector

notation when there is no risk of confusion. For example, we will write