The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 445 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Continuous-State, Continuous-Time Models 445

In the spirit of positive economics, continuous-time financial models

are mathematical devices used to predict, albeit in a probabilistic sense,

financial observations made at discrete intervals of time. Stochastic

gains predict trading gains only at discrete intervals of time—the only

intervals that can be observed. Continuous-time finance should be seen

as a logical construction that meets observations only at a finite number

of dates, not as a realistic description of financial trading.

Let’s consider processes without any intermediate payoff. A self-

financing trading strategy is a trading strategy such that the following

relationships hold:

i i i i i

θθθθtSt = ∑θtSt = ∑



θi 0 S 0 + ∫

t

θtdSt , t ∈ [ 0 , T]

i i  0 

We first define arbitrage in the absence of a payoff-rate process. An

arbitrage is a self-financing trading strategy such that: θ 0 S 0 < 0 and θTST

≥ 0, or θ 0 S 0 ≤ 0 and θTST > 0. If there is a payoff-rate process, a self-

financing trading strategy is a trading strategy such that the following

relationships hold:

i i i i i

θθθθtSt = ∑θtSt = ∑



θi 0 S 0 + ∫

t

θtdGt , t ∈ [ 0 , T]

i i  0 

where G i i

t

i = S

t + Dt is the gain process as previously defined. An arbi-

trage is a self-financing trading strategy such that: θ 0 S 0 < 0 and θTST ≥

0, or θ 0 S 0 ≤ 0 and θTST > 0.

ARBITRAGE PRICING IN CONTINUOUS-STATE,

CONTINUOUS-TIME

The abstract principles of arbitrage pricing are the same in a discrete-

state, discrete-time setting as in a continuous-state, continuous-time set-

ting. Arbitrage pricing is relative pricing. In the absence of arbitrage, the

price and payoff-rate processes of a set of basic assets fix the prices of

other assets given the payoff-rate process of the latter. If markets are com-

plete, every price process can be computed in this way. In a discrete-state,

discrete-time setting, the computation of arbitrage pricing is done with

matrix algebra. In fact, in the absence of arbitrage, every price process

can be expressed in two alternative ways: