The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 444 Wednesday, February 4, 2004 1:08 PM

444 The Mathematics of Financial Modeling and Investment Management

θ ⋅ S to represent the scalar product θ ⋅ S. If a payoff-rate process is asso-

ciated with each asset, we have to add the gains consequent to the pay-

off-rate process. We therefore define the gain process

G i i

t

i = S

t + Dt

as the sum of the price processes plus the cumulative payoff-rate pro-

cesses and we define the trading gains as the stochastic integral

t t

T i

t = ∫θθθθsdG = ∑∫θsdG

i

s s

0 i 0

How can we match the abstract notion of a stochastic integral with

the buying and selling of assets? In discrete time, trading gains have a

meaning that is in agreement with the practical notion of buying a port-

folio of assets, holding it for a period, and then selling it at market

prices, thus realizing either a gain or a loss. One might object that in

continuous time this meaning is lost. How can a process where prices

change so that their total variation is unbounded be a reasonable repre-

sentation of financial reality? This is a question of methodology that is

relevant to every field of science. In classical physics, the use of continu-

ous models was assumed to reflect reality; time and space, for example,

were considered continuous. Quantum physics upset the conceptual cart

of classical physics and the reality of continuous processes has since been

questioned at every level. In quantum physics, a theory is considered to

be nothing but a model useful as a mathematical device to predict mea-

surements. This is, in essence, the theory set forth in the 1930s by Niels

Bohr and the School of Copenhaghen; it has now become mainstream

methodology in physics. It is also, ultimately, the point of view of posi-

tive economics. In a famous and widely quoted essay, Milton Friedman,

recipient of the 1976 Nobel Prize in Economic Science, wrote:

The relevant question to ask about the “assumptions” of a theory

is not whether they are descriptively “realistic,” for they never are,

but whether they are sufficiently good approximations for the pur-

pose in hand. And this question can be answered only by seeing

whether the theory works, which means if it yields sufficiently

accurate predictions.^2

(^2) Milton Friedman, Essays in the Theory of Positive Economics (Chicago: University
of Chicago Press, 1953).