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The fundamental theorem of asset pricing is a theorem of financial engineering
that relates the existence of an equivalent martingale measure to the
no-arbitrage condition. It has origins in Cox and Ross' (1976) method of
risk neutral valuation, which was
formalized by Harrison and Kreps (1979),
Harrison and Pliska (1981,
1983) and
Back and Pliska (1991).
Those authors generalized the notion of a risk neutral measure to that of
an equivalent martingale measure. Today, their methodology and the
fundamental theorem of asset pricing represent the primary
approach financial
engineers use to price
derivatives. This article explains these concepts.
Let's start by asking: in what units should we measure wealth or the value of assets? In ancient Rome, silver coins were the unit of account. Some
agrarian societies measure wealth in heads of cattle. Today, values are
often expressed in terms of USD
or some other currency.
In financial engineering, a unit of account is called a numeraire.
If no numeraire is specified, the numeraire is implicitly understood to be
one unit of some currency. Often, calculations can be simplified by
employing some other
traded asset as a numeraire. All that is really required is that a numeraire
have[1] a
strictly positive value. The numeraire may have a value that, when
measured in units of currency, is constant, variable but deterministic, or
even random over time. Examples of each of these cases are:
 A
unit of currency is a numeraire with constant value.
If
interest rates are assumed constant, a
discount instrument such as a
Treasury bill is a numeraire with variable but deterministic value.
If
interest rates are assumed stochastic, the accumulated value of a dollar
invested in a money market account is a
numeraire with a random value.
Let t represent time and consider a market comprising a numeraire and d other
traded assets. We may denote their prices in units of currency as
 .
For convenience, let
be the numeraire. Then prices of the assets in units of the numeraire are
calculated as
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[1] |
Two probability measures are said to be
equivalent if they have the same null spaces.
That is, the set of events that have probability 0 under one measure is
the same as the set of events that have probability 0 under the other
measure.
For a given numeraire
, a
probability measure
is a martingale measure relative
to
if all
asset prices measured in units of
are
martingales under
.
This means that, if an asset
 matures
at future time T,
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[2] |
where
indicates an expectation taken with
respect to probability measure
.
The asset's price today in units of the numeraire is simply the (measure
)
expectation of its value at maturity in units of the numeraire. This is
highly analogous to Cox and Ross' (1976)
basic idea of risk neutral valuation. The risk neutral measure becomes
.
Accumulating or
discounting values at the risk free rate is replaced by dividing
prices by values of the numeraire at different points in time. Now it
should make more sense why we might use assets such as discount
instruments or money market accounts as numeraires. The numeraire can play
the role of a
discount factor.
Let
be the "real world" probability measure. We call
an equivalent martingale
measure if
and
are equivalent, and
is a martingale measure relative to some numeraire
.
Building on groundbreaking
discrete-time work of Harrison and Kreps (1979),
Harrison and Pliska (1981)
derived the fundamental
theorem of asset pricing, which states that
a
market is arbitrage free if and
only if there exists an equivalent martingale measure
,
and
a
market is complete if and only if there exists a unique equivalent
martingale measure
.
It is a standard assumption of economics that markets are
arbitrage free. If we make that assumption, the
fundamental theorem of asset pricing tells us there is an equivalent
martingale measure
,
and we can use
to calculate asset prices as expectations according to [2].
More often than not, this is how financial engineers approach
option pricing or other financial engineering
problems today.
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 arbitrage-free pricing
The approach to pricing instruments that underlies essentially all of financial
engineering.
Brownian
motion A simple continuous stochastic process that is widely used in physics
and finance for modeling random behavior that evolves over time.
law of one price
The notion that, if two assets have identical cash flows, they should have the
same market value.
martingale
A type of stochastic process that has zero drift.
option
pricing theory An introductory article.
random walk
hypothesis Financial model based
on the empirical observation that stock and commodity prices behave like a
random walk. |
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Back (2005)
and Bingham and Kiesel (2004)
are excellent financial engineering texts.
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 Back,
Kerry and Stanley R. Pliska (1991). On the fundamental theorem
of asset pricing with an infinite state space, Journal of
Mathematical Economics, 20, 1-18.
Cox, John C. and Stephen A. Ross
(1976). The valuation of options for alternative stochastic
processes, Journal of Financial Economics, 3, 145-166.
Harrison,
J. Michael and David M. Kreps (1979). Martingales and
arbitrage in multiperiod securities markets, Journal of
Economic Theory, 20, 381-408.
Harrison,
J. Michael and Stanley R. Pliska (1981). Martingales and
stochastic integrals in the theory of continuous trading,
Stochastic Processes and their Applications, 11, 215-260.
Harrison,
J. Michael and Stanley R. Pliska (1983). A stochastic calculus
model of continuous trading: Complete markets, Stochastic
Processes and their Applications, 15, 313-316. |
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