he
is a reference filtration and
is a column of
adapted
independent standard Brownian motions. Consider a column of prices of several
traded assets
:
where
and
are column and matrixvalued processes adapted to the filtration
.
For simplicity we restrict our attention to assets without dividends. We
assume a possibility to construct a portfolio
such that the value of the portfolio is deterministic during the next
infinitesimally small time
interval:
The
is another columnvalued process adapted to
.
It represents trading strategy. By noarbitrage assumption we conclude that
such riskless portfolio must earn the riskless rate
:
Equivalently,
where the
is stochastic
adapted
riskless rate. We summarize the above argument with the following
statement
We think of columns
,
of the matrix
as elements of a linear space of some finite dimension. We introduce the
linear span
of the set
.
Let
be an orthogonal projection to the linear span
.
The statement (*) reads as
We would like to conclude that
.
Indeed, suppose such conclusion is false and there is a nonzero element
We
have
by the form of the
.
We arrive to a contradiction by taking
in the statement (**). Therefore
.
Equivalently, there are
adapted
processes
such that
or


(Market price of risk)

This result is remarkable because the
is dependent only on one index. The formula
(
Market_price_of_risk
) states that the
excess return of the asset over the riskless rate
is proportional to the volatilities associated with the driving Brownian
motions
and the coefficient of proportionality
does not depend on the type of the asset but is only dependent on the source
of risk. Therefore, the coefficients
are
called "market prices of risk". We
have


(Risk neutral Brownian motion)

According to the proposition (
Girsanov
theorem
), there exists a change of the probability measure that makes the
process
given by
a standard Brownian motion. Such probability measure is called "the risk
neutral measure". Note that
hence the discounted price of any traded asset
is a martingale with respect to the risk neutral measure
:


(Risk neutral pricing)

The above relation is the foundation of classic derivative pricing.
Consider the following special
situation:
where the
and
are some deterministic functions of time. The
(
Market price of risk
) reads
as
or
Hence, in the setting (***), if
is a deterministic function then the market prices of risk are deterministic
functions as well.
A particular vector
may be represented as a linear combination of
in many different ways if the number of vectors
is bigger then the dimensionality of their linear span. If there are more
sources of uncertainty than traded assets then the risk neutral measure is not
unique. This means that there may be a variety of opinions about prices of
contingent claims but still there is no arbitrage.
