he goal of
this section is to understand the change of numeraire from trading point of
view. We perform the delta-hedging argument and connect the change of
numeraire to the change of variables in the backward Kolmogorov's equation.
The state of the world is given by the processes
and
suitable as numeraires, see (
Suitable
numeraire
).
Thus
and
There is a traded derivative priced at
.
Assuming that the derivative is defined by the final payoff
,
the function
has two descriptions. The notations are explained below.
First
description:
Second description:
We assume above that
,
are described by the
SDEs
with respect to any numeraire
.
The notation
stands
for
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(XY bracket)
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According to the formula (
Change of
Brownian motion
), the
terms are connected by the
relationships
Similarly,
We would like to see connection of these results to the delta hedging
argument.
We consider a derivative
given by the state variable
.
We form a
portfolio
and perform
hedging
calculation (see the argument of the chapter (
Delta
hedging
)):
We change the unknown
function
calculate the
derivatives
and substitute these into the PDE
:
We
conclude
By symmetry of
and
variables, we
obtain
We compare with the proposition
(
Multidimensional
backward Kolmogorov equation
) and derive
These results agree with the previously stated
goals
and
Hence, from PDE point of view, change of numeraire is the particular
multiplicative change of the unknown function. From
-hedging
point of view, it is a change of units of measure. For example, an interest
rate contract may be priced in units of currency or in units of treasury
bills.
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