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 is given by the processes
and
suitable as numeraires ( Suitable
numeraire). There is a traded derivative with the price
 .
We
have
and
Assuming that the derivative is defined by the final pay off
 ,
the function
has two descriptions. The notations are explained below.
First
description:  
Second description:
  
We assume above that in the numeraire
the SDEs
are 
The
stands for a scalar product of columns. The notation
stands
for | | (XY bracket) |
According to the ( 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 have a derivative
in the market given by the state variable
 .
We form a
portfolio
and perform the
 hedging
calculation (see the argument in the chapter ( Delta
hedging)):  
We perform the change of the unknown
function
so we calculate
derivatives   
substitute these into the
PDE
and
simplify 
By the symmetry of
and
variables  
Hence, we
obtain   
These results agree with the previously stated
goals
and
Hence, from the PDE point of view, change of numeraire is the particular
multiplicative change of the unknown function. It also may be regarded as a
change of units of measure. We change from
measured in units of currency to the
measured in units of the price of a traded instrument
 .
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