e use the
notation
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(MMA numeraire)
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The
is the worth of one unit of reference currency invested in MMA at time
.
It is a traded asset suitable as a numeraire, see
(
Suitable numeraire
).
According to the martingale result
(
Risk neutral pricing
), price
of a call stroke at
written on the traded asset
is given
by
Let
,
.
We rewrite the last formula for
:
The expressions
and
may be regarded as parts of kernels for numeraire changes, see
(
Change of numeraire
definition
). Indeed,
where
and the
Prob
is the probability changed to the numeraire
.
Similarly,
where
is the price of riskless bond with maturity
as observed at time
and
Prob
is the probability with respect to the numeraire
.
We compute the probabilities as
follows.
where the
is a martingale under
Prob
.
where the
is a martingale under
Prob
.
We are unable to proceed further without some assumptions about the
distribution. Under log-normal assumption, we compute both of the
probabilities by
setting
where the variables
are standard normal with respect to the corresponding measures and the
volatility
is the same in both of the expressions as shown in the
(
Change of measure recipe
section
).
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