ariance swap is a pathdependent
contract that pays at maturity
an amount depending on the quantity
called realized variance. The times
are daily closure prices (or something reasonable, so that
are small for all
),
,
.
We assume everywhere in this section that the process
is a diffusion (does not
jump):
and is either considered in its martingale measure or being discounted as in
the section (
optimal
utility
). Under this assumption we proceed to show that
is the measure of volatility during time interval
.
Indeed,
Hence,
If the payoff of the contract is a linear function then we are interested in
the risk neutral
expectation
We next show how the last quantity could be approximated with a linear
combination of European claims (static hedge).
Since we assume the diffusion
process,
Hence,
We also assumed "nodrift",
hence,
We
obtain
The last expectation is the European payoff and was considered in the section
(
Log contract
).
We consider next the process with deterministic drift and dividends.
