Example of an incomplete model
where .
Suppose there is portfolio having stocks, then the portfolio’s value follows
From the risk-neutral measure perspective, the discount process is
then
Note that
Denote , and
Define
then
and thus
and is a martingale under because
To verify the 1st FTAP (i.e. the mode has no arbitrage), we claim that there are infinite number of risk-neutral measures because we can define
and define
Note that
which are two independent brownian motions. And is also a risk-neutral measure because
but ; in fact, there are risk-neutral measures for different .
And thus the market is not complete.
From the replication perspective:
We cannot hedge a European call under this setting.
under the measure , we have
The call price at time t is
Consider the replication:
Therefore, this equation does not have a solution.
Stochastic integrals of deterministic integrands
Note that not every stochastic integral is normally distributed, for example
Proof:
Since this integral is always bigger than , it cannot be normally distributed because the normal distribution has positive probability of being smaller than .
However, for some deterministic function , we have
Proof: denote
Then
Thus
Dupire’s Formula
Suppose
We want to choose so that the model can match all European option prices.
Define the transition densities as
The SDE (*) falls in a more general form
The Kolmogorov equations claim that
- backward (BKE):
- forward (FKE):
Note that
to find s.t. the model matches the observed prices, Dupire claims that
Example: where
plug into the Dupire, i.e. , the solution is
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