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
which is a martingale. 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):
For European call options, we have
to find s.t. the model matches the observed prices, Dupire claims that
Example: where
plug into the Dupire, i.e. , the solution is
so that
Loading Comments...