Continuous-time finance theory, especially for option pricing boils down to the following components:
(a) Underlying asset process modeling. The key issue here is that there’s some Fokker-Planck type evolution equation for a probability density function which for tells us the probability that the return starting at will move to in time . This is the most important part of fiance theory.
(b) All derivatives of an asset can be priced easily once an equivalent martingale measure is found on a path space. This is a measure for which the underlying return process is a martingale. Then the derivative prices will just be the present value of future cashflows, for example, the price of a European call struck at will be with respect to the equivalent martingale measure .
With Levy processes, there is an issue of incomplete markets, i.e. non-uniqueness of the equivalent martingale measure that is handled by choosing among differing martingale measures using for example the Esscher transform. For the Meixner distributions Schouten calculated that the equivalent martingale measure is where
In this case, then option prices can be calculated using formulae and numerical integration.
Generally, the Escher transform is as follows. Let
Then the Esscher transform is defined by:
Then we seek that ensures that the underlying price is a martingale, so
This parameter can be found by numerical optimization for general for example for nonlinear modifications of the Meixner process.