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## EQUIVALENT MARTINGALE MEASURE FOR MEIXNER PROCESS BY SCHOUTENS 2002

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 $p_t(x,y)$ which for $x,y \in \mathbf{R}$ tells us the probability that the return starting at $x$ will move to $y$ in time $t$.  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 $K$ will be with respect to the equivalent martingale measure $E_Q[ (X_T - K)_+]$.

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 $Meixner(a,b,d,m)$ Schouten calculated that the equivalent martingale measure is $Meixner(a, a\theta+b, d,m)$ where

$\theta = -\frac{1}{a} ( \frac{-\cos(a/2) + \exp(m-r)/2d}{\sin(a/2)})$

In this case, then option prices can be calculated using formulae and numerical integration.

Generally, the Escher transform is as follows.  Let

$M(z,t) = \int_{-\infty}^\infty e^{zx} f(x,t) dx$

Then the Esscher transform is defined by:

$f(x,t,h) = \frac{e^{hx} f(x,t)}{M(h,t)}$

Then we seek $h=h^*$ that ensures that the underlying price is a martingale, so

$1 = e^{\delta} M(1,1;h^*)$

$\delta = \log(M(1,1,h^*)$

This parameter $h^*$ can be found by numerical optimization for general $p_t(x,y)$ for example for nonlinear modifications of the Meixner process.