This is a conjecture based on Merton (1980) and empirical fractional Poisson jumps observed in daily volatility jumps using 1900 stocks.
The option prices with fractional Poisson jump process with parameters should satisfy some sort of fractional Black-Scholes partial differential equation.
where is the probability of events in and satisfies
In order to reach (*) the basic idea is to take a time derivative of which splits up by product rule and chain rule into three parts. One is trying to use the fact that satisfies
Take one time derivative and use on one piece and then apply the integral and use (**) on another part and then exploit (***).
INGREDIENTS FOR ANALYSIS
Recall that the Caputo derivative is
(a) from (behgin-orsinger-fpp) and therefore
(b) Integration by parts: when then
(c) Now let
Now apply . The first term (I) is
by (a). Integrate by parts to resolve terms (II) and (III).
So now we use the equation satisfied by to work out the FPDE that must satisfy:
So, for example