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## A TIME-FRACTIONAL BLACK-SCHOLES PARTIAL DIFFERENTIAL EQUATION FOR FRACTIONAL POISSON JUMPS

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 $(\mu,\lambda)$ should satisfy some sort of fractional Black-Scholes partial differential equation.

$F(S,t) = \sum_{n=0}^\infty P^{\mu,\lambda}_n(t) EW( e^{-\lambda k t} X_n, t; \sigma,r)$ (*)

where $P^{\mu,\lambda}_n(t)$ is the probability of $n$ events in $[0,t]$ and satisfies

$D^\mu_t P_n = -\lambda P_n + \lambda P_{n-1}$ (**)

In order to reach (*) the basic idea is to take a time derivative of $F(S,t)$ which splits up by product rule and chain rule into three parts.  One is trying to use the fact that $W$ satisfies

$\frac{1}{2}\sigma^2 V_n^2 W_{11} + r V_n W_1 - W_2 - rW = 0$ (***)

Take one time derivative and use $SF_S(S,t) = \sum P_n V_n EW_1$ on one piece and then apply the integral $\int_0^t ds (t-s)^{-1-\mu}$ and use (**) on another part and then exploit (***).

INGREDIENTS FOR ANALYSIS

Recall that the Caputo derivative is

$D^\mu_* f(t) = \frac{1}{\Gamma(1-\mu)} \int_0^t (t-s)^{-\mu} f'(s) ds$ when $0 < \mu <1$

(a)  $D^\mu_t P_n = -\lambda P_n + \lambda P_{n-1}$ from (behgin-orsinger-fpp) and therefore

$\sum_{n=0}^\infty D^\mu_*P_n(t) G(X_n) = \lambda P_1 G(X_1)$

(b)  Integration by parts:  when $F(0)=0$ then

$\int_0^t (t-s)^{-\mu-1} F(s) ds =$

$\frac{1}{\mu \Gamma(1-\mu)} D^\mu F(t) + t^{-\mu}/\mu F(t)$

(c)  Now let $F(S,t) = \sum_{n=0}^\infty P_n(t) EW(V_n)$

$F_t = \sum_{n=0}^\infty P'_n EW -\lambda k \sum P_n V_n EW_1 + \sum P_n EW_2 = I + II + III$

Now apply $\int_0^t (t-s)^{-\mu-1} ds$.  The first term (I) is

$\lambda P_1 EW$ by (a).  Integrate by parts to resolve terms (II) and (III).

$D^{\mu+1} F = \lambda DP_1 EW - \lambda^2 k P_1 EW_1 + \lambda P_1 EW_2 +$

$t^{-\mu}/\mu ( \sum P_n ( -\lambda k V_n EW_1 + EW_2)$

So now we use the equation satisfied by $W$ to work out the FPDE that $F$ must satisfy:

$\frac{1}{2}\sigma^2 V_n^2 W_{11} + rV_n W_1 - W_2 - rW=0$

So, for example

$D^{\mu+1} F - t^{\mu+1}/\mu \sum P_n( \frac{1}{2}\sigma^2 V_n^2 EW_{11} + (r-\lambda k) V_n EW_1 + rEW) =$

$\lambda DP_1 EW - \lambda^2 k P_1 EW_1 + \lambda P_1 EW_2$