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## ROBERT MERTON’S 1980 POISSON JUMPS MUST BE UPDATED TO FRACTIONAL POISSON PROCESS

I love to complain about the Aristotelian orthodoxy of Black-Scholes-Merton.  Well, so they are smart people obviously and it was Merton (optionpricingwhenunderlingstock who introduced Poisson jumps to the log-normal model of Black-Scholes (1973).  This should be updated based on my results which show that at least in the equities volatility jumps do not follow a Poisson jump process but a fractional Poisson jump process.

Merton’s formula for option pricing with jumps:

$F(S,t) = \sum_{n=0}^\infty \frac{e^{-\lambda\tau} (\lambda\tau)^n }{n!} E[ W( SX_n e^{-\lambda k\tau},\tau,E,\sigma^2,r)]$

can be translated for this empirical finding using fractional Poisson process

behgin-orsinger-fpp