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:
can be translated for this empirical finding using fractional Poisson process