I read that a jump in a stock price can be modeled as a Poisson process. But I have also read that a Poisson process is a good model for a counting process (i.e. number of hits to a website per unit time).
Intuitively, I am not able to see the link between a jump process and a counting process. How are both these processes modeled by a Poisson process?
The asset prices were initially modeled as a Brownian motion. One property of a Brownian motion is that it's a (a.s.) continuous process. It may sound a bit crazy given how it is usually shown on graphs, but it's true.
Some people have noticed that asset prices don't seem to be always continuous. They seem to jump out of their continuities from time to time. That's how jump-diffusion models were introduced, where you have Gaussian diffusion combined with Poisson jumps. Poisson process is discrete, so it's discontinuous.
I borrowed the next picture from this site. This kind of plots are often the motivation for jump-diffusion models. I think that these jumps are somewhat more obvious in energy trading. In equity prices they are not easy to establish. Since we get data at finite time intervals, it's not so obvious to tell whether a price change is part of the diffusion or a jump.
Conveniently the jump-diffusion process happens to be a special case of Levy processes, which have a nice theory developed. That's why they're quite popular, and not because there's a clear intuition as to why the jumps have to be Poisson in asset prices.
On the other hand, Poisson distribution was specifically developed for counting processes. For instance, it can be applied to radiation detector measurements, such as Geiger counters. Poisson fits these things perfectly, and intuition is there.
Hence, I would say the link is solely through usage of Poisson distribution: in one case out of convenience, and in another naturally.
