**Candidate: Brownian motion**

- Markovian
- "Tractable"(-ish, depending on how you transform it)
- Excess kurtosis = 0
- Market completeness (if number of hedging instruments >= number of sources of randomness)

**Poisson**We've already met Gauss last time. Now we'll introduce a contemporary of him, Siméon Denis Poisson

**.**While Brownian motion is a diffusion stochastic process (a

*random walk*down the street), the Poisson process is a

*counting process*that is associated with jumps. When we add a Poisson counting process to the diffusion equation, the stochastic differential equation becomes a jump-diffusion equation that allows for jumps, or "gapping up/down". Why is this desirable or required? Just talk to any trader, and they'll tell you that price jump is a reality, especially for markets with substantial close hours (i.e. the majority of the markets other than FX, S&P futures...). So instead of a random walk down the street, perhaps after all it is more like parkour down the street?

**You Jump, I Jump**What are the advantages of modeling asset prices with a jump-diffusion process over a diffusion only process?

**Candidate: Jump-diffusion Process**

- Markovian
- Not tractable (unless under restrictive assumptions)
- Excess kurtosis > 0
- Market incomplete (unless under restrictive assumptions)

*fat tail*feature, hence excess kurtosis. Finally, market completeness is in general lost (there are two sources of randomness - diffusion and jump, but only one hedging instrument - the asset itself), and one has to estimate a market price of risk of the jumps. Incidentally, in Merton's model he assumes that such a market price of risk is zero because of diversifiability.

The implementation of jump-diffusion model can be challenging. The numerical calibration is of course more computationally intensive, but the more subtle and fundamental issue is this: how do we identify jumps? When there is a "sudden" gap in the time series, how can one be sure that it is NOT just an extreme value, yet still being drawn from the good old Gaussian distribution?

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