Saturday, November 23, 2013

How do you model the dynamics of asset prices? A self-reflection (2 of 3)

Last time we have set a stage for the discussion on modeling the dynamics of asset prices. Gaussian process is the "canonical" way to go, and we've seen the characteristics/limitations of it. To recap:

       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)
This view is usually summed up and referred to as "random walk" (or diffusion, if you are from the natural science branch of the academia). But does the financial market really walk? Or does it prefer doing something else?

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) 
So, only Markovian-ness stays. Derivative prices under jump-diffusion models are generally not tractable unless assumptions are made. For example, Merton's jump-diffusion model assumes that the jump size is lognormally distributed. The possibility of jumps gives the asset return distribution a desired 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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