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

Last time we added jumps to the recipe to enrich the original random walk dynamics. Jump diffusion model is able to fit to market implied vol much better than the diffusion only counterpart. Before moving on, just two reminders:

1. The numerical pricing procedure becomes more involved, especially for grid/finite differencing methods (because we end up with an integro-differential equation, as opposed to a PDE)
2. Although we usually speak of "the" jump diffusion model, we are free in specifying the probability distribution of the jump amplitude

How about good old Gaussian?
So is jump diffusion the ultimate holy grail? And is pure diffusion model done with? For the first question, we will see later in the series that there are many other approaches that supersede jump diffusion, but for now let's take a step back and address the second question.

It turns out that the use of mixtures is able to tackle at least some of the shortcomings of the original random walk model. A mixture, not surprisingly, is formed by combining more than one 'sub-distributions.' (usually, but not necessarily, of the same kind) Here we focus on Gaussian mixture. There are a number of possibilities regarding how the sub-distributions are combined, the use of Markov chain being a prominent one. Essentially, every time we need a new random variable to be generated, we first toss a coin to determine which sub-distribution out of the mixture to draw from. In the case of a mixture of two Gaussians, if one sub-distribution has a larger variance than the other, then the mixture would have fatter tail, although the individual sub-distributions do not.

       Candidate: Mixture

• Markovian
• Most likely not tractable
• Excess kurtosis
• Market incomplete

The use of mixture models in option pricing is rare, possibly because not everyone likes the idea of the hidden Markov chain that comes with the same package. Justifying and interpreting it is easier in the context of econometric modeling that deals with longer time frames, in which case the state changes can be perceived as a change in regimes. For something like intra-day option pricing, the model may require that we draw from alternate sub-distributions every 5 minutes or so, and it's not trivial to make sense of what that means. (other than the fact that it fits the data well)

Another French Mathematician
If producing a fatter tail is all we care about, then what about substituting Gaussian process with something with excess kurtosis? The resulting model would likely not have analytic solution, but we can fall back on numerical methods as our last resort. For example, ask the computer to generate 10000 random numbers from a Student's t distribution and use those in a Monte Carlo simulation.

The idea sounds neat, but one has to be cautious about it. We may not be aware of it, but when we were modeling asset prices with Gaussian or Poisson processes, both of the processes are closed under linear transformation. In other words, aggregating multiple Gaussian (or Poisson) random variables would result in yet another Gaussian (or Poisson) variable. That means when we use Gaussian process to model asset prices, even though it may not be a very faithful description of reality, at least we are being consistent and we know what we are getting ourselves into: namely, if the one-day distribution is an N(0.01, 0.05), then the one-month distribution would be an N(0.25,0.25), etc. Other distributions in general do not have this property. If in our hypothetical Monte Carlo simulation we draw from a t distribution with a degree of freedom of 10 for daily return, how would the monthly return be distributed? How about quarterly?

These problems can be avoided by going for non-Gaussian processes while staying within the territory of Levy processes, which is a broad family of processes that possess the nice property of being closed under linear transformation. (obviously both Gaussian and Poisson processes are members of this family)