# Toy models of asset returns

When making simple agent-based models of banking systems to look at global properties (say systemic risk) one of the basic decisions you have to make is how to model returns on external (to the banking network) assets. The goal is to have as simple (and general) of a model as possible, without it being fundamentally unreasonable. What are some standard approaches?

If I was trying the first thing that comes to mind then I would assume that returns on different assets are independently and normally distributed. This is tempting because of the central-limit theorem and how easy Gaussians are to work with, but unfortunately it is also considered one of the most dangerous assumptions in finance.

Instead, it seems that distributions with fatter tails are preferred. For instance this recent paper (which is written at a level of abstraction that I'd be interested in working at) uses student's t-distribution (with df = 1.5) to sample returns. However, I've been told by some quants that even this distribution doesn't have a heavy enough tail. Further, distributions with full support on $(-\infty, +\infty)$ simply don't make sense as models of returns for me. How could you possibly lose more money than your original investment? But if you simply truncate the distribution on the left though then the fat tail you cut off will result in a significant effect (as compared to truncating something like a Gaussian).

What are typical distributions used to model returns in work that needs to discuss external assets but is not primarily focused on them? Is there a good survey paper discussing advantages and limitations of simple return models?

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I have no idea but perhaps some you could consider: A skewed-t distribution... Or, a gaussian distribution for the centre and generalized pareto distribution (or generalized extreme value distribution) for a parametric distribution of tails (see extreme value theory). Then you just tweak the tail parameter to get whatever tail area you want. You could maybe use a multivariate parametric Vine copula to paste the univariate distributions together and create your economy. –  Jase Feb 27 '13 at 9:47
en.wikipedia.org/wiki/Agent-Based_Computational_Economics , and suitable process which generates density with excesse kurtosis could be obtain by sum of a random number of i.i.d. Gaussians - sum of random number of trades each trade moves price iid Gaussian, number of trades is random and arise from interaction between agents - I've seen a few such papers; for general overview of agent models in quantitative finanse you could check this survey hal.archives-ouvertes.fr/docs/00/62/10/59/PDF/reviewII.pdf ps. zero intelligence models are easier = starting point –  Qbik Mar 5 '13 at 22:48
@Qbik thanks! I usually work with evolutionary game theory, so zero intelligence is right up my alley! –  Artem Kaznatcheev Mar 6 '13 at 7:21

## 1 Answer

Some of the used heavy-tail distributions are:

• Log-Cauchy and Log-Gamma
• Lévy
• Burr and Weibull
• Mixed normal

Here two papers that cover some of them and others:

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