Monte Carlo: house of cards?

May 8, 2009 in Finance,Math,Risk

The WSJ recently ran apiece on Monte Carlo risk management:

Here is how a typical Monte Carlo retirement-planning tool might work: The user enters information about his age, earnings, assets, retirement-plan contributions, investment mix and other details. The calculator crunches the numbers on hundreds or thousands of potential market scenarios, guided by assumptions about inflation, volatility and other parameters...

These models were supposed to help quantify and manage the risks of mortgage-backed securities, credit-default swaps and other complex instruments. But given the events of the past couple of years, it appears that the models often gave big institutions, as well as small investors, a false sense of security.

I can't stand idly and let another "models are just the tool" opportunity go by. Monte Carlo simulations are among the most powerful and versatile tools available. As an empiricist, I prefer their discretized approximation of reality to grappling with overly simplified closed form models. There are few substitutes to a Monte Carlo expectation for determing the outcome of a complex process, given a reasonable set of starting assumptions.

Yes: given a reasonable set of starting assumptions.

A Monte Carlo simulation is nothing more than a bunch of random numbers which, taken together, determine a path through time.  At each step along the path, more random numbers are generated. The rules by which those numbers are generated - what we refer to as their marginal distributions and pairwise correlations - are not part of the Monte Carlo process, strictly speaking. They are purely at the discretion of the operator. And their specification is critical to the success of the model (which, in their absence, is just a big random number generator).

Monte Carlo methods are really useful because of the existence of path-dependent outcomes. Path-dependent means that the end value depends on intermediate values. For example, a stock price is not path dependent because having a price X on day 1 does not preclude having a price Y on day 2. But what if a bond can be called at par if - and only if - it trades above par for 20 days? Then the price of the security depends heavily on its price in the past. Monte Carlo analysis solves such problems because it actually generates a price path; other methods can not account for such securities.

So why are these risk mangement companies using Monte Carlo methods to value stocks? Beats me - a closed form expression would do just as well (given the same assumptions as the current set up) without the need to waste cycles on each Monte Carlo iteration. I have a sneaking suspicion, however, that it's because it's nice to show investors "probable stock paths", something which Monte Carlo can do and "math" can not.

This author actually comes close to the mark:

Critics emphasize that the problem isn't Monte Carlo itself, but the assumptions that go into it. Since no standard approach exists, one user might plug in a range of assumptions on interest rates, inflation or volatility that is different from another user.

But this implies that a solution is to have a "standard approach," which would presumably take the form, "Using 2% interest rates, 2% inflation, 30% volatility gives you the right answer." Obviously, that doesn't exist.

Now, some of the assumptions going into these Monte Carlo calculators are wrong because the end-user has mis-specified them (ridiculous interest rates, for example). But the real mis-specification is that the Monte Carlo models described in the article use normal distributions as the heart of their engine. Even the best end-user assumptions will be rendered useless if the distributional assumptions are wrong.

And here's the rub: a non-path-dependent normally-distributed Monte Carlo simulation can be perfectly replicated by... a Gaussian copula.


"Gaussian copula" has become a dirty word in finance, because we recognize the havoc that these models have wreaked on the CDO industry. We - by which I mean countless journalists - have railed against this model because it was not suited to the task (in my opinion, this is a little like blaming Porsche when you fill your tank with 83 and ruin your car). And yet here it is again in another guise! If I write an article pointing out that normal Monte Carlo is just a discrete Gaussian copula (and I guess I am), can I start a new crusade against Monte Carlo models?

I hope not, because I think Monte Carlo models are wonderful tools. Like the Gaussian copula, they are well suited to answering a variety of questions. Unlike Gaussian copulas, they can be adjusted to account for non-normality (that's sort of a cheap shot, since a Gaussian copula is by definition normal; a non-normal copula could be substituted instead). It's sad that at the moment such models are needed most, we choose to blindly rage against the method rather than aknowledge the deficiency of its current implementation and just simply, astoundingly simply, correct it.

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