May 10

Monte Carlo simulations. Part 4: FRM exam tips

by David Harper, CFA, FRM, CIPM

FRM | Risk | Quant |

In previous posts, I introduced the Monte Carlo simulation (MCS) approach. If you are sitting for the FRM exam, you need to understand at a bare minimum:

  • The generic steps required in MCS to compute value at risk (VaR)
  • The typical approach to simulating a stock price path
  • The challenges in applying MCS and coping techniques
  • The basic interest-rate path models; instead of GBM, what do we use to model interest-rate dynamics

The generic steps

Rather than memorize the four-step list, understand the idea of how VaR is determined under MCS:

Typical approach to stock simulation

The classic stock price path model is GBM, where the stock drifts per its expected return (the linear component) but is also randomized by the shock component:

 

Application challenges

The benefit of MCS is obvious once you understand it: awesome flexibility. We get to specify the behavior model and distribution of the random variable. Further, we decide the number of trials.

There's also the rub (Jorion's "speed versus accuracy"): more trials implies greater accuracy (if the model is good!). But more trials are computationally intensive. In the case of many variables (a portfolio with multiple assets, assets with multiple risk factors, or more realistically, both!), the computational grid grows exponentially.

Several accelerating "coping mechanisms" are discussed. These often involve ways to collect smaller trial samples that are still relevant to the risk question; e.g., only sample near the tail, where extreme losses occur. 

Interest-rate models

We can't use the GBM above for interest-rates because GBM is not mean reverting: an upward shock can easily be followed by another upward shock. There is no gravitational pull toward a mean. Bonds on the other hand, by definition, are mean-reverting: if you hold to maturity, your effective rate may fluctuate in the interim, but lacking a default, as you approach maturity, your rate must converge to equilibrium (the rate implied, at purchase, when the price will eventually meet the face).

A few interest-rate models are discussed including a two-factor model. Here a short and long rate are modeled separately, but they both are mean-reverting (the omegas are the long-run values):

Otherwise, this model is no so drastically different from the GBM. Each rate has two components, the first is a "drift" toward the long-run average. The second is a random variable, a normally distributed yield change (the "random shock").

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