Jun 18

Intro Stochastic Toolkit for Risk Management (Paper)

by David Harper, CFA, FRM, CIPM

Here is a really helpful introduction to a set of financial time series that are popular in risk measurement. The authors, of Fitch Ratings (Damiano Brigo, Antonio Dalessandro, Matthias Neugebauer, and Fares Triki), classify stochastic series according to two features:

  • Normal versus fat-tails
  • Mean reversion versus no reversion

For FRM candidates, the relevant reading is Linda Allen Chapter 2. She reminds us that different asset classes exhibit different behaviors; e.g., interest rates tend to be mean-reverting. The authors share this typology:

  Normal tails Fat tails
No mean reversion arithmetic Brownian motion GARCH
Mean reversion Vasicek CIR, Vasicek with Jumps

I was initially confused by the classification of GARCH as non mean reverting, but Mr. Damiano (who is a quantitative rock star) was kind enough to clarify for me: "in the paper we talk about mean reversion in the asset dynamics, not in volatility. In a GBM with GARCH volatility the asset volatility is mean reverting but the asset itself is not." Okay, so distinguish between two kinds of mean reversion:

  • Mean reversion in the asset
  • Mean reversion in the volatility (corresponds to mean reversion "in return volatility" in Linda Allen's Table 2.4)

I recommend this for the its highly accessible discussion of geometric Brownian Motion (GBM) and GARCH. Highlights for an FRM candidate:

"In GARCH the variance is an autoregressive process around a long-term average variance rate. The GARCH(1,1) model assumes one lag period in the variance. …The GARCH(1,1) model can be generalised to a GARCH(p, q) model with p lag terms in the variance and q terms in the squared returns. The GARCH parameters can be estimated by ordinary least squares regression or by maximum likelihood estimation [note: Jorion appears to be incorrect that MLE is required], subject to the constraint ω + α + β = 1, a constraint summarising of the fact that the variance at each instant is a weighted sum where the weights add up to one. As a result of the time varying state-dependent volatility, the unconditional distribution of returns is non-Gaussian and  exhibits so called "fat" tails. Fat tail distributions allow the realizations of the random variables having those distributions to assume values that are more extreme than in the normal/Gaussian."

This is more succinct than our assigned reading. Note especially how a conditional variance (the variance estimate that is conditional on prior information, namely prior variance and squared return) by virtue of varying through time creates unconditional fat-tails. And then,

"The particular structure of the GARCH model allows for volatility clustering (autocorrelation in the volatility), which means a period of high volatility will be followed by high volatility and conversely a period of low volatility will be followed by low volatility. This is an often observed characteristic of financial data"

They review NGARCH , which is a clever way to amplify amplify lagged negative return (bad news implies greater volatility) yet mute a lagged positive return (good news implies less volatility). This overcomes a symmetry in GARCH.

And then some great review on the Vasicek model, which captures the mean reversion of short interest rates. For some reason, this important topic appears to have been dropped from the 2008.

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