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# Regime-Switching Volatility Model

#### RiskRat

##### New Member
Subscriber
It is said that :

A typical distribution is a regime-switching volatility model: the regime (state) switches from low to high volatility, but is never in between. A distribution is “regime switching” if it changes from high to low volatility.

Further, following are the assumptions of regime-switching volatility model:

1. different market regimes exist with high or low volatility.
2. The mean is assumed constant,
3. If interest rates are drawn from one regime, the distribution is normally distributed.
4. If interest rates are drawn from more than one regime, this unconditional distribution need not be normally distributed.

Can someone explain the reasons for assumptions 3 & 4.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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Hi @RiskRat This is from Linda Allen Chapter 2:
"This type of distribution is often called a “regime-switching volatilty model.” The regime switches from low volatility to high volatility, but is never in between. Assume further that market participants are aware of the state of the economy, i.e., whether volatility is high or low. The econometrician, on the other hand, does not have this knowledge. When he examines the data, oblivious to the true regime-switching distribution, he estimates an unconditional volatility of 7.3bp/day that is the result of the mixture of the high volatility and low volatilty regimes. Fat tails appear only in the unconditional distribution. The conditional distribution is always normal, albeit with a varying volatility." [endnote 5: This is a popular model in recent econometric literature, invoked in order to better understand seemingly unstable time series. This is also a particularly useful example for our purposes. See James Hamilton (1990, 1995).]. Thanks,

#### nikogeorgiev

##### Member
David,

I am not quite clear as to why under the regime switching volatility model the probability of fat tails is much lower? Can you please clarify?

Many thanks!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI @nikogeorgiev This is Linda Allen's point and I don't think she is quite saying that the "probability of fat tails is much lower" in regime-switching volatility but rather than the unconditional distribution of such a volatility exhibits fat tails. Image the two regimes are normal times such that asset returns have mean of 3.0% with 8.0% volatility but can occasionally shift abruptly into a high-volatility regime during which the volatility is 20.0% and the mean is something different. Suspending disbelief, it might be possible for the conditional returns to be perfectly normal: during the calm regime, volatility is 8.0%, skew = 0, kurtosis = 3. During the high vol regime, volatility is higher but kurtosis is still 3.0. Hence the importance of conditional versus unconditional: the returns might be normal during the regime (conditional on the current phase). Now step back and image we are analyzing a dataset that span multiple regime shifts but we are not aware of the regimes (sub-periods)! This dataset commingles the regimes, not unlike how a normal mixture distribution "mixes" two normals with a resulting distribution that always has fat tails (!)--well, actually in fact, its actually the same idea as a normal mixture. The unconditional distribution, because it actually commingles the phases, is fatter than normal because it is a mixture of two sub-normals, one for each regime. I hope that's helpful!

#### nikogeorgiev

##### Member
Thanks David. This is helpful.

Schweser are saying "the probability of large deviations from normality occurring are much less likely under the regime-switching model. The regime-switching model captures the conditional normality and may resolve the fat tail problem".

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @nikogeorgiev Well, I perceive that to be a matter of my minor disagreement with the semantics: I think the first sentence is hard to interpret. Start with the second sentence: "The regime-switching model captures the conditional normality and may resolve the fat tail problem." That's perfectly consistent with my post above, and IMO matches Linda Allen. The fat tail problem refers to: why are returns observed as fat-tailed, when if they are i.i.d. they (per the CLT) should actually be approximately normal. That's the "problem," but it's not really a problem because we don't really believe i.i.d. But okay, if i.i.d. applied, then (daily) returns should be approximately normal. Why would they instead be so fat (which they empirically typically are)? The second sentence here is correct because it says: conditionally they are normal, but the regime shifts, which creates an unconditional (i.e., by "seeing through" the regimes) non-normal fat distribution. Right? So the second sentence, I think, is very plain.

I think the first sentence means: in an accurate regime shifting model, where each regime is characterized by its own normal distribution, then the probability of large deviations are "more normal" (i.e., less likely) than they are under a non-regime shifting model, which is an unconditional model so that it has fatter tails, and the large deviations are more likely. So I just think the first sentence is potentially imprecise, but at the same time, hopefully you can see that I think it's trying to say the same thing. Thanks!

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#### nikogeorgiev

##### Member
Thanks David. You have this knack for explaining complicated stuff easily so that everyone understands... Much appreciated.