Mean Reversion

[Liming]

Dear David,

How are you? May I have your explanation on the following points?

I don't understand the necessary link between having a negative correlation and having mean reversion. When I am looking at the auto regressive functions such as GARCH(1,1) and the generic form of autoression (y(t)= alpha + beta*y(t-k) + error(t) ), and try to compare them with the one-factor model used to simulate bond yields where parameter k governs the seed of mean reversion and will create a negative drift toward long run value when r is too high or a positive drift when r is too low, I don't see that a long-run value exists or a similar self-adjusting 'pulling' mechanism exists in GARCH(1,1) and the generic auto regression.
That is to say that: I think that no matter beta is negative or positive in GARCH(1,1), the estimated variance will always drift away from its long run value since the formula is a simple addition,unlike the one-factor model, where a pulling mechanism exists whenever r is too low or too high.

Can you kindly explain why these two functions still qualify as mean reversion?

Thank You!

Cheers!
Liming

 

[David Harper CFA FRM]

Hi Liming,

I am good, thanks, trust you are doing well (based on your quality questions, it appears you are nicely along in studies!)?

You may find this link helpful: http://www.bionicturtle.com/learn/article/what_is_mean_reversion_in_financial_time_series/
(this post contains links to two papers:
1. an excellent intro comparison of GBM vs. GARCH and
2. a review of 5 or 6 definitions of mean reversion. Yikes!)

so, the first issue is, there are (unfortunately) different meanings of "mean reversion." In the FRM, we care about basically two:

1. Mean reversion of asset *returns* (which is *not* a feature of GARCH); e.g., in the paper above, the authors say the GARCH is not mean reverting. How can they say that? Because they are using a definition that is consistent with your point; i.e., mean reversion of "asset dynamics" Under their definition, Vasicek or CIR for interest rates is mean reverting but GARCH is not...but we are not so strict to this definition....
2. Mean reversion of variance (which *is* a feature of GARCH, and is the key feature of GARCH that distinguishes it from EWMA)

mean reversion in the *returns* is something we can also call negative auto- or serial correlation (i.e., a high return is likely to be followed by a low return)

in GARCH(1,1), var estimate = LR variance * gamma + alpha * recent return^2 + beta * recent variance
although you are correct, this is this first term is simply a weighted LR variance, it still constitutes a "gravitational pull" on the series; the LR variance here is called the unconditional variance: if we forecast estimated volatility using GARCH(1,1) the forecast curve converges to the unconditional variance (i.e., LR variance)

... Hull shows an alternative formulation which puts a more intuitive focus on the mean reversion (i.e., of variance, not returns, not price levels) aspect
dV = a*(LR Variance - V)dt + alpha*SQRT(2)*V*dz (page 482; emphasis mine; it is not hard to show this follows from GARCH(1,1))

...so the point is not necessarily to comprehend this (stochastic) process in detail but rather to see that, given a*(LR Variance -V), as Hull says "the variance has a drift that pulls it back to [the long run variance] at a rate of a."

Hope that helps, David

 

[Liming]

Dear David,

Thanks for your answer and your positive opinion of my study. However, I'm feeling that the more I study, the more there is for me to study. :coolsmile: :coolsmile:

Can I just confirm with you about the following point, which I guess is the key for understanding the mean reversion in variance as modeled by GARCH(1,1)?

As mentioned in the article http://www.bionicturtle.com/learn/article/what_is_mean_reversion_in_financial_time_series/ ," Negative autocorrelation refers to the fact that a high variance is likely to be followed in time by a low variance. The reason it's tricky is due to short/long timeframes: the current volatility may be high relative to the long run mean".
After reading this, I think that the sign of beta determines the type of autocorrelation. A negative beta means negative autocorrelation, which will necessarily translates into a mean reversion to its long run variance. The reason for this translation is that when beta is negative and when the previous variance(the third term in the formula) is very high, the third term will be a large negative number (after multiplication of beta and the previous variance), and this combined large negative number will sort of "cancel" the upward increase brought about by previous variance and even bring down the new variance. As a result, the new variance reverts to its long run value.

I hope I've expressed my question clearly. Thanks

Cheers!
Liming
28/09/2009