Random Walk/Mean Reverting (Quant A)

[sathyat]

Hi David,
Thanks for your reply on my previous question.

Another request - Will you be able to upload a spread sheet example to illustrate Randon walk/Mean reversion.?

Regards,
Sathya

 

[David Harper CFA FRM]

Hi Sathya,

Yes, i am working on an improved Monte Carlo that will show both (and more), but is not ready yet.

To tide you over, here is last year's simple stock price simulator:

Note this is classic version, found in countless texts, of discrete-time Brownian motion for a stock price. (More here for the subtle relationship btwn Brownian motion & random walk,but this is outside FRM scope). Not to geek out on you, but our concern (per Hull) is the idea this is a Markov/Weiner process (I'll use these interchangably): it has no memory or "the past is irrelevant". Purists will note distinctions but we can loosely say random walk = markov = Weiner = no memory of the past; i.e., tomorrow's stock is a function of today's stock price

so, in the spreadsheet this is simulated as discrete steps:

price = yesterday's price + (expected return*time interval) + [NORMSINV(RAND())*volatility*SQRT(time)]

this equation is very instructive; i.e., what is NORMSINV(RAND()) doing? Also, a common test question is: why can't we use this for fixed income securities? Questions for you: why can't we? What can we use? Is it still Markov?

Thanks, David

 

[humheehum]

Hi David - I have a question on mean reversion and its impact on LR vol.

Both in your lecture slides and Allen, there is a table which summarizes the impact of mean reversion on LR vol. The table splits the issue into :-
1) Mean reversion in returns
2) Mean reversion in return volatility.

I have understood pt 1 using the 2-period variance example in Allen.

I have not understood pt 2. Eg: - why does the square root rule overstate volatility if Current vol is higher than Long run volatility.

I think Allen has tried to explain this by stating is b=1 i.e a random walk exists then square root rule holds ( an implicit assumption in the square root rule).
If b<1, then there is mean reversion - this would lead to a negative covariance (I have not understood this) leading to a lower vol. Is this because the square root rule now become

variance (2 period) = 2*variance - 2*covariance (??)

Could you please expand on what pt 2 is saying ?

Thanks
Ashim

 

[David Harper CFA FRM]

Hi Ashim,

Yes, that's tough. As I re-read this section of Allen, I think you have a great observation. IMO, much confusion (for all of us) stems from multiple meanings of "mean reversion" and the difference between "long-run mean reversion" and autocorrelation.

First, to answer you question, Allen's formula (yours above) is demonstrating (1). By use of the 2-period variance, she is showing that the presence of autocorrelation implies the square root rule will overstate. It would be easier, IMO, if she dropped "mean reversion" in this context and only referred to autocorrelation.

Second, to (2) above: please think of the first term in GARCH(1,1):
GARCH(1,1) = (weighted)(L.R. variance) + (weighted)(lagged variance) + ...

The first term, (gamma)(L.R. variance) is the "mean reversion in return volatility" (#2 above);
The second term is autoregression in the volatility (i.e., the 'A' in GARCH); a.k.a., volatility clusters.

This (#2, mean reversion in return volatiliyt) can be understood b/c the square root rule scales over n-periods; if volatility is identical over n-periods, then we are scaling over, say:
volatility = 2%, volatility=2%, volatility =2%,....

But if volatility has a long-run average of, say, 1%, and we are currently at 2%, then we are scaling:
volatility = 2%, volatility=1.9%, volatility =1.8%,....

Hopefully, you can see how, the first will give a higher n-period volatility than the second.

You have surfaced a challenging idea that reveals the potential ambiguity of mean reversion. Summary of key points:

Mean reverting already has (at least) two meanings in financial time series. For example, GARCH(1,1) is sometimes (often?) said to be NOT mean reverting because:

1. GARCH does not mean revert in the return (a.k.a., asset dynamics). GARCH(1,1) does not mean revert per Allen's #1; e.g., it does not say stock returns revert to +6% BUT
2. GARCH does indeed mean revert in return volatility; e.g., it does include a weight for say, the long run volatility is +30%. It does mean revert per Allen's #2 above.

EWMA models volatility clustering (autoregression) but not "mean reversion in (return) volatility"
GARCH models BOTH!

Finally,
* We can speak of mean reversion in the asset dynamic (i.e., stock returns revert to 10%)
* We can speak of mean reversion in volatility, per the gamma-weighted param GARCH(1,1) and Linda Allen's #2 above
* And we can speak of autoregression in returns (as does Allen's discussion) or autoregression in volatility, per GARCH(1,1) but per the beta-weighted param not the gamma-weighted param

 

[humheehum]

David - thanks. That was a great post. It took me some time to get my head around everything - but I think I have understood what you have written.

Allen has turned out to be a very a dense and a challenging reading. Any other readings you would recommend is a must within the Quant section?(any particular chapters of Gujarati?)

Is there a list of readings for all the sections/topic areas that you would recommend reading or is it something you mention in your webcasts/lectures (in which case as I go along I will read the ones you mention are important)

Thanks again.

Regards,
Ashim

 

[sipanivishal]

Hi David,

http://www.bionicturtle.com/learn/article/what_is_mean_reversion_in_financial_time_series/
I think with regard to mean reversion in return...you have made a small error.....
"In returns If current volatility is greater than (>) long run mean (LRM) volatility,
SRR will overstate"

From your discussion above only I learnt that If there is mean reverson wrt Return.....it will always overstate the long run variance because Covariance will always be negative.Correct me if wrong

Thanks
Sipani

 

[sipanivishal]

Hi David,

can you please derive the 2 period Variance ?

Thanks
SIpani

 

[David Harper CFA FRM]

Hi Sipani,

Thanks for the correction.

Regarding, "If there is mean reverson wrt Return.....[square root rule] will always overstate the long run variance because Covariance will always be negative" Yes per Linda Allen Chapter 2, that is correct. Can you refer to L. Allen, please, for the derivation. It is given on p 70 - 71

Thanks, David

 

[[email protected]]

>>Yes, i am working on an improved Monte Carlo that will show both (and more), but is not ready yet.

Hi David,
did you finish the " improved Monte Carlo" with Mean reversion?
If so, may I/we have a look at it?
Cheers!
MZ

 

[David Harper CFA FRM]

Hi MZ, no i haven't (I need to finish the updated formula sheets first), thanks, David