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Hull.21.05

mgarcia

Member
Hi,
The answer states: The square root rule overstates when there is mean reversion in returns (i.e., negative autocorrelation). Therefore, the actual daily volatility is greater than (>) 4.89%.

question: why is the actual daily vol greater? If the square root overstates the vol and 4.89% in the upper limit, why we say is greater than 4.89%? Is this because we apply the square root to the annual vol to get the daily vol?

Thanks
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi MGT,

I input into XLS 1.a.1 (adjusted) to illustrate because this issue seems to often come down to perceived language interpretation; I am not saying my sentence is the best, maybe it's not good. But please note here:

http://db.tt/LG3bdTd

Column C is daily i.i.d. with annual volatility of 30%
Column C is autocorrelation of -0.20 (mean version, in this context) also with annual volatility of 30%

… under the mean-reversion, a daily vol of 2.32% scales to annual 30%; i.e., when "scaling up" mean reversion implies the SRR will overstate: 2.32% * SQRT(250) = 36.7% > 30%

But now scale down: scaling down 30% under SRR gives 30% * SQRT(1/250) = 1.90% which understates the 2.32% under mean reversion ... i.e., the actual 2.32% is greater than the SRR 1.90%, so the SRR understates. Such that we conclude:
* Under A/C < 0 (A/C > 0), SRR vol/var overstates (understates) when scaling UP
* Under A/C < 0 (A/C > 0), SRR vol/var understates (overstates) when scaling DOWN

Hopefully the math makes sense and shows why this seems to often confuse "merely" as a matter of language/interpretation vis-a-vis scale up versus scale down. Please let me know if you perceive a better way?


Thanks, David
 
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