Quantifying randomness: variance ratio

[mysticman]

Stephen,
In your opinion if the variance test has proven itself as a good indicator of trendiness (and thus of non-random behavior in the markets), why are the proponents of randomness still around?
And why are all the pricing models still using assumptions of randomness?
And how does this test compare with the Hurst exponent which does the same thing?

 

[stephencrowley]

The markets are mostly random.. 95% or more.
If they were 100% random then none of us should be trading at all.. better off playing roulette.

It depends on your defintion of random I think... randomness is not a binary value e.g. completely deterministic or completely random but somewhere in between.

The efficient market hypothesis has been debunked many times.. at least most forms of it.

Also, the hurst exponent is interesting and is very similiar to the variance ratio.. except that you can only calculate one value for the exponent for the whole series.

Using variance ratio you can plot the variance of any number of time scales compared to the smallest scale.

A quick check on my data shows that the slope of the variance ratio profile and that of the hurst exponents are very similiar.

Quote from mysticman:

Stephen,
In your opinion if the variance test has proven itself as a good indicator of trendiness (and thus of non-random behavior in the markets), why are the proponents of randomness still around?
And why are all the pricing models still using assumptions of randomness?
And how does this test compare with the Hurst exponent which does the same thing?

 

[g0d0t]

Quote from gbos:

If you are not interested in options I wouldn't recommend buying the book.

Out of curriocity I made a quick calculation using the Parkinson number in a sample of DJI data and also in a sample of a strong uptrending index.

DJI gave an almost random data series with a ratio 1.73 (1.67 = random)

The uptrending market gave a ratio of 0.89.

hi gbos,

I have a question about the zip file. There is the formula on column F row 22:

=0.601*SQRT(SUM(E2:E22)*252/21)

Is the number "252" a fixed number? How do you calculate that number if you want to calculate the Parkinson Number for 5 days interval instead of 20 days like you did?


Thanks,
g0d0t

 

[mysticman]

Quote from g0d0t:
Is the number "252" a fixed number? How do you calculate that number if you want to calculate the Parkinson Number for 5 days interval instead of 20 days like you did?

And to follow up, the 252/20 or 252/5 are common ratios in all volatility formulas to calculate the volatility of periods less than a year, but the volatility formulas always use the square root of the days in the ratio. For example, the volatility of a 25 day period would be sq rt 25 divided by sq rt of 252, or 5/16 of the annual volatility. Are you saying that the Parkinson's formula does not use the square root of the days?

 

[gbos]

The Parkinson formula is

N is the number of time periods you use for the calculation (for example number of days)
H is the high of the i-th period
L is the low of the i-th period

For annualizing the number you also have to multiply it by the square root of number of trading days per year (252).

The quantity sqr(1/(4*ln(2)) is approx the 0.601factor you see in the cell calculation.

If the market is following the ¡bell curve¢ then this number must equal 1.67 times the historical volatility. If it is more than this (for example 2 times historical volatility) then the market maker has an extra edge in that market. If it is less then it is better to follow a trend.

A small catch is this: the Parkinson number was calculated for a 24-hour no closing market. It can compare 24-hour high/low to data sampled every day at the same time. If you are going to use it in an equities market then for the historical volatility it is better to use open-to-close volatility.

As to your question why is ¡bell curve¢ math is used in pricing models the answer is that they are the only precise math tool available for quantify uncertainty .

 

[mysticman]

gbos,
Thanks for a very complete explanation of Parkinson's volatility. I just cannot help ask where the following information comes from:

Quote from gbos:
the Parkinson number was calculated for a 24-hour no closing market. It can compare 24-hour high/low to data sampled every day at the same time. If you are going to use it in an equities market then for the historical volatility it is better to use open-to-close volatility.

Not doubting that you have a source for this, but it seems inconceivable that in 1980 he could have used any data other than stocks. The only data that would qualify as 24 hour would be currencies. All the papers I have seen so far use daily high-low for stocks and commodities from regular trading hours. Where can I find out more?

 

[gbos]

It¢s a note made by Taleb in his Dynamic Hedging book. He explains that the math to arrive at the Parkinson number and the equation

Parkinson Number = 1.67 * HV

only applies in a 24 hour market. In an equities trading market

Parkinson Number = 1.67 * (altered HV)

were for the altered HV you use the open-to-close returns and not the close-to-close returns.

In the Parkinson number you always use the high and low of the day. It is just that if trading is discontinuous you don't expect this number to equal 1.67 the actual day to day historical volatility.

 

[mysticman]

I haven't read the book, obviously, but from what I know of the guy he seems quirky. 1.67 seems like an arbitary number, perhaps the average of all stocks. We all know that stocks differ greatly, each having a personality. You can take two, for example, that have the same HV, with one having a high intraday dispersion and the other a low range.
I have done studies comparing daily range to HV which reveal very different relationships. So I think the relationship of Parkinson's # to HV would be different for each stock.
Still, I'm willing to graph out a couple of stocks and see what the relationship is.
Parkinson's is supposed to be a more efficient, quicker, and accurate way at arriving at HV, so it seems somewhat backward to see it being made dependent on HV or altered HV.

 

[gbos]

1.67 is the ratio when the stock returns follow the normal distribution. No one claims that an arbitrary stock will have this ratio. It use it as an indicator for examining if a stock/market/etc is mean reverting or have more trends than expected.

I have some papers that discuss the probability density functions of the extremes but the math is very heavy to verify quickly his results. Anyway I know that Taleb holds a phd in statistics and I doubt he could be that wrong.

 

[nononsense]

Hi All,

Some make it seem as if the market has to be tackled somewhat like a problem in Statistical Mechanics.

In my experience, mathematics can indeed be useful. However, the main thing is to fret out some rather subtile properties well hidden from view of the masses, irrespective of how learned some may seem to be in this mass.
I'm sure that if you happen to stumble on something like this, you will not start with dispensing mathematical formulae related to your find.