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I tried it in several symbols and timeframes with the same result:

$$\frac {mean(HIGH-LOW)}{mean(|CLOSE-OPEN|)}$$

Symbol       Result
------       ------
EURUSD:W1    1.9725  
EURUSD:D1    2.0023 
EURUSD:H1    2.1766  
USDJPY:W1    1.9949 
USDJPY:D1    2.0622  
USDJPY:H1    2.2327 
SAN.MC:D1    2.0075  
BBVA.MC:D1   2.0075    
REP.MC:D1    2.1320
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Are W1, D1, H1 time frames? You might want to make that clearer. – Louis Marascio Jun 16 '12 at 13:21

2 Answers 2

up vote 18 down vote accepted

There is a very good reason why the ratio $$\frac {mean(HIGH-LOW)}{mean(|CLOSE-OPEN|)} \approx 2$$ on various financial series. If the price of a security evolves according to a Wiener process beginning at the opening bell and throughout the day, and the drift is negligible for that period of time, i.e.$\mu=0$, then the denominator of the above ratio closely approximates the average absolute deviation, $$AAD=\frac{2\sigma}{\sqrt{2\pi}}\int_0^\infty xe^{-x^2/2}dx=\sqrt {2/\pi}\cdot\sigma$$ for a normal distribution, where $\sigma$ is the standard deviation. On the other hand $$\mathbb E(HIGH-OPEN) = \sqrt{2/\pi}\cdot\sigma$$ $$\mathbb E(LOW-OPEN) = -\sqrt{2/\pi}\cdot\sigma$$(See the running maximum of a Wiener process on Wikipedia.) So we have for such an idealized Wiener process: $$\frac {\mathbb E(HIGH-LOW)}{\mathbb E(|CLOSE-OPEN|)} = \frac{\sqrt{2/\pi}\cdot\sigma-\left(-\sqrt{2/\pi}\cdot\sigma\right)}{\sqrt {2/\pi}\cdot\sigma} = 2.$$ It should not be too surprising to see this more or less borne out by observation.

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Unfortunately, I cannot answer fully your question. Though I'll give you my partial answer.

First of all, using entire price history of an index (from Yahoo), this is what I got:

                   Daily   Weekly   Monthly
DJIA               2.91    2.37     2.33
NASDAQ 100         1.74    1.94     1.91
NYSE Composite     1.61    1.59     1.85
S&P 100            1.75    1.90     1.97
S&P 600 Small Cap  1.59    1.82     1.99

Based on this, I don't think we can claim that the ratio is always 2.

So you agree, that the ratio is not always 2. But still, you want to know why it is equal to 2.91 or 1.59 or whatever.

This is how I would proceed in answering the question. First, the ratio can be expressed as

$$ \frac{E[max(P_1, P_2,...,P_n)-min(P_1, P_2,...,P_n)]} {E[|P_1 - P_n|]}$$

Second, I would start expanding the fraction in order to get a better picture of what exactly influences the ratio. I hope to expand the fraction, and then have some terms cancel each other out and then obtain as an answer 2, $\sigma / \mu$, or something else concise and beautiful. The problem is, it is extremely hard (at least for me) to obtain analytical expression for the expected value of a maximum (or minimum) of a sequence of correlated non-normal variables--the prices. I do not think anyone can give you the analytical expression for that. So this is where it ends as for analytical answer.

You can also use numerical methods, something like Monte Carlo simulation. Assume some model for prices, simulate them, and do some sensitivity analysis in parameters of the model to see how they affect the ratio.

Finally, one thing I do not get is why you are interested in that ratio. Shoundn't you instead be interested in $$ E(\frac{max(P_1, P_2,...,P_n)-min(P_1, P_2,...,P_n)]} {|P_1 - P_n|})$$

For example if the true expected value of the above ratio is equal to 10, and during a trading day the price is such that the ratio is 20, you would start buying the asset because you expect the close price to be higher than the current price. You would then sell the asset for a profit. Using your definition of the ratio, I see no usefulness in it. Care to explain?

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I didn't try indexes, but DJIA components are closer than the index itself (2.91) : MMM 2.0118 AA 1.8223 AXP 1.9819 T 1.9381 BAC 1.8917 BA 1.9329 CAT 1.9299 CVX 1.9701 CSCO 1.9582 KO 1.8967 DD 1.8921 XOM 1.9578 GE 1.9642 HPQ 1.8642 HD 1.9523 INTC 1.9294 IBM 1.8367 JNJ 1.9227 JPM 1.9468 KFT 2.0584 MCD 1.9899 MRK 1.9604 MSFT 1.9045 PFE 1.9475 PG 1.9921 TRV 1.8728 UTX 1.9770 VZ 1.9525 WMT 2.0081 DIS 1.8454 – jla May 16 '11 at 11:57
Did you use the entire price history? You might have used a period in which the ratio was close to 2. Why are you interested in that ratio? – Dmitrii I. May 16 '11 at 20:19
I took the entire daily price history available from Yahoo. I'm not interested in that ratio in itself, only as property value that many symbols have in common. – jla May 17 '11 at 11:13

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