Why Average High–Low Range Can Be About Twice the Open–Close Move
Summary
The document examines why the average intraperiod high–low range is often close to twice the average absolute open–close change. It presents observations across several currency pairs, timeframes, and Spanish equities, with ratios clustered near two in many cases. An idealized explanation models prices within a period as a zero-drift Wiener process: the expected absolute close-to-open move is proportional to volatility, while the expected upward and downward excursions from the open contribute equal amounts to the high–low range. This yields a ratio of two under those assumptions.
A second response cautions that the ratio is not universal, showing index-history estimates that vary across markets and sampling intervals. It notes that deriving the ratio for correlated, non-normal price observations is difficult and suggests simulation under explicit price models to study parameter effects. It also distinguishes the ratio of average ranges from the average of period-by-period range-to-move ratios, which answer different questions. The Wiener-process result is a benchmark, not a guarantee or trading signal; the document does not establish predictive or profitable use for either statistic.
Key ideas
- A zero-drift Wiener process implies an expected high–low range about twice the expected absolute open–close change.
- The derivation depends on symmetric excursions from the opening price and negligible drift within the period.
- Observed ratios vary across instruments and sampling intervals, so the value two is not universal.
- The ratio of averages differs from the average of individual period ratios.
- Simulation can help examine how assumptions about price dynamics affect the ratio.
Tags
Full text
# Why is the ratio of Hi-Low range to Open-Close range close to 2?
# Why is the ratio of Hi-Low range to Open-Close range close to 2?
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
```
## Answer by JL344 (score 21, accepted)
https://quant.stackexchange.com/a/3625
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.
## Answer by Lepto Kurtič (score 3)
https://quant.stackexchange.com/a/1170
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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.