Square root of time: from a 2.4 pip M5 standard deviation to the hourly and daily range

[Fairman]

Square root of time: from a 2.4 pip M5 standard deviation to the hourly and daily range

If M5 price changes behaved like a random walk, volatility would scale with the square root of the number of bars. This gives you expected ranges for longer windows from one measured number.

Measure it: take 500 M5 closes, compute the change from each close to the next in pips, and take the population standard deviation. Suppose it is 2.4 pips.

One hour is 12 bars: 2.4 x sqrt(12) = 8.3 pips of standard deviation.
Four hours is 48 bars: 2.4 x sqrt(48) = 16.6 pips.
A full day is 288 bars: 2.4 x sqrt(288) = 40.7 pips.

The standard deviation describes where the price ends up, not the high to low range. For a random walk the expected range is about 1.6 times the standard deviation. So the hour is expected to cover about 13.3 pips, four hours 26.5, and the day 65.

Compare with the actual average daily range of your pair over the last three months. If the predicted 65 pips is far below what you see, the day contains trends or volatility bursts. If it is far above, there is mean reversion in the data, or the quiet hours are dragging down the sample.

[FXS HongKong Scalper]

Useful rule of thumb. FX returns are not truly random walks, so the daily figure often overstates the actual range on quiet days and understates it on news days. Measuring the actual daily range alongside the square root estimate shows how much trending or mean reversion is present for each pair.

[Amanhasnoname]

I tried this on GBPUSD a few months ago and the gap between the estimate and the actual range was interesting. The square root figure came out around 70 pips for the day, while the real average daily range over three months was closer to 95. Most of that difference came from London and the overlap with New York, where the moves trend much more than a random walk would suggest.

What I found more useful was doing it per session. I measured the M5 standard deviation separately for Asia, London and New York, and the London figure was nearly double the Asian one. Scaling each session on its own gave me ranges that matched reality much better than one number for the whole day.

I now use the hourly estimate to set realistic targets. If the expected range for the next hour is about 12 pips and I'm aiming for 15 from a standing start, I know I need a trend day to get there. It's stopped me placing targets that only had a small chance of being reached in the time I hold trades.

[Fairman]

Comparing the real hourly figure to the prediction gives a variance ratio. Suppose you take 500 hourly closes and the standard deviation of hourly changes is 11 pips, not the 8.3 implied by the M5 data. The ratio of variances is (11 / 8.3) squared, about 1.76. A ratio above 1 says moves tend to continue from one 5 minute bar to the next, so hours are wider than the random walk expects. Below 1 says moves tend to reverse, so hours are narrower. Many pairs show ratios near 1 on short horizons, with differences appearing by session, so test it on your own pair and in the hours you trade.

[Fairman]

The same scaling gives a quick estimate of how likely a stop is to be touched. With an hourly standard deviation of 8.3 pips, the chance that price reaches 10 pips above the start at some point in the hour is about 2 x (1 - Phi(10 / 8.3)). Phi(1.2) is 0.885, so the probability is 2 x 0.115 = 23%. A 10 pip stop on one side has a 23% chance of being touched in an hour of pure noise, with no information at all. Stops tighter than the noise in your holding period get hit often by chance, which is one reason many stop outs happen in trades that later turned out to be right.

[Fairman]

The rule works best inside a single regime. M5 volatility at the London open is not the same as at 13:00 or during the Asian session. Suppose the 07:00 to 09:00 M5 standard deviation is 3.5 pips while the 02:00 to 04:00 one is 1.2. Multiplying either by sqrt(288) produces a daily number that never happens, because no day is made of one regime. Scale within the window you care about: for a two hour London window, 24 bars, 3.5 x sqrt(24) = 17.1 pips. Build a table by hour from your own data, so each estimate uses its own session's measured volatility.

[Fairman]

Options traders use the same scaling in the other direction. An annualised implied volatility is converted to a daily one by dividing by the square root of 252 trading days. If EURUSD implied volatility is 7% (an illustrative level), the daily standard deviation is 7 / 15.87 = 0.44%, which on 1.0850 is about 47.8 pips. That is a market based estimate you can compare with your own measured 40.7 pips. If implied is much higher than what the recent bars gave, the market is pricing an event or a regime change, and a sheet that only looks at past bars would be too calm.