In this model, we assume asset price follows a log-normal distribution and the log return follows a normal distribution.

Note: Normal distribution is just an assumption, it's not the real distribution of return

Setting "Estimation Period Selection" is for selecting the period we want to construct the prediction interval.

For "Current Bar", the interval is calculated based on the data of the previous bar close. Therefore changes in the current price will have little effect on the range. What current bar means is that the estimated range is for when this bar close. E.g., If the Timeframe on 4 hours and 1 hour has passed, the interval is for how much time this bar has left, in this case, 3 hours.

For "Future Bars", the interval is calculated based on the current close. Therefore the range will be very much affected by the change in the current price. If the current price moves up, the range will also move up, vice versa. Future Bars is estimating the range for the period at least one bar ahead.

There are also other source selections based on high low.

Time setting is used when "Future Bars" is chosen for the period. The value in time means how many bars ahead of the current bar the range is estimating. When time = 1, it means the interval is constructing for 1 bar head. E.g., If the timeframe is on 4 hours, then it's estimating the next 4 hours range no matter how much time has passed in the current bar.

Note: It's probably better to use "probability cone" for visual presentation when time > 1

Volatility Models :

Sample SD: traditional sample standard deviation, most commonly used, use (n-1) period to adjust the bias

Parkinson: Uses High/ Low to estimate volatility , assumes continuous no gap, zero mean no drift, 5 times more efficient than Close to Close

Garman Klass: Uses OHLC volatility , zero drift, no jumps, about 7 times more efficient

Yangzhang Garman Klass Extension: Added jump calculation in Garman Klass, has the same value as Garman Klass on markets with no gaps.

about 8 x efficient

Rogers: Uses OHLC, Assume non-zero mean volatility , handles drift, does not handle jump 8 x efficient

EWMA: Exponentially Weighted Volatility . Weight recently volatility more, more reactive volatility better in taking account of volatility autocorrelation and cluster.

YangZhang: Uses OHLC, combines Rogers and Garmand Klass, handles both drift and jump, 14 times efficient, alpha is the constant to weight rogers volatility to minimize variance.

Median absolute deviation: It's a more direct way of measuring volatility . It measures volatility without using Standard deviation. The MAD used here is adjusted to be an unbiased estimator.

Volatility Period is the sample size for variance estimation. A longer period makes the estimation range more stable less reactive to recent price. Distribution is more significant on a larger sample size. A short period makes the range more responsive to recent price. Might be better for high volatility clusters.

Standard Deviation One shows the estimated range where the closing price will be about 68% of the time.

Standard Deviation two shows the estimated range where the closing price will be about 95% of the time.

Standard Deviation three shows the estimated range where the closing price will be about 99.7% of the time.

Note: All these probabilities are based on the normal distribution assumption for returns. It's the estimated probability, not the actual probability.

Manually Entered Standard Deviation shows the range of any entered standard deviation. The probability of that range will be presented on the panel.

People usually assume the mean of returns to be zero. To be more accurate, we can consider the drift in price from calculating the geometric mean of returns. Drift happens in the long run, so short lookback periods are not recommended. Assuming zero mean is recommended when time is not greater than 1.

When we are estimating the future range for time > 1, we typically assume constant volatility and the returns to be independent and identically distributed. We scale the volatility in term of time to get future range. However, when there's autocorrelation in returns( when returns are not independent), the assumption fails to take account of this effect. Volatility scaled with autocorrelation is required when returns are not iid. We use an AR(1) model to scale the first-order autocorrelation to adjust the effect. Returns typically don't have significant autocorrelation. Adjustment for autocorrelation is not usually needed. A long length is recommended in Autocorrelation calculation.

Note: The significance of autocorrelation can be checked on an ACF indicator.

ACF

The multimeframe option enables people to use higher period expected move on the lower time frame. People should only use time frame higher than the current time frame for the input. An error warning will appear when input Tf is lower. The input format is multiplier * time unit. E.g. : 1D

Unit: M for months, W for Weeks, D for Days, integers with no unit for minutes (E.g. 240 = 240 minutes). S for Seconds.

Smoothing option is using a filter to smooth out the range. The filter used here is John Ehler's supersmoother. It's an advance smoothing technique that gets rid of aliasing noise. It affects is similar to a simple moving average with half the lookback length but smoother and has less lag.

Note: The range here after smooth no long represent the probability

Panel positions can be adjusted in the settings.

X position adjusts the horizontal position of the panel. Higher X moves panel to the right and lower X moves panel to the left.

Y position adjusts the vertical position of the panel. Higher Y moves panel up and lower Y moves panel down.

Step line display changes the style of the bands from line to step line. Step line is recommended because it gets rid of the directional bias of slope of expected move when displaying the bands.

Warnings:

People should not blindly trust the probability. They should be aware of the risk evolves by using the normal distribution assumption. The real return has skewness and high kurtosis . While skewness is not very significant, the high kurtosis should be noticed. The Real returns have much fatter tails than the normal distribution, which also makes the peak higher. This property makes the tail ranges such as range more than 2SD highly underestimate the actual range and the body such as 1 SD slightly overestimate the actual range. For ranges more than 2SD, people shouldn't trust them. They should beware of extreme events in the tails.

Different volatility models provide different properties if people are interested in the accuracy and the fit of expected move, they can try expected move occurrence indicator. (The result also demonstrate the previous point about the drawback of using normal distribution assumption).

Expected move Occurrence Test

The prediction interval is only for the closing price, not wicks. It only estimates the probability of the price closing at this level, not in between. E.g., If 1 SD range is 100 - 200, the price can go to 80 or 230 intrabar, but if the bar close within 100 - 200 in the end. It's still considered a 68% one standard deviation move.

Release Notes:
Update Multi time frame function default setting