Bands in VWAP: standard deviation explained

First the VWAP: the centre of the bands

VWAP, for Volume Weighted Average Price, is the average price of the period, where each price counts in proportion to the volume traded at it. TradingView's help page on VWAP defines it as a tool used to measure the average price weighted by volume, and gives its calculation: for each candle, a typical price, (high + low + close) ÷ 3, multiplied by volume; then VWAP = cumulative (typical price × volume) ÷ cumulative volume.

The running total restarts at each anchor. On TradingView, the Anchor Period setting specifies how frequently the calculation is reset. So VWAP is not a moving average: it has no rolling window, it grows by one candle with every candle until the next reset. The same page writes that it is best suited for intraday analysis, and that the closer it is to the day's close, the more lag it has.

Why this price matters. Large orders executed by algorithm are often sliced to stay close to the volume-weighted average price, which the lesson on large orders describes. VWAP therefore serves as an execution benchmark: bought below it, an order got a better fill than the session average; above it, a worse one. It is a benchmark, not a level that holds price.

The formula for bands in VWAP: a volume-weighted standard deviation

A band is VWAP plus or minus a multiple of a measure of dispersion. Sierra Chart's documentation on VWAP with standard deviation lines writes the calculation out in full. Call X the price input of each candle and V its volume, and sum from the start of the period.

Step 1: VWAP. VWAP = sum of (X × V) ÷ sum of V.

Step 2: the weighted variance. For each candle, take the gap between its price and VWAP at that candle, square it and multiply it by the candle's volume. Variance = sum of [(X − VWAP)² × V] ÷ sum of V. A deviation therefore counts for more when it happened on heavy volume.

Step 3: the standard deviation and the bands. Standard deviation = square root of the variance. Then, with a multiplier b: upper band = VWAP + b × standard deviation and lower band = VWAP − b × standard deviation. Sierra Chart draws up to four pairs of bands, at b, 2b, 3b and 4b.

What the other help pages say. TradingView's help only writes that, when a band is selected, the indicator calculates the standard deviations since the last anchor. The thinkorswim reference for the VWAP study writes that standard deviations are based upon the difference between the price and VWAP. Neither page details the weighting: do not expect to find the same bands, to the tick, from one piece of software to another.

VWAP bands worked by hand on four candles

The prices and volumes are invented. The period starts at candle 1, and each candle is summed up by one price and one volume. Candle 1: 100, volume 1,000. Candle 2: 102, volume 3,000. Candle 3: 101, volume 2,000. Candle 4: 104, volume 2,000.

VWAP, candle after candle. Candle 1: 100,000 ÷ 1,000 = 100. Candle 2: (100,000 + 306,000) ÷ 4,000 = 101.50. Candle 3: 608,000 ÷ 6,000 = 101.33. Candle 4: 816,000 ÷ 8,000 = 102. The simple average of the four prices is 101.75: VWAP is higher because the prices 102 and 104 carried more volume than the price 100.

The squared deviations, weighted. Candle 1: (100 − 100)² × 1,000 = 0. Candle 2: (102 − 101.50)² × 3,000 = 0.25 × 3,000 = 750. Candle 3: (101 − 101.33)² × 2,000 = 0.111 × 2,000 = 222. Candle 4: (104 − 102)² × 2,000 = 4 × 2,000 = 8,000. Sum: 8,972.

The standard deviation and the bands on candle 4. Variance: 8,972 ÷ 8,000 = 1.12. Standard deviation: square root of 1.12 = 1.06. Bands at one standard deviation: 102 + 1.06 = 103.06 and 102 − 1.06 = 100.94. Bands at two standard deviations: 104.12 and 99.88. Candle 4, at 104, sits above the first band and just under the second, but it is the candle that has just pushed them apart: on its own, it accounts for 8,000 of the 8,972 in the sum.

VWAP band settings, platform by platform

TradingView. The VWAP indicator offers three band multipliers, switched on one by one, and a bands calculation mode that sets the units of the distance. In percentage mode, the help writes, a multiplier of 1 means 1%: the band is then no longer a standard deviation, it is a fixed distance around VWAP. The source of the calculation is also a setting; the help notes that the bar's average value is traditionally used as the source.

Sierra Chart. Four methods to choose from: the variance, the standard deviation, a fixed offset or a percentage of VWAP. By default, the calculation is based on the bars in the chart and uses the last trade price of each bar; one input lets you base it on the more detailed underlying data. The documentation warns that the standard deviation lines can be different on different timeframe bars, because they are calculated in part using the chart bar values.

thinkorswim. Two bands only, an upper and a lower one, with a number of deviations set separately for each, and a time frame on which the cumulative calculations are performed.

None of the three sources read recommends one multiplier over another, or measures what a setting produces.

How to read VWAP bands

Which side of VWAP. TradingView makes it the first marker: prices trading above VWAP characterise a bullish trend, prices below it a bearish trend. The bands add a scale to that reading: how far from the average price are we, measured with the dispersion of the session itself.

The bands as overbought and oversold levels. This is the reading thinkorswim gives: the upper band serves as the overbought level, the lower band as the oversold level. It suits a session that swings around its average price. In a session that runs one way, nothing stops price from riding a band for hours: VWAP and its bands follow price, they do not hold it.

No percentage attached to a band. You often read that one standard deviation holds about 68% of prices and two standard deviations 95%. Those figures belong to a normal distribution, and the distribution of a session's prices never quite is one, which the lesson on auction market theory already says about the value area. A two standard deviation band is therefore not a probability of a return to VWAP.

None of these readings is an order to buy or sell. They are conventions, to be tested on your market, your timeframe and your anchor, with the method in the article on how to backtest a trading strategy.

The limits of VWAP bands

They are worth nothing at the start of the period. On the first candle, the standard deviation is zero and the bands sit on VWAP. Over the next candles, they rest on a handful of values and jump apart with every move, as on candle 4 of the example. The bands only become readable once the session is well under way.

They lag late in the session. The running total grows all day: the bigger it is, the less a new candle moves it. TradingView writes this about VWAP, and the same arithmetic applies to the standard deviation.

They depend on the anchor, the timeframe and the software. Changing the anchor changes the whole calculation. Sierra Chart warns that the lines differ from one bar timeframe to another, and the help pages read do not all describe the same formula. Two traders looking at 'the second band' are not necessarily looking at the same price.

They are worth what the volume is worth. On spot forex, there is no centralised volume: VWAP and its bands are calculated on tick volume, as the lesson on the forex market explains. And none of the sources read publishes a measurement of what these bands are worth as a signal: no success rate, no comparison of settings. If you use them, note in your journal where price stood relative to VWAP and its bands at entry, and let your own trades answer.

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