# Volatility and Bollinger Bands

It is a common knowledge that Bollinger Bands (price standard deviation added to a moving average of the price) are an indicator for volatility. Expanding bands – higher volatility, squeezing bands – lower volatility. A bit of googling and you get the idea. In my opinion – that’s wrong, unless, one uses a twisted definition of volatility.

Let’s consider two possible scenarios:

• Prices go up 1% for 10 days in a row.
• Prices go up 1%, down 1% for the same 10 days.

Which one is more volatile in your opinion? What do you think is the conclusion based on Bollinger Bands indicator?

Looking at the figure above, it’s clear that according to our indicator, the volatility is way higher in the first scenario. The figure also reveals the reason – the standard deviation is simply the squared distances from the mean. In the first case, the squares of the large distances over the beginning and the ending of the period add a lot. All in all – exactly the opposite conclusion of what I would have thought – I would prefer my indicator to determine that the volatility is higher in the second case. I can settle for the volatility being approximately the same. But a factor of six is way too much:

```aa = rep(1, 10)
for(ii in 2:10) aa[ii] = aa[ii-1]*1.01
aa
# 1.000000 1.010000 1.020100 1.030301 1.040604 1.051010 1.061520 1.072135 1.082857 1.093685

bb = rep(1,10)
for(ii in 2:10) bb[ii] = bb[ii-1]*(1+0.01*(-1)^ii)
bb
# 1.0000000 1.0100000 0.9999000 1.0098990 0.9998000 1.0097980 0.9997000 1.0096970 0.9996001 1.0095961

sd(aa)
# 0.03151583

sd(bb)
# 0.005271538
```

Pretty clear – when prices don’t go too far from the mean (the second case), the bands contract.

The reason for this behavior is that the price series is not stationary:

I am not saying Bollinger bands are useless. They could be used to determine ranging prices (contraction in the bands without significant contraction in the returns). Or to define some extreme profit targets (in strong trends the Bollinger bands widen a lot, thus, taking profits once a widened Bollinger band is touched makes sense). Yes, they are useful, just not for the purpose they are usually advertised for.

1. Pete Werner says:

Nice post, its good to highlight that there’s lots of ways you can define volatility. I like to think of it as how much price moved around over a given period, and usually use high – low either as it is, or relative to the open. That way it captures periods with large intraday moves. Close to close returns can miss large range days, and imo if you use stops the highs/lows are more important than closes. ATR is another option and can be useful as it will capture gaps. I rarely use it but I mostly focus on FX, equities gap a lot more. Ultimately it depends on what you are doing and what you want to call volatility.

1. quintuitive says:

In practice, I use various methods from this paper: http://www.todaysgroep.nl/media/236846/measuring_historic_volatility.pdf to measure volatility. All of them are available in R’s TTR package.

2. Christian says:

I think you got volatility wrong here. Volatility should be measured as the standard devation of returns (i.e. movements of price from one period to the next), not of prices.

1. quintuitive says:

Absolutely, that’s exactly the point.

3. Helmuth Vollmeier says:

To compare apples with apples you should apply your code to a rolling window, which is how BBs are calculated, like so ( for a period of 10 observations for aa and bb): “rollapply(aa,10,sd)” . Now the result is exactly as you expected, namely that the first series which increases 1% per day gets a far higher and increasing sd.