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Bollinger Bands on Log Returns and the Role of Distribution Assumptions

Article Quant Q&A · Author: John S.
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Summary

The document asks whether a rolling Bollinger Band strategy could be applied to log returns rather than price levels. The proposed rationale is that price returns may have fat tails or asymmetry, while a log return series that passes a stationarity test and shows mild mean reversion might offer a basis for trading short-term deviations. The questioner suggests trying this on liquid, higher-frequency data but supplies no strategy results or out-of-sample evidence.

The accepted response corrects a misconception: asymmetry does not prevent a distribution from having a standard deviation, and fat tails affect its behavior without making it undefined. It also points out that treating returns as normally distributed is an assumption; under that assumption, prices are lognormally distributed and have finite variance. Consequently, switching from price levels to returns does not by itself avoid the distributional assumptions behind a standard-deviation band strategy. The exchange offers a conceptual clarification, not evidence that either implementation is profitable, and the stationarity and Hurst observations alone do not establish a trading edge.

Key ideas

  • A standard deviation can exist for distributions that are asymmetric.
  • Fat tails affect standard-deviation behavior but do not rule out its existence.
  • Normal log returns imply lognormal prices under the stated distributional assumption.
  • Using log returns does not automatically remove the assumptions behind Bollinger Bands.
  • Stationarity tests and mild mean reversion do not by themselves demonstrate strategy profitability.

Tags

Mean reversionTechnical indicatorsStatisticsBacktesting
Full text
# Trading based on the log return series


# Trading based on the log return series












A common strategy in trading is to use a bollinger band system. Simply put, we bet on reversion to the mean and take the opposite trade to the current movement under the assumption a move is overdone.

However implicit in this is the idea of standard deviation which really only applies to symmetrical distributions. Generally, stocks, futures, etc have long tails. So it would seem regardless this strategy would likely fail any serious scrutiny outside of pairs trading.

We can take the log return of the price series. If returns are distributed log normally then log returns are distributed normally. Suppose this series passes the ADF test with p < 0.01, and a hurst exponent test reveals mild mean reversion properties (0.45 <= H < 0.5).

I've searched and searched and I have not found any information regarding trading this series. Since no one has written about it, it seems like I'm walking into a waste of time. Why couldn't we apply a simple bollinger band strategy to the log return series (rolling) and trade this instead of the price? In the case of "higher frequency data", for example 15 minutes or less, in a highly liquid market it would seem this would have some value.

## Answer by Max van Leeuwen (score 4, accepted)

https://quant.stackexchange.com/a/71532

Firstly, distributions don't have to be symmetrical to have a standard deviation. In addition, I think you mean that financial instruments (stocks, futures, ect.) have FAT tails not long tails. This affects the value of the standard deviation but doesn't prohibit its existence.

About your proposed strategy; I think it's important to realize that a lognormal distribution for the price series (which implies the normal distribution for the return series) is an assumption. Since a lognormal distribution has a variance you are implicitly doing the assumption that allows for the first strategy to be applied as well.

TLDR;

- Neither fat tails nor asymmetry prevents a price series to have a standard deviation.

- If the return series is normally distributed, the price series is lognormally distributed and consequently has a variance. Both strategies thus rest on a similar assumption.

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.