# 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.

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;

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

2. 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.

• I see I hadn't made the connection that by doing one you are implicitly doing the other. It does make sense in hindsight. Since returns can't be decoupled from the price (obviously) trading one well would mean trading the other well. Jul 8, 2022 at 14:47
• Hi: if you use the log price ( to calculate the mave and std ) with BBs and you use log price on the vertical axis of the BB plot, then the vertical difference between log price at time t+n and the log price at time t, is ( approximately ), the n period return. So, if the log price has crossed over its upper $k \times sigma band$, since n periods ago, it's saying that the return has gotten larger in the last n periods. But it's not the case that, when generating entries and exits, using the return and it's variance is in any way equivalent to using the log price and it's variance. Jul 8, 2022 at 21:52
• What I'm saying above ( in a confusing way ) is that if you take a stock's price series and use A) the log price and its variance to construct BB's and then also B) use returns and its variance to construct BB's, you will get two very different results when looking at the entries and exits. Jul 8, 2022 at 21:56