Why Lognormal Prices Imply Normally Distributed Log Returns

Summary

The document clarifies the link between a lognormal price assumption and the distribution of returns. A simple return is the ratio of an ending price to a starting price, minus one. Taking the logarithm of one plus that return therefore gives the logarithm of the price ratio.

If the two prices are modeled as lognormally distributed, the answer states that their ratio is also lognormal. By definition, the logarithm of a lognormal variable is normally distributed, so the log return is normal under that assumption. This explains the stated relationship, but it does not establish that real market prices or returns follow these distributions. The source frames lognormality as a modeling assumption that may not hold for a particular price series, and gives no empirical test or broader comparison of return measures.

Key ideas

• A simple return plus one equals the ratio of ending price to starting price.
• The logarithm of that ratio is the log return.
• Under the stated assumption, a ratio of lognormally distributed prices is lognormal.
• Taking the logarithm of a lognormal variable yields a normally distributed variable.
• The distributional conclusion depends on the lognormal price assumption.

Tags

# Why should we use log returns? Log normality


# Why should we use log returns? Log normality












According to this link, there are some reasons we have to use log returns.

But I can not understand the first reason provided in the link:

> First, log-normality: if we assume that prices are distributed log normally (which, in practice, may or may not be true for any given price series), then $\log(1 + r_i)$ is conveniently normally distributed, because: $$ \tag{1} 1 + r_i = {p_i \over p_j} = e^{\log\left({p_i \over p_j}\right)}$$

I can't understand how equation $(1)$ is related to the normal distribution.

Anyone can help?

## Answer by AdB (score 1, accepted)

https://quant.stackexchange.com/a/43161

Saying that prices are lognormally distributed here means that $p_i, p_j$ are assumed to be lognormally distributed. Then it is easy to verify that $\frac{p_i}{p_j}$ is also lognormal. Hence, by the definition of the lognormal distribution, $\log \left( \frac{p_i}{p_j} \right)= \log \left( 1 + r_i \right)$ is normally distributed.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.