Logarithmic Midprice as the Average of Log Bid and Ask

Summary

The note clarifies how to define a logarithmic price from bid and ask quotes. The intended expression is the arithmetic average of the two log prices, equivalent to the log of the geometric mean of the bid and ask. This choice gives a central price represented on a logarithmic scale, which is useful when working with log returns because returns are differences between log prices.

The alternative of taking the log of the bid-ask difference describes the spread on a log scale, not a midpoint price. The response notes that this alternative can produce negative values when the spread is below one, making it unsuitable as a price indicator in that setting. The discussion is brief and offers no empirical comparison or broader treatment of quote-based price measures; its clarification applies to the definition in the cited paper.

Key ideas

• A logarithmic midpoint is the average of the logarithms of the bid and ask.
• That average equals the logarithm of the geometric mean of the two quotes.
• Taking the logarithm of the bid-ask difference measures a transformed spread rather than a midpoint price.
• Log prices are compatible with calculating returns as differences between successive log prices.

Tags

# Logarithmic price defined as the midpoint of the log bid and ask : Simple Clarification


# Logarithmic price defined as the midpoint of the log bid and ask : Simple Clarification












Guys I would like a simple clarification. The paper by McMillan and Speight (2012) at the data section, defines the logarithmic price as the midpoint of the logarithmic bid and ask. Is that translates to [log(BID)+log(ASK)] / 2 or as log (BID-ASK) ? Thank you very much.

## Answer by Jan Sila (score 3, accepted)

https://quant.stackexchange.com/a/28359

I reckon it's the $\frac{log(ASK)+log(BID)}{2}$, just simple arithmetic average as it makes sense, also considering logarithmic returns, when you can only take a differences from log prices.

Also, the second alternative would yield negative 'prices' for spreads lower than 1, which cannot serve as a 'price' indicator.

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