How Dividend Yield Came to Be Denoted by q in Option Pricing
Summary
This note traces the use of q for continuous dividend yield in derivative pricing. It contrasts Merton’s use of D in an early treatment of continuous dividends with later use of q in work by Hull and White and in Hull’s textbook. Other papers used d or the Greek letter delta, suggesting that the convention took time to become established.
The explanation offered is that Hull’s influential textbook helped spread q. The author speculates that q may have been chosen because it sits alphabetically near r, the risk-free rate, making the parameters easy to introduce together. A second, more tentative suggestion connects q to subtracting dividend yield from r in risk-neutral stock dynamics. The document does not identify a definitive origin: the proposed reasons are explicitly guesses, and the cited literature provides evidence about usage rather than a confirmed account of why the notation began.
Key ideas
- Dividend yield is commonly represented by q in derivative pricing models.
- Earlier literature used other symbols, including D, d, and delta.
- Hull’s textbook is proposed as a major source of q’s widespread adoption.
- The suggested links between q and r are hypotheses, not established historical explanations.
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Full text
# Why do we use the letter $q$ for dividends?
# Why do we use the letter $q$ for dividends?
In derivative pricing models, we often use the letter $q$ to designate the dividend yield i.e.: $$\textrm{d}S_t=S_t((\mu-q) \textrm{d}t+\sigma\textrm{d}W_t)$$ for the price process $S$ of the stock.
Is there some historical reason for this notation convention? Or maybe a reference to some term I don’t know?
I can imagine the letter $d$ might be avoided due to the use of differentials $\textrm{d}t$.
This specific parameter catches my attention because it is the only common model parameter or function which uses a Roman letter without direct reference to a common term such as $r$ for rate, $S$ for stock, $V$ for value or $C$ for call.
## Answer by Bob Jansen (score 5, accepted)
https://quant.stackexchange.com/a/75545
It seems it didn't take long before the case of continuous dividends was considered in the literature. Robert Merton's 1973 paper "Theory of Rational Option Pricing" considers the case of dividends in section 7 and denotes it with a $D$.
In "An Overview of Contingent Claims Pricing" from 1988 by John Hull and Alan White $q$ is used and the 2nd edition of Hull's "Options, futures, and other derivatives" (can't find the first one) does as well. Unfortunately, they do not cite the source of this notation and I didn't find any interesting leads in the references section of the paper.
The notation didn't immediately catch on:
In the 1990 paper by David C. Leonard, Michael E. Solt "On using the Black-Scholes Model to Value Warrants" the dividend yield is still $d$. The 1996 "American Options on Dividend-Paying Assets" by Mark Broadie, Jérôme Detemple used $\delta$. In "American options with stochastic dividends and volatility: A nonparametric investigation" by "Mark Broadie, Jérôme Detemple, Eric Ghysels, Olivier Torres" from 2000 it's the same.
To answer the question why: Because Hull is doing it for a very long now time and many people read his book. I made a guess why Hull choose $q$ in the comments:
> If I had to guess it’s because $q$ is alphabetically close to $r$
In the 2nd edition Hull introduces $q$ as below. A few pages back $r$ is introduced in a similar way and this 'closeness' of definitions might have suggested this convention to Hull for didactic purposes.
A more far fetched explanation is that the source is the discussion in "An Overview of Contingent Claims Pricing" also shown below. The subtraction of little $q$ from $r$ is necessary to have the correct drift under the $\mathbb{Q}$ measure. This is even more far fetched since Hull and White don't discuss risk neutral valuation in terms of the risk neutral $\mathbb{Q}$ measure.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.