# 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.

• If I had to guess it’s because $q$ is alphabetically close to $r$. May 14 at 20:35
• I don't know, but I'd be surprised if it has anything to do with the convention of appending Q to an equity ticker after bankruptcy. As @BobJansen said, r is the risk free intetest rate, while the next letter q is the dividend yield. May 14 at 20:37
• History: As far as I can tell, Fischer Black and Myron Scholes, The Pricing of Options and Corporate Liabilities Journal of Political Economy, Vol. 81, No. 3 (May – Jun 1973), pp. 637-654, doesn't mention dividends at all, while Robert Merton, Theory of Rational Option Pricing,The Bell Journal of Economics and Management Science, Vol. 4, No. 1. (Spring, 1973), pp. 141-183 polymer.bu.edu/hes/merton73py538.pdf uses d to denote continuously compounded dividend yield. May 14 at 20:48
• How about strike K? It's not common either. (It originally from baseball actually). May 14 at 21:36
• Cool to see that you are still curious, Daneel, even as a veteran of this site :) May 15 at 20:22

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 Unfortunately, they do not cite the source of this notation and I didn't find any interesting leads in the references section of the paper.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.  