Should I adjust historical data for dividends when estimating drift?

Question

I'm building a Geometric Brownian Motion model which incorporates future dividends which vary over time. Since these should reduce stock price when paid, I can incorporate that into the model, however, I just realized that when I estimated drift from historical data, that data already incorporates dividends. Should I adjust the historical to remove the impact of the dividends?

Answer 1

In a Black-Scholes framework, the Geometric Brownian Motion specifying the dynamics of a stock continuously paying dividends is given by:

$$ dS_t = (\mu_{\text{Total}}-q)S_tdt+\sigma S_tdW_t $$

Under the model above, the stock price $S_t$ is continuously paying a dividend equal to $qS_tdt$, and as a consequence the stock price depreciates continuously at the same rate $-$ the same way in stock markets, stock prices plunge immediately after a dividend payment by an amount equal to the dividend paid.

Let us define $\mu_{\text{Price}} = \mu_{\text{Total}}-q$. Now, depending on what you mean by:

"[...] data already incorporates dividends."

We have:

• If your stock data is not adjusted for dividends $-$ "not adjusted" meaning that stock prices do not incorporate paid dividends $-$ then if you estimate the drift on that data series you will be estimating $\mu_{\text{Price}}$ i.e. the price return drift which only captures returns from variations in the stock price independent of dividend payments.
• If your stock data is adjusted for dividends $-$ "adjusted" meaning that stock prices are corrected by adding back paid dividends $-$ then if you estimate the drift on that data series you will be estimating $\mu_{\text{Total}}$, i.e. the total return drift which incorporates dividend return.

You can manually adjust your data series to "remove the impact of the dividends" $-$ I am unsure whether you mean subtracting paid dividends from the stock price, or adding them back $-$ and you will be able to estimate both drifts $\mu_{\text{Total}}$ and $\mu_{\text{Price}}$ from each data series, adjusted and unadjusted. From there you can also estimate the continuous dividend yield $q$.

For discrete dividend modelling, things get messier. You can take a look at Wikipedia's article on the Black-Scholes model, where they consider extensions with dividend-paying stocks.