I want to simulate the price path of a stock for one quarter using geometric Brownian motion. The stock has a continuous dividend yield of 5% based on the annual dividend yield. However, historically this dividend is paid out once a year in the same quarter that I model. Thus, I want to reflect the annual dividend yield in this exact quarter. I am not sure how to do this. I believe that using the continous dividend yield of 5% would underestimate the effect. Another thought I had was to scale the continous dividend yield with 4 to make it 20%, but I dont think this makes sense either. Another solution I have thought of is to model it as a discrete dividend, but I want to avoid this as the exact date of the dividend is unknown.

Please share your insights if you have an idea on how to reflect annual dividends in one quarter.


1 Answer 1


What follows is a discrete dividend version of my answer in this post:

When the stock pays dividends it is not true that the deflated stock price $$ e^{-\int_0^tr(s)\,ds}S_t $$ is a martingale. But instead (see [1]) no arbitrage theory dictates that the process $$ M_t:=e^{-\int_0^tr(s)\,ds}S_t+D_t $$ is a martingale where $D_t$ is the pathwise present value of all dividends $d_i$ paid until time $t\,$: $$ D_t=\sum_{t_i\le t} d_i\,e^{-\int_0^{t_i}r(s)\,ds}\,. $$ To understand this a bit better note that the portfolio consisting of the stock plus its past dividends, when they got put into the money market account, is $$\tag{1} \Pi_t=S_t+\sum_{t_i\le t} d_i\,e^{\int_{t_i}^tr(s)\,ds}\,. $$ This is an asset that does not pay dividends. Hence $e^{-\int_0^tr(s)\,ds}\Pi_t$ must be a martingale, and it obviously equals $M_t\,.$

A natural assumption is that the stock $S_t$ jumps down by $d_i$ at the dividend payment date $t_i\,:$ $$ S_{t_i}-S_{t_i-}=-d_i\,. $$ Since the second term in (1) jumps up by $d_i$ at $t_i$ it follows that $\Pi_t$ is a continuous process. It is sometimes called cum dividend process of $S_t\,.$ A simple model you can assume is that $\Pi_t$ now follows a GBM with a certain constant vol $\sigma_{cum}\,:$ $$\tag{2} \Pi_t=\Pi_0\exp(\textstyle\int_0^tr(s)\,ds +\sigma_{cum}W_t-\frac{\sigma^2_{cum}t}{2})\,. $$ Ignoring dividends that have been paid before $t=0$ yields $\Pi_0=S_0\,.$ To calculate the forward price of the stock we can use the martingale property: $$ E[e^{-\int_0^tr(s)\,ds}\Pi_t]=\Pi_0=S_0\,. $$ This gives $$ E[e^{-\int_0^tr(s)\,ds}S_t]+\sum_{t_i\le t} d_i\,P(0,t_i)=S_0 $$ where $P(0,t_i)=E[e^{-\int_0^{t_i}r(s)\,ds}]$ are discount factors. This yields the formula for the forward price of the stock: $$\tag{3} \boxed{F_t=\frac{E[e^{-\int_0^tr(s)\,ds}S_t]}{P(0,t)}=\frac{S_0-\sum_{t_i\le t} d_i\,P(0,t_i)}{P(0,t)}\,.} $$ If you compare this with the traditional $$ F_t=S_0e^{(r-q)t} $$ you can get the levels of $d_i$ from your continuous dividend yield $q=5\%\,.$ For a single $d_i$ or for $d_1=...=d_n$ (this is what you seem to assume) this is very easy. Clearly the level of $d_i$ depends on how many dividends you assume to be known in advance.

BTW: The model assumption (2) was not used to derive (3). Only the martingale property of $M_t=e^{-\int_0^tr(s)\,ds}\Pi_t\,.$ In other words: (3) holds for every arbitrage free model of the stock that only assumes a couple of dividends $d_i$ are known in advance.

[1] D. Duffie, Dynamic Asset Pricing Theory. Princeton University Press, 1991.


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