How Continuous Dividends Affect Stock Portfolios and Option Pricing

# Utterly confused by "continuous dividends" in financial math


# Utterly confused by "continuous dividends" in financial math












I get extremely confused by 'continuous dividends' paid on stocks. As long as there are no dividends, I understand the arguments, but when there are dividends all of a sudden I no longer comprehend it.

So, consider some fixed stock. To be very concrete, let us assume a binomial model, where at time t=0 the stock price is S_0. One time-step later the price might be either $u S_0$ (the "up-state") or $d S_0$ (the "down-state"). Let's assume the next step in the model happens after $\Delta t$ years time. Suppose I have 6 shares of stock at t=0 (for a total value of $6 S_0$); then $\Delta t$ years later I either have a portfolio valued at $6 u S_0$ or $6 d S_0$ (depending on whether the up or down-state is reached). So far, everything is without any dividends.

Now, let's assume the stock in question pays continuous annual dividends at a rate of q. Let's still assume I got 6 shares of stock at time t=0. Clearly since one share's value is $S_0$ my portfolio is still valued at $6 S_0$ in this case. Now, let's say that in the next time-step (i.e., $\Delta t$ years later) we reached the up-state. What is the value of my portfolio now? I would think that I still have my 6 shares of stock, so my portfolio value would be $6 u S_0$, but this doesn't take into account any dividends. Would my portfolio be valued at $6 u S_0 e^{q \Delta t}$ (or perhaps $6 u S_0 e^{-q \Delta t}$) !? I am not sure which one (if any) of this is correct, since I do not understand what is going on here; I'm utterly confused.

It would be very nice if someone could clarify this, and give a clear and meticulous explanation.

## Answer by Rylan (score 1)

https://quant.stackexchange.com/a/81128

Continuous dividends, like continuous interest compounding, is there to make continuous-time computations easier, not to represent reality. With this in mind, let's first consider a discrete dividend.

Suppose you have a portfolio that is just one stock \$10. It pays a dividend of \$1. If you hold it when you receive the dividend, you will then have \$1 cash and a stock worth \$9.

From a portfolio perspective, there's no real difference. You had \$10 before the dividend and you have \$10 after the dividend, and if you want to have \$10 in stock still, you just reinvest your \$1 dividend in stock.

The same idea holds for continuous dividends; you receive the cash and can reinvest it if you like. (I will say that I haven't personally seen continuous dividends be used with a discrete tree; we can perhaps assume when we mix them that the dividend accrues continuously and is paid at a given timestep in the tree?)

While I said the dividend isn't impactful from a portfolio perspective, it does in fact matter from the perspective of derivatives pricing. Loosely, if you buy a call on a stock that pays dividends, the current price of the stock "includes" some dividends that are going to be paid to whoever owns the stock, but not to people who hold the option.