Including Dividends

In this lesson, we looked at pricing options in the cases where the underlying security pays out dividends every period that are proportional to the underlying stock price.  Like so:

The replicating portfolio will change as well to include the dividend:

The risk-neutral probabilities, q and (1-q), must also be recalculated to price options:

We can also calculate the value of the dividends.  Consider an option on a security that does not pay dividends on a call option with K=0:

Then consider the same security except now it does pay dividends:

So the value of the dividends is: