Black Scholes Model and Dividends

Question

My question can be summarised as such:

• Consider a portfolio. Say it has a price $\Pi = x$.
• Portfolio consists of a stock and a sequence of call options underlying on the stock.
• It has been announced that a dividend will be paid in half year. However, assume that the stock price does not change today.
• How will the value of the portfolio change today?

My argument:

1. If the stock price does not change today due to announcement, then we can assume the dividend is already priced into the stock value.
2. In order to use the Black-Scholes-Merton option pricing model, the underlying stock price must only consist of a risky component, and not the certain dividend component as it must be assumed that stock prices follow a geometric Brownian motion.
3. Since the stock price used for the model decreases (subtracting the present value of the dividend in half year), and the delta of the portfolio is positive, the value of the portfolio must decrease.

Where is the flaw in my argument (if there is one) ?

Answer 1

The option portfolio would not change its value because the dividend also has been already priced in as you can assume that option markets have the same information on the underlying as the stock market.

A dividend in general decreases the value of the call options because these are foregone relative to holding the asset as soon as the market has as an expectation about it. You should assume though that the maturity of your call options is greater or equal the dividend payment date.

The fact that the dividend was already priced in upon the announcement is not relevant because it may change the underlying price (or not) $S_t$, but not the calculation of the underlying for the option pricing $S_te^{-d(T-t)}$.