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Dividends and Its Effect on Option-Pricing

Dividends matter to options because they reduce the forward price of the underlying. On the ex-dividend date the stock drops mechanically by roughly the dividend, and options must price that in advance.

The direction follows immediately: a lower expected future price means calls are worth less and puts are worth more.

In the formula

With a continuous dividend yield qq:

C=Se−qTΦ(d1)−Ke−rTΦ(d2)C = Se^{-qT}\Phi(d_1) - Ke^{-rT}\Phi(d_2)d1=ln⁡(S/K)+(r−q+12σ2)TσTd_1 = \frac{\ln(S/K) + (r - q + \tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}

The e−qTe^{-qT} discounts the stock for dividends the option holder will not receive. An option confers no ownership, so no dividends, and the price must reflect that.

Continuous yield is a reasonable approximation for an index. For a single stock paying a few discrete dividends, the better method is to subtract the present value of the dividends from the spot price and price the option on the remainder.

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