# When valuing a vanilla option on an index, should we take dividend into account?

When valuing a vanilla option on an index (eg FTSE 100), should we take index dividend yield into account?

$$c=Se^{-q\tau}N\left(d_1\right)-Ke^{-r\tau}N\left(d_2\right)$$ $$d_1=\frac{\ln\left(\frac{S}{K}\right)+\left(r-q+\frac{1}{2}\sigma^2\right)\tau}{\sigma\sqrt{\tau}}$$ $$d_2=d_1-\sigma\sqrt{\tau}$$

Using the above formulae, should $q$ be 0 or "equivalent index dividend yield" when $S$ is an index (eg FTSE 100)?

• The black-Scholes equation is not an sacred oracle, it is simply an approximated equation. Then, first make the computation with q = 0; after this make the computation with q = dividend yield. Finally compare the two results and you will obtain a variation interval for the price of the vanilla option. – Juan Ospina Aug 17 '15 at 14:48
• Use of Black Scholes is with forward and zero-coupon bond $$c = B(0,\tau) \left(F_{0,\tau}N(d_1)-KN(d_2)\right)$$ $$d_1 = \frac{\ln\left(\frac{F_{0,\tau}}{L}\right)+\frac{\sigma^2}{2}\tau}{\sigma\sqrt{\tau}}$$ $$d_2 = d_1-\sigma\sqrt{\tau}$$ – MJ73550 Apr 14 '16 at 9:17

Ftse100 would not have a smooth dividend yield, as your formula has, it would be discrete, being much higher on certain days of year than others. In pricing options on ftse, u need to take into account implied dividends (dividends that are implied by put call parity)

To expand on Randor's answer, the standard Black-Scholes formula as you've given it assumes a constant continuous dividend yield of $q$. To adapt this to cope with discrete deterministic (absolute) dividends $d_i$ at known times $\tau_i$, you could recast the formula in terms of the "dividend-free" stock price:

$$S^* := S - \sum_i d_i e^{-r\tau_i}$$

and set $q$ to zero.

N.B. You only include dividends between spot and the option expiry.

When valuing a plain index option, there are two options in terms of index dividend:

(1) The underlying price is a spot price like in the FTSE 100 case (option is valued off the index): you can use continuous dividend yield. You can imply a dividend yield from a linearized call-put parity:

The present value of the dividend payment is

$PV(div)=P-C+(S-K)+K(e^{rt} - 1)$,

then the implied dividend yield is $d = \frac {PV(div)}{T*S}$

(2) The underlying instrument is a future contract on the index (for example, IBEX35 or TAIEX), you'd set index dividend yield to zero and use future price of corresponding maturity as an underlying price.

For the purposes of calculating option prices or implied volatilities, the use of a dividend forecasting model based on projected actual dividend growth rates can lead to an option model which is internally inconsistent. In contrast, the use of a model based on constant dividend yields is not only consistent, but also easier to implement.