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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 ...

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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 ...

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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)

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