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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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The term of art in our industry for this type of option pricing formula is a series solution. As Farahvartish indicates in the comments, a series solution is not considered to be an "analytical solution" due to the reliance on a converging infinite sum for actual numeric output.(*) Series solutions have been employed at least since the 1990s, when they ...

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If you assume that IV of different expiration options is equal, then it mathematically follows that you are correct. Weeklies would give you the maximum theta decay. That is the theoretical answer. In practice, you may not have weeklies on every stock or index and you might have them but they trade too thinly. Wide bid/ask spreads etc. Also sometimes there ...

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