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While many systems like to treat dividends as a continuous yield when pricing equity options, it works quite poorly for short-dated options.

In the short run, deterministic dividends are clearly the way to go, since the upcoming dividend is usually known with fairly high precision. In the medium term, we may start to think of those dividends as being linked to the stock price, but still want to treat them discretely so as to get early exercise dates right. In the long term, tracking all those discrete dividends becomes a pain and it feels nicest to go back to a yield.

Advanced option pricing frameworks allow for mixtures of these 3 treatments. What are some good ways of selecting a reasonable mixture of dividend treatments in any given circumstance?

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Time to expiration is what should guide the choice.

A tractable approach is to make the distinction between discrete and yield at the LEAP boundary (or simpler options with expiration more than 1yr into the future).
When the options are long dated, like LEAPs for example, then the simplicity of the yield approach is usually 'good enough'.

It usually makes sense for to ONLY use the the discrete dividend approach for options near expiration.

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This article discusses the topic well, you can model the short term discrete dividends from discrete fixed and the medium/long term with discrete proportional dividends. If you know the annual estimated dividend (Factset estimates/dividend swaps etc), you can use historical payment amounts and dates to weight and project the discrete fixed dividends and then calibrate the subsequent proportional dividends after some horizon (1 or 2 years) from the historically weighted projections.

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1141877

The paper incorporates credit risk as well but you can set the hazard rate to zero for your analysis.

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Theoretically, it's "complicated". Build a spreadsheet of all the index constituents, their expected payouts, associated dates, and real-time stock prices, and index weights. Then hours of struggle later, cross your fingers, and hope you haven't made any calculation mistakes ;-)

The quick and easy way... simplify your equations with r=0 and replace the spot price of the index with the forward index value from the futures market. In effect, you're using an ATMF versus an ATM framework. This will have the same expiry as the options, plus embed both the interest rate and expected dividend components.

The eagle-eyed observer might correctly observe monthly expiries on options versus only quarterly on futures. It's a fair point. However, it is trivial to interpolate ATMF from put-call parity on the Oct/Nov/Jan/Feb/Apr/May/Jul/Aug options.

Put simply, you can get the market to all the hard work for you here!

For options on single stocks rather than indices, the principle holds. Put-Call parity gives you a forward price of the same tenor as the associated options. For a stock worth 100, if the market is pricing in a dividend payment of 1 and you are sure it will be say 1.1, then reprice your calls/puts for a future worth 0.1 more than currently priced.

Finance 101, under no-arbitrage conditions:

Forward = Strike + Call - Put
Forward + Cash = Spot + Dividend Dividend = Strike + Call - Put + Cash - Spot

[True pedants might quibble about the time value of a future dividend surprise. However, compare a few weeks of interest on a dividend surprise (a few % of a few % of price) to the materiality of price and implied vol uncertainties; and life is just too short]

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