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What methods are available to calculate IV and greeks for an american option with discrete dividends, and how do they compare?

Should I use Roll-Geske-Whaley and solve for a given option price?

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If you have an American-exercise option and want to treat discrete dividends properly, you will have to apply some sort of technique for determining the exercise strategy. Mathematically this is phrased in terms of an optimal stopping time. Computationally it is handled in a grid technique like a tree or other PDE solver.

Basically, you set up a finite grid of potential stock prices $S_{n,t}$ and corresponding options values $V_{n,t}$, where $V$ starts out unknown. You know what $V$ looks like at the option expiration time $T$, and so you fill that in and then work backwards from $T$, filling in $V$ values as you go.

The American exercise comes in when you figure out for which cases $V_{n,t}$ would be bigger by exercising early.

Without discrete dividends, this all well-handled in the Leisen-Reimer trees that Matt Wolf points to in this answer to a question about real-time pricing.

With discrete dividends, things get a lot trickier. If you are willing to assume a completely fixed value for each dividend, then this example appears to show how to treat the option in QuantLib. If you want the dividends to be partially fixed and partially proportional, I think the only choice is this R package I wrote.

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