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Pricing American Options with Discrete Dividends Using Grid Methods

Article Quant Q&A · Author: Victor
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Summary

The document outlines how to price American options when the underlying pays discrete dividends. Because exercise may be optimal before expiration, the problem is treated as an optimal-stopping problem. A numerical grid method, such as a recombining tree or PDE solver, evaluates option values from expiration backward through possible stock-price states and compares continuation value with the value from exercising early.

The response says discrete dividends make the computation more involved than the no-dividend case. It points to tree methods for the simpler setting and notes that fixed cash dividends can be handled in a cited software example. A mix of fixed and proportional dividends may require a different implementation. The discussion gives a high-level computational approach, but does not compare methods quantitatively or specify convergence, calibration, or accuracy criteria. Implied volatility and Greeks would be obtained from the resulting pricing model, though the response does not detail those calculations.

Key ideas

  • American exercise requires evaluating whether early exercise is preferable to holding the option.
  • Tree and PDE grid methods can work backward from expiration to value the option.
  • Discrete dividends complicate the treatment of early exercise.
  • The dividend model may distinguish fixed cash amounts from proportional components.
  • The response does not provide a quantitative comparison of implied-volatility or Greek calculations.

Tags

OptionsDerivatives pricingVolatilityStatistics
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# Implied volatility and greeks for american option with discrete dividends


# Implied volatility and greeks for american option with discrete dividends












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?

## Answer by Brian B (score 3)

https://quant.stackexchange.com/a/32502

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.