Approximating American Put Prices with Discrete Dividends

Summary

The question asks how to adapt Black’s approximation, commonly used for American calls on dividend-paying stocks, to price puts when dividends are discrete. The response does not derive a put-specific version of Black’s approximation. Instead, it points to established approximate methods for American options with dividends: the Bjerksund–Stensland approximations and the Barone–Adesi–Whaley approximation.

The answer suggests comparing model outputs against a reference implementation and cautions that some published equations for Bjerksund–Stensland may contain typographical errors. It recommends checking the accompanying implementation for consistency. These are practical pointers rather than a quantitative comparison: the post supplies no equations, assumptions, numerical examples, or evidence ranking the methods. Pricing accuracy will depend on the option and market inputs, and the discussion does not specify how to handle dividend timing or validate exercise boundaries.

Key ideas

• The answer recommends Bjerksund–Stensland and Barone–Adesi–Whaley as alternative American option approximations.
• It does not provide a direct adaptation of Black’s approximation for puts with discrete dividends.
• Comparing a pricing implementation against a reference can help identify inconsistencies.
• Published approximation formulas should be checked against implementations because typographical errors may occur.

Tags

# Black's Approximation - Discrete dividend for Put Options


# Black's Approximation - Discrete dividend for Put Options












I am currently trying to price and option chain for dividend paying stocks (american style exercise). I am able to calculate the Net Present Value (NPV) of dividends until maturity and then apply Black's approximation to compute the value of the call option.

However, when now trying to apply the same procedure to price Put options, I obtain inconsistent results.

My question is: assuming Black's approximation is a good way to price Call options with discrete dividends being paid, how should I proceed to get a similar approximation for the Puts?

In all the great books I only find reference to pricing the calls.

Thank you for your help in advance!

## Answer by peterram (score 1, accepted)

https://quant.stackexchange.com/a/46227

In the past few days I tried pricing Put options using other methods other than the Black's approximation. So far i came to the conclusion that the best methods are those presented in Haug "The complete guide on option pricing formulas":

- Bjerksund and Stensland Approximation (1993,2002) both price pretty well

- Barone-Adesi and Whaley Approximation

a good website to compare your results to is: https://rdrr.io/rforge/fOptions/man/BasicAmericanOptions.html

Finally a warning that the book referenced above contains several small mistakes in the equations of the Bjerksund and Stensland approximation. These small typos can be misleading. I reccommend always checking the VBA code section for consistency

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