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!

  • $\begingroup$ Depending on how you're planning to use it, you could simply use put-call parity. $\endgroup$
    – Chris
    Jun 19, 2019 at 16:45
  • $\begingroup$ That's a good idea, however, I am already using put-call parity to retrieve the implied interest rate, which i then feed to my model to price the calls & the puts, you see? I'm afraid if I used put-call parity again, I would be eating my own tail, so to speak. any thoughts? $\endgroup$
    – peterram
    Jun 19, 2019 at 16:58
  • 2
    $\begingroup$ Not sure what you mean by inconsistent results but there is no such thing as put-call parity for American options. Your problem is most likely that one. $\endgroup$
    – Ivan
    Jun 20, 2019 at 17:40

1 Answer 1


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

  • $\begingroup$ Note than neither B-AW nor BS handle discrete dividends (unlike RGW). With some work BS (at least the 1993 version) could probably be modified to handle a single discrete dividend by shifting the exercise barrier to a higher level when the dividend occurs. As you have discovered Black's Approximation is for calls only and cannot be turned into a put method. $\endgroup$
    – Alex C
    Jun 21, 2019 at 19:51
  • $\begingroup$ Thanks Alex, that's a good point! My market data source is very reliable at giving me the net present value of dividends until maturity of the option, but not so reliable for dividend yields. So I am using the NPV value to calculate a dividend yield to then plug in the models as you said. I wonder to what extent it is correct to consider a continuous dividend yield on a dividend paying stock for example. It doesnt seem right to me that you use the same yield for an option that pays no dividends durings it's lifetime, vs an option than does. How can you address that? $\endgroup$
    – peterram
    Jun 21, 2019 at 20:28
  • $\begingroup$ Very basic stuff here: you make the yield time dependent. Now of course it will break down around the dividend date as you can not model a discrete jump by a continuous model. But then Black-Scholes breaks down for vanillas as well and you end up going down the rabbit hole. $\endgroup$
    – jherek
    Nov 20, 2020 at 12:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.