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I have been trying to understand how to price FX options with barriers. In Hull (Options, Futures, and Other Derivatives), there are closed formulas how to calculate the premiums of European Call and Put options with barriers. However, I am not able to find either closed formula or a model of computing the price of FX options with American barriers. I am not experienced with pricing of FX options, so I apologise if that seems to be an elementary question.

Could anyone please advise where I can read or find more information?

Thank you in advance!

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    $\begingroup$ What is an American barrier supposed to be? I wonder if @Candidate means "American-exercise barrier option". $\endgroup$
    – Brian B
    Apr 30 at 22:07
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    $\begingroup$ I’m voting to close this question because the term "American barrier" does not exist in the literature or in markets. $\endgroup$
    – noob2
    Apr 30 at 22:24
  • $\begingroup$ I strongly disagree with the comments here. An American barrier is a widely used term in option pricing. It is part of the official [fpml ](fpml.org/spec/fpml-5-6-5-rec-1/html/confirmation/…) syntax. $\endgroup$
    – AKdemy
    May 2 at 20:29
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    $\begingroup$ You can price all barrier options in Black-Scholes analytically as an approximation. More information on the formulas can be found in financial literature for example Espen Haug's book "The Complete Guide to Option Pricing Formulas", 2nd Ed. pp.152-onwards. As a general guide, it will underprice it from what you will get quoted. $\endgroup$
    – AKdemy
    May 2 at 20:43
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    $\begingroup$ @AKdemy, thank you for your insights, they are really useful! I will research the two sources (Wystup and Haug) you outlined. And I agree that "American Barrier" is a widely used and valid term (that is why I used it in the post). In any case, thank you once again for your help! I am happy to see that there are people like you - prone to help. :) $\endgroup$
    – Candidate
    May 6 at 20:14
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Chaper 24 of the book "FX Derivatives Trader School" (much better than you might think from the title!) covers this topic.

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  • $\begingroup$ Thank you so much for your reply! In my view, the book provides quite good explanations. I read chapter 24, but it seems like they refer to “pricing tools”, which are exactly what I need. Of course, I am not interested in any implemented code – I would like to see the formulas used to price the option. I looked at the rest of the book, where they develop other “pricing tools” in excel, but as far as I see those models refer to pricing of different options. Am I missing such a tool from the book or there is a chapter where the “pricing tools” from chapter 24 is implemented? Thank you! $\endgroup$
    – Candidate
    Dec 2 '20 at 19:31
  • $\begingroup$ I think there some hints in the chapter about pricing ("mixed" volatility model which should be stochastic-local volatility). Quick skim of Clark (Foreign Exchange Option Pricing: A Practitioner's Guide) and Castagna (FX Options and Smile Risk) suggested nothing in either on pricing American barriers. $\endgroup$
    – user42108
    Dec 4 '20 at 20:29
  • $\begingroup$ Sorry for the late reply! Thank you for your help, you are the only one who responded my question. I also checked the book of Clark, but it seems that there is again very scarce information about pricing fx options with american barriers. Honestly, I cannot find information about this and I don't know where to search. Anyway, thank you again! If I find something useful, I will share it. Best! :) $\endgroup$
    – Candidate
    Dec 21 '20 at 19:50

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