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this is my first time asking questions here. I want to look for some calculation method to price a very exotic option.

The FX Euro-American Knockout Option (EAKO) is an option that has an American style up-and-out call feature and an European style down-and-in put feature. Throughout the option's life, if the spot rate rises above the upper barrier rate, the option is immediately exterminated, and the buyer loses the opportunity to receive any payoff from the option. If the upper barrier is not touched, and if the lower barrier is touched at maturity, the buyer may enjoy a payoff of a strike rate - spot rate from the issuer. Note that the strike rate is larger than the lower barrier rate.

With this payoff, I have always been thinking of it as a path-dependent product, so I incline to use monte carlos method to get an expected price of it. However, I found that in Bloomberg OVML page, the EAKO is priced with a default Vanna-Volga method. As such I want to know if anyone of you have any insights for this. Or is there any paper that I can rely on. To be frank, I don't see how Vanna-Volga method can be used to price a barrier option with mixed exercising style.

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  • $\begingroup$ Forget VV - Bloomberg chooses that for any exotic although they have SLV. I am not sure if EAKO is supported in SLV but I can check at work tomorrow. Otherwise, withing Bloomberg, you will need to script it in DLIB and use SLV with Monte Carlo. VV is an adjustment to BS that is like an "overhedge". However, I recommend to not use it for anything. $\endgroup$
    – AKdemy
    Commented Jan 27, 2022 at 17:19
  • $\begingroup$ Thank you for your prompt and your insight reply. Yes i do agree that BS is never a good option to chase with. However with the SLV, I am not sure with the parameters. Therefore, for preliminary solution, it is a BS for me currently. Anyways, I do forward to hearing from you!!! $\endgroup$
    – Fangy
    Commented Jan 27, 2022 at 19:06
  • $\begingroup$ What do you mean with you are not sure with the parameters? Leaving shifted LMM aside (which isn't FX), Bloomberg's SLV is by far the best model the firm offers and it's even calibrated to traded barrier option prices. $\endgroup$
    – AKdemy
    Commented Jan 27, 2022 at 20:03
  • $\begingroup$ I am trying to say is that I am trying to replicate those model by myself for learning, so if I am using the SLV, i think I have to search for details for the stochastic factors? $\endgroup$
    – Fangy
    Commented Jan 28, 2022 at 2:26

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I don't think you learn something useful by replicating VV. The model BBG uses has not been updated for about 20 years, and the latest white paper is from 2007 ("Variations on the Vanna-Volga Adjustment").

That said, when I tested it today, it is actually quite robust in the EAKO case. I think the reason is that the nature of the product is quite simple. The EKI and AKO are essentially independent structures. If you price a normal EKI and set EAKO AKO to 0 (it does not take 0 directly, just make it sufficiently close), you see the price is identical. If you now price a Down and out American barrier and compare by how much the price declines in % from the vanilla case, you can compute the adjustment needed in the EKI to get to the EAKO (same percentage drop roughly). That holds within 0.05% of notional for all variations I tested.

Since you write you are trying to replicate those models for learning purposes, I assume you are able to do the "simple" stuff already? E.g.

  • match the vanilla model in BBG exactly? or
  • solve delta for strike (especially for delta premium included pairs) - this will in fact be needed for VV?
  • Compute a vol surface from ATM DNS, RR and BF quotes?
  • Compute the price of a digital (BBG uses dK = 1% in the call spread)? or
  • compute the CME triangulation between the 3 books (PQO, FUT, VQO)? which is an interesting example outside of BBG
  • compute probabilities of hitting touch / no touch in BS world? (also needed for VV)

In my opinion, instead of trying to compute VV, implementing a PDE solver for American options or computing a Heston model is more useful. Replicating the SLV in BBG is impossible because you lack the input data needed to calibrate (barrier option prices specifically).

If you still want to do it, Uwe Wystup - Mathfinance provides a good explanation of VV. I found an online source for the BBG doc.

What is particularly needed is the following equation: enter image description here (sorry for copy pasting a pic, but that saved quite a few minutes which helps a parent of 2 lovely young kids).

  • compute strikes and prices for 3 vanilla with BS (ATM, 25d call and 25DP)
  • reprice again using ATMF vol to compute theoretical BS value (no smile) and also get the Vega, Vanna and Volga (matrix on the left of the BBG whitepaper screenshot above)
  • compute difference to BS price, vega, vanna and volga in USD values (right hand side vector $Y_{ATM}$ ...)
  • solve the linear system of equations to obtain the VV coefficients ($v_{vega}$, $v_{vanna}$, $v_{volga}$ in the middle of the screenshot)
  • Compute the BS vega, vanna and volga for the KO using the ATMF Vol (just use the BBG pricer (OVML) if you want to be quick)
  • compute the probability of not hitting the barrier (use BBG pricer and a down and out no touch for example - again using BS; which assumes you did the "simple" stuff already before tackling this). I call this p_sym like in the BBG paper although strictly speaking, this is wrong. The reason the author uses p_sym is that BBG uses an average of the probabilities of hitting the barrier in FOR ccy numeraire and DOM ccy numeraire in order to be symmetric. However, normally these two probabilities should not be far apart and this simplification should work just fine.
  • Compute the VV adjustment using the BS greeks of the KO ($X_{vega}$...), p_sym and the coefficients computed above

enter image description here

$X^{BS}$ is just the BS price, the term to he right is the VV adjustment. I suspect the reason BBG defaults to this is that it is only a handful of computationally "cheap" steps. SLV is a lot more complex (which is why you need to tell OVML manually to calc the price by pushing the button). If you now have thousands of clients pricing products, that quickly makes a big difference.

Last but not least, why do you look at EAKO? In my experience, this is not really something traded much. TARFs (and all sorts of variations like Pivot, Chooser, Conditional etc) are probably the most important options to know in FX (aside from vanilla, and standard barriers and digitals). Followed by accumulators.

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  • $\begingroup$ First, thank you very much for your detailed answer!! I do see that the EAKO is very near EKI at the payoff section in Bloomberg. In my previous research, when I am looking into EAKO in Google, I actually cannot find any related documents. So I am not sure how to decompose it/ or to find any analytical formula for this pricing. Thank you again for such a detailed explanation!! Responding to your question at the end, I look into EAKO as I hear my friends saying that it is quite popular among Asian investors. And yes, I have touched TARF too, which can be priced with simulations. $\endgroup$
    – Fangy
    Commented Jan 30, 2022 at 6:56
  • $\begingroup$ I think my EAKO comment is a bit biased. Interesting to hear it is popular in Asia. $\endgroup$
    – AKdemy
    Commented Jan 30, 2022 at 7:53
  • $\begingroup$ However, as i furthur review your answer, I think there is some misunderstanding here: If you now price a Down and out American barrier and compare by how much the price declines in % from the vanilla case, you can compute the adjustment needed in the EKI to get to the EAKO (same percentage drop roughly). This is because the american option is actually an up-and-out, so I wonder how can the adjustment be done in this scenario? $\endgroup$
    – Fangy
    Commented Jan 30, 2022 at 16:11
  • $\begingroup$ I looked at the default on my BBG which is European up and in, American down and out. In the opposite case, you simply price an American up and out (instead of down and out). $\endgroup$
    – AKdemy
    Commented Feb 1, 2022 at 20:14

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