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KI/KO Probability of a window barrier

January 10th, 2024, 3:35 pm

Hello everyone,

Thank you all for reading this topic and helping me on this !

I have a problem on calculating the KO/KI proba of a barrier for WINDOW options... 

So I am working on a accurate EURUSD vanilla B&S pricer in order to get the KI/KO proba for different types of options (EU Barriers, Am Barriers, Window Barrier).
I precise that I said "accurate" because I compare the prices I get with EURUSD Curncy OVML on BBG. I re-created the vol smile, the implicit rate etc... all this so I can have barely the same vanilla price as I see on BBG.

The vanilla prices I have are correct and the difference with OVML is just about 0.1 bp. So now that I have correct vanilla prices I just need to replicate digit options or one touch options in order to have the EU/Am/Window barrier proba ! Here is the process for an EU barrier: 
  • Depending on the barrier level compared to the spot S I price a CallSpread or PutSpread with two strikes : x-e/2 and x+e/2 with e = 10bp for me
  • I have now my CallSpread  / PutSpread price which is P, now the proba is just given by PROBA=P/e 
My PROBAS for EU Barrier are quite accurate so it's okay, same for AM Barriers (for now my AM barriers probas are just calculated with the proxy : PROBAEU *2) and the differences with ovml probas are just around ±1%.

But when it comes about window barrier options things get more complicated, the process for a barrier  x between (mm/dd/yyyy) date1= 03/10/24 and date2=06/10/24:
  • I price 2 call spreads / put spreads with date1 maturity and date2 maturity. Each spreads with strikes : x-e/2 ; x+e/2.
  • Pdate1 is the spread price maturity date1 and Pdate2 is the spread price maturity date2. Now I just divide each price by e and multiply by 2 for Am barriers so I have : PROBA1*2 ; PROBA2*2.
  • I just have to do PROBA2*2 - PROBA1*2 in order to have the "residual proba" for the window period ? WinPROBA
But my WinPROBA is really different with what I see on OVML and I don't know why, where is the problem or maybe I have a wrong input ? If the OVML proba is different of this calculation, there is arbitrage opportunity, isn't it ? 

Thank you again for reading this and sorry for my bad english..