Hopefully someone knows the answer to this before I start crying...For cases of zero drift, or for barriers on the forward, not spot rate, an example Put call transformation is stated as: Call_UpIn (S,X,H,T,r,v) = X/S*Put_DnIn (S,S*S/X,S*S/H,T,r,v) For OTC FX markets, all barriers are on spot, as you probably know. The above relationship breaks down when you have significant drift over long time periods. An example would be USD/JPY:Spot = 120.0x = 120.0H = 125.0t = 360 daysrd = 3rf = 0.1If you try to use the above relationship with spot barrier models (Reiner/Rubinstein), you will end up with negative values in some cases. If you try to use Fwd barrier models, whilst adjusting the barrier for drift, the prices don't correctly converge with the actual spot barrier model at the trigger point.Can someone outline whether there is an exact general solution for transformations of barriers on spot with drift? refs; Haug 99 Barrier Put-Call transformations, Haug 2001 First then knockout options

Gao Huang Subrahmanyan 1999,2000 Valuation of American Options using Decomposition TechniqueI'll answer my own question, since I found part of the answer.in Gao Huang Subrahmanyan, they state that a CALL_do (S,K,H,r,d) = PUT_uo (K,S,KS/H,d,r). You can see that the interest rates are reversed to account for the different direction of the drift. This also applies to Call_di = PUT_ui etc ..I'm still trying to understand if this is applicable to First then barrier options, since I'm having trouble making them work for non-zero drift situations.

Are you just interested in re-using code or are you trying to develop static hedges of barrier options usingvanilla options? For the former it is no issue to switch rates but for the latter you can't.If it is for the latter thenin Bowie and Carr Risk 94 and in Carr Ellis Gupta static hedging of exotic options JF 98, we developed bounds on barirer options in terms of vanillas.In Chou and Carr, Breaking Barriers Risk 98, we developed exact replications with drift but it involves options of all strikes.You can find the last two papers on my web sitewww.math.nyu.edu/research/carrp/papers/pdfbestp

First then barriers & Non-zero driftHi, sorry for the (following) long reply, to your short question:I'm specifically trying to make the first then barrier option type work with non-zero drift. F-T-B options specifically use the result in G-H-S '99 (Gao etc: Val of Amer barrier opts), to derive a Call DnIn-then-Upin. Starting with the following symmetry: C_ui (S,X,H,T,r,v) = P_di (X,S, SX/H, r,v)He (haug,2001) derives a Call DnIn-then-Upin(S,X,L,U,T,r,v) as a special case of this relationship when S = L (I assume that's how)resulting inCall DnIn-then-Upin(S,X,L,U,T,r,v) = X/L*Put_DnIn(S, L*L/X, L*L/U, T, r,v)Problem is that this result only works for cases where barrier is a FUT or there is zero drift. As you are well aware in the FX markets, you don't get very far without handling spot barriers. I tried to introduce drift by reversing the rates (r & d) as per G-H-T:Call DnIn-then-Upin(S,X,L,U,T,r,d,v) = X/L*Put_DnIn(S, L*L/X, L*L/U, T, d, r,v)But this results in bad values (larger than the equivalent Call DnIn) - so obviously I screwed some relationship up, but I have not been able to figure out what...regardsJulian

Did the American plain vanilla put-call transformation in Espen Haug's paper help at all?http://home.online.no/~espehaug/Barrier ... ations.pdf

The put-call barrier transformations also hold for case with drift (as described in link below). However putorcall has a good point, when using put-call symmetry to static hedge plain barrier options or barrier-symmetry/transformation to static hedge more complex barriers (like first-then-barriers) things get more complicated and I have only been able to do this in case of "zero drift" (b=0), first-then-barrier options on futures/forwards. However there should be a trick around this??Using reflection principle one can come up with nice closed form with drift, the drawback of this method; it is less intuitive and gives no static/semi static hedge. I use reflection principle in my American Barrier paper and also for our paper on Knock-in-out Margrabe that you can downloade from wilmott front page. Let me see what I can do when I return from my vacation.

You might want to check out the following paper: it contains all the formulae and symmetries between the various barrier type options, single, double (exponentially curved) barriers under geometric Brownian Motion.J.K.Hoogland and C.D.D.Neumann, Local scale invariance and contingent claim pricing II: path-dependent contingent claims, IJTAF, vol.4, no.1 (2001), p.23-43It can be downloaded here: Article or ArticleJiri

LOCAL SCALE INVARIANCE AND CONTINGENT CLAIM PRICING II:PATH-DEPENDENT CONTINGENT CLAIMSThanks for the reference: I took a quick look at the paper you suggested. Section 5.1 (Single barriers) definitely shows down-out to up-out transformations etc e.gUpOutPut (S; P;K;B; r; t) = DnOutCall (P; S;1/k;1/B; -r; t)I replaced lander with r, since that was suggested at the top of the section.I have not yet checked if they solve the issue of generalising First-then-barriers .

By the way, do someone know how to generalize the common symmetry relationship K1*K2 = F^2 (Ki : strikes, F forward) in order to take the smile into account.Remember that when the vol is flat, a portfolio long put short call (risk reversal) :Ptf = Put (S, K1 = F^2/K1, vol) – F/K2 * Call (S, K2, vol) ahs a zero gamma (and also premium = 0)