QuoteOriginally posted by: JanDashHi ali8,I haven’t written anything down, and I may be wrong or have overlooked something, but I think that the problem of a piecewise constant rebate paid at touch for a single barrier option (with the usual caveats of lognormal dynamics, no skew etc.) has a closed-form analytic solution. The solution is a sum of bivariate integrals {Bk}. I don't disgree with this, but perhaps piecewise constant is not necessary.If H(t) is the hitting density (the defective prob. density to first hit the barrier at t),and R(t) is the rebate, then isn't the option value:H(t) is a fairly standard computation for GBM (and is also known forgeneral Levy processes and some others).regards,

Hi Alan –Yes I think I agree. I would suppose the answer for continuous R(t) is the limit of zero time interval DTk -> 0 of what I said before. The piecewise constant rebate over the {DTk} intervals becomes a continuous curve in the limit; you need DRk = R(k) – R(k-1) -> 0 such that DRk/|DTk| -> R’[t(k-1)] where |Dtk| = t(k) – t(k-1). Instead of a sum over bivariate integrals I believe you’ll get a time integral of bivariate integrals in the form you suggested. I still haven’t written anything down, but I think it’s OK.-------

Find the analytical solution in:http://www.wilmott.com/pdfs/020310_skachkov.pdf