how to compute: Prob(X-Y-K>0) with copulas

pierrelefou · September 10th, 2007, 1:04 pm

I would like to compute the price of a Digital option on the spread of CMSLet X and Y be two CMS rateLet K be a strike We have the payoff: E(1{X-Y-K>0})can I say that: E(1{X-Y-K>0})=B(0,T)*Proba(X-Y-K>0)=B(0,T)*Proba(X-Z>0)=B(0,T)*Int(Proba(X>x,dx=0..infinity) ?? where Int the the integraleand 1{} is the indicatrice function

Olya · September 12th, 2007, 9:18 am

May be it's a problem of notation but I don't see where is dependence on Z (or equivalently Y) in your last expression. In general, it should be Proba(X-Z>0) = where fxz(x,z) - joint density function.If you model your CMS rates as lognormal, then defining x'=X(T)/X(0), y'=Y(T)/Y(0), you'll get fx'y'(x',y') bivariate normal. The X-Y-K>0 condition becomes x'>ln(Y(0)/X(0)exp(y')+K/X(0)), i.e. Proba(X-Y-K>0) = Is that what you are looking for?