First Order Replication of Spread Options

Given a spread option with underlyings X,Y and a copula model, we construct explicitly payoff functions f and g such that all first order greeks of the spread option with respect to the marginal distributions are matched by those of the replication portfolio {f(X),g(Y)}.Standard replication techniques allow to replicate f(X) and g(Y) by vanilla calls and puts on X and Y. In combination, we construct explicitly a replication of spread options by vanilla calls and puts such that all marginal first order Greeks are matched. In the case of CMS spread options, this yields a replication by the corresponding swaptions. We test our method in this setting using the SABR model for the marginal distributions of the swaptions which is a standard approach in the market. We show that all first order greeks of the CMS spread option with respect to the SABR parameters are matched by those of the replicating swaption portfolio