Option Valuation in Multivariate SMM/SABR Models (with an Application to the CMS Spread)

We consider the joint dynamic of a basket of n-assets where each asset itself follows a Swap Market Model or SABR stochastic volatility model. Using the Markovian Projection methodology we approximate it by a univariate displaced diffusion SMM/SABR dynamic for the basket to price caps and floors in closed form. This enables us to consider not only the asset correlation but, in the case of the SABR model, as well the skew, the cross-skew and the decorrelation in our approximation. If for example spread options are considered the latter is not possible in alternative approximations. We illustrate the method by considering the example where the underlyings are two constant maturity swap (CMS) rates. Here we examine the influence of the swaption volatility cube on CMS spread options and compare our approximation formulae to results obtained by Monte Carlo simulation and a copula approach