CMS spread options

We analyze European options on CMS spreads, obtaining closed form formulas for the values of these instruments. There are three key steps in this analysis. The first step is to create a hybrid numeraire in which the spread $\tilde {R}_{1}\left ( T\right ) -\tilde {R}_{2}\left ( T\right ) $R~1T−R~2T is a Martingale. Like other CMS calculations, this reduces valuation to the valuation of standard European options plus quadratic convexity corrections. The second step is to combine the volatility smiles of the individual swap rates $\tilde {R}_{1} $R~1 and $\tilde {R}_{2} $R~2 to obtain the smile of the spread $\tilde {R}_{1}\left ( T\right ) -\tilde {R}_{2}\left ( T\right ) $R~1T−R~2T. We do this by modeling the volatility smiles of both swap rates by normal $\left ( \beta =0\right ) $β=0 SABR models. (In practice this may require using a shifter to convert SABR models with $\beta \neq ~0 $β≠ 0 into SABR models with $\beta =0 $β=0.) We then show that the volatility smile of the spread $\tilde {R}_{1}\left ( T\right ) -\tilde {R}_{2}\left ( T\right ) $R~1T−R~2T is also governed by a normal SABR model, and derive the SABR parameters for the spread. The third step is to use the closed-form formulae for quadratic options under the SABR model to obtain explicit formulae for the valuation of European options on CMS spreads. These formulas are not exact, but they are accurate up to $O\left ( \varepsilon ^{2}\right ) $Oε2, the same accuracy as the original SABR formulas. They also satisfy call-put parity exactly, and are exactly consistent with the valuation of CMS options on the component swap rates $\tilde {R}_{1} $R~1 and $\tilde {R}_{2} $R~2.