Convexity adjustment for constant maturity swaps in a multi-curve framework

In this paper we propose a double curving setup with distinct forward and discount curves to price constant maturity swaps (CMS). Using separate curves for discounting and forwarding, we develop a new convexity adjustment, by departing from the restrictive assumption of a flat term structure, and expand our setting to incorporate the more realistic and even challenging case of term structure tilts. We calibrate CMS spreads to market data and numerically compare our adjustments against the Black and SABR (stochastic alpha beta rho) CMS adjustments widely used in the market. Our analysis suggests that the proposed convexity adjustment is significantly larger compared to the Black and SABR adjustments and offers a consistent and robust valuation of CMS spreads across different market conditions.