CMS Options & Swaption-Based Hedging

Constant Maturity Swap products sit at the intersection of rates modelling and vol trading: the CMS rate is not a martingale under its natural payment measure, so a convexity adjustment is needed to price any CMS-linked coupon correctly. Getting that adjustment right requires both a model for the swaption smile and a static replication argument that connects CMS convexity to the full strip of swaption prices.

This course covers the two dominant industry approaches — Hagan's static replication formula and the linear Terminal Swap Rate approximation — and pairs them with the SABR stochastic volatility model to supply the swaption smile inputs. You calibrate SABR to a real swaption vol cube, compute CMS convexity adjustments, price a CMS spread option via Kirk's approximation, and build the swaption strip hedge that replicates the CMS payoff.

By the end you will be able to explain exactly why a CMS cap is more expensive than a co-terminal vanilla cap, compute the hedge notionals from first principles, and analyse the vega risk and hedge cost across tenors. These skills are directly applicable to any role involving structured rates products, exotic payoffs or vol-surface arbitrage.

The project starts with SABR calibration. You are given a swaption vol cube — implied vols across expiries, tenors and strikes — and you calibrate the SABR parameters (α, β, ρ, ν) for each expiry-tenor pair by minimising the sum of squared errors against the Hagan et al. approximate vol formula. The calibrated surface becomes the input for all downstream calculations.

With the vol surface calibrated, you compute CMS convexity adjustments for a range of CMS tenors using both the Hagan static replication formula (numerical integration of g''(K) × swaption prices over the strike grid) and the linear TSR approximation (closed-form). You compare the two approaches and quantify the difference in adjustments across the curve.

You then price a CMS spread option on the 10Y−2Y swap rate differential using Kirk's approximation, construct the swaption strip hedge for the CMS cap by computing the replicating notionals N(K) = g''(K), cost the hedge portfolio, and compute delta and vega Greeks for the full CMS position — giving you a complete view of the product's risk profile.