Valuation models
CMS swaps, caps and floors
CMS swaps, caps and floors on any swap tenor, with Hull's convexity adjustment and the payment-delay timing adjustment, vols read from a swaption cube at the CMS tenor or a flat normal vol, with PV01 and vega.
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Methodology
Conventions
- Each period pays the CMS swap rate fixed at its start, at the period end.
- CMS rate = forward swap rate + convexity adjustment (bond-yield approximation) + timing adjustment.
- A swaption cube is read at the CMS tenor; caplets use Black or Bachelier on the adjusted rate.
References
- Hull, Options, Futures, and Other Derivatives, ch. 30
- Hagan (2003), Convexity Conundrums
Formulas
- Convexity
- CA = −½ σ² t G″(y) / G′(y)
- Timing
- TA = −ρ σ_y σ_R τ R t / (1 + τ R)