Valuation models
CMS spread options
Caps and floors on a geared spread of two CMS rates, each convexity and timing adjusted, a normal spread vol from the two rates' vols and their correlation, and the value change for a lower correlation.
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Methodology
Conventions
- Each period pays max(gearing (CMS long − CMS short) − K, 0) for a cap.
- Each CMS rate is its forward swap rate plus convexity and timing; the spread is normal.
- The spread vol comes from the two rates' normal vols and their correlation.
References
- Brigo and Mercurio, Interest Rate Models, ch. 13
- Hagan (2003), Convexity Conundrums
Formulas
- Spread vol
- σ_S = g √(σ₁² + σ₂² − 2ρ σ₁ σ₂)
- Caplet
- c = N τ D [(S − K) N(d) + σ_S √t φ(d)]