Valuation models

Inflation caps and floors

Zero-coupon and year-on-year CPI caps and floors off a zero inflation curve, by Bachelier, shifted Black, YoY SABR smile or the Jarrow-Yildirim three-factor model with its convexity adjustment, with each caplet's forward, vol and value.

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Methodology

Conventions

  • Forwards come off the zero inflation curve; payments are discounted on the nominal curve.
  • Year-on-year caplets pay N τ (y − K)⁺ on the period's rate y = (I(tᵢ)/I(tᵢ₋₁) − 1)/τ; a ZC cap pays once at maturity.
  • Bachelier and shifted Black (or SABR for the smile) treat each rate on its own; Jarrow-Yildirim adds the convexity of a ratio paid at its fixing.

References

  • Mercurio (2005), Pricing Inflation-Indexed Derivatives
  • Jarrow and Yildirim (2003), Pricing TIPS and related derivatives, JFQA
  • Kenyon (2008), Inflation is Normal
  • Hagan et al. (2002), Managing Smile Risk

Formulas

YoY caplet
V = N τ D(tᵢ) Bachelier(y, K, σ√tᵢ)
ZC cap
V = N D(T) Black(I(T)/I(0), (1 + K)^T, σ²T)
Jarrow-Yildirim
dn = (θₙ − aₙn)dt + σₙdWₙ, dr = (θᵣ − aᵣr)dt + σᵣdWᵣ, dI/I = (n − r)dt + σ_I dW_I