Valuation models

Swaptions

European payer and receiver swaptions with Black-76, shifted Black or Bachelier, physical or cash settled, and Bermudan swaptions on a Hull-White model calibrated to the co-terminal Europeans with QuantLib.

Inputs

Results

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Methodology

Conventions

  • European: Black-76 or Bachelier on the forward swap rate, times the annuity.
  • Bermudan: Hull-White fitted to the curve, its vol calibrated to the co-terminal Europeans, on QuantLib finite differences.

References

  • Hull, Options, Futures, and Other Derivatives, ch. 29
  • Brigo and Mercurio, Interest Rate Models

Formulas

Forward swap rate
F = (P(T0) − P(Tn)) / A
Payer
V = N A [F N(d1) − K N(d2)]
Hull-White
dr = (θ(t) − a r) dt + σ dW