Interest-rate options

An interest-rate option gives its holder a payment or exercise right tied to future rates. Caps and floors protect individual coupon periods, while a swaption gives the right to enter a swap. Their valuation uses the underlying schedules and curves together with an option-pricing model.

QuantSupport supports optionlet valuation from Black or Normal volatility markets and Gaussian short-rate models for caplets, caps and swaptions. The choice determines the required volatility inputs and the interpretation of model sensitivities.

Optionlets and Black/Normal markets

A caplet fixes at \(T\), accrues over \([T,S]\), and pays \(N\tau(L-K)^+\). A floorlet replaces the payoff by \(N\tau(K-L)^+\). MakeCapFloor builds a strip of these periods. Under Black-76,

\[ C=N\tau P_d(0,S) \left[F\Phi(d_1)-K\Phi(d_2)\right],\qquad d_1=\frac{\log(F/K)+\tfrac12\sigma^2T}{\sigma\sqrt T}, \quad d_2=d_1-\sigma\sqrt T. \]

Projection determines \(F\), the policy-selected curve determines \(P_d\), and the index surface supplies volatility at the fixing/strike coordinates. Strike can be absolute, ATM or relative to the forward. The Black and Hull–White caplet/cap pricers interpret Relative(p) as \(F(1+p)\), unlike the additive \(F+p\) returned by the generic Strike::resolve helper. An absolute strike avoids this consumer-specific ambiguity. VolatilityType::Normal selects Bachelier rather than forcing a lognormal formula on absolute-rate volatility:

\[ C=N\tau P_d(0,S) \left[(F-K)\Phi(d)+\sigma_N\sqrt T\,\phi(d)\right], \quad d=\frac{F-K}{\sigma_N\sqrt T}. \]

A cap/floor sums the corresponding optionlets. Market sensitivities can therefore reach projection and discount quotes and the neighboring surface pillars. The units of Black vega and Normal vega are not interchangeable.

Gaussian bond-option representation

A one-curve caplet can be expressed as a put on the accrual-end bond:

\[ C=N(1+\tau K)\, \operatorname{PutBond}\left(T,S,\frac{1}{1+\tau K}\right). \]

Hull–White supplies an affine bond price and its option volatility. For constant sigma and nonzero mean reversion \(\alpha\),

\[ B(t,S)=\frac{1-e^{-\alpha(S-t)}}{\alpha},\qquad \sigma_P=\sigma B(T,S) \sqrt{\frac{1-e^{-2\alpha T}}{2\alpha}}. \]

The fixed-parameter Hull–White caplet and cap pricers take alpha and sigma directly. Their risk reports treat these supplied parameters as fixed, while configured calibration is responsible for connecting model parameters to market volatility quotes.

Swaptions and Jamshidian decomposition

MakeSwaption creates a EuropeanSwaption with distinct expiry and underlying swap maturity, payer/receiver direction, strike, rate convention and schedule. A payer has the right to pay fixed. The specialized closed-form pricer uses Jamshidian’s bond-option decomposition.

At expiry \(T_0\), choose the critical rate \(r^\ast\) satisfying

\[ \sum_i c_iP(T_0,T_i\mid r^\ast)=1, \qquad X_i=P(T_0,T_i\mid r^\ast). \]

Here \(c_i\) includes the fixed coupons and final principal in the decomposition. A payer becomes a weighted sum of bond puts with strikes \(X_i\); a receiver uses calls. The converged critical-rate dependency is reconnected by implicit differentiation rather than differentiation of branch choices in the scalar root solver.

ClosedFormHullWhiteSwaptionPricer::from_configuration accepts a typed EuropeanSwaption/HullWhiteClosedForm assignment and a Hull–White model for the selected index. Fixed parameters give a constant sigma. A calibrated source selects caplet-surface or swaption-cube contracts and resolves a piecewise sigma schedule. A custom provider must supply source quote details as well as constructed volatility objects and curves.

The schedule assembles independent constant-sigma fits; it is not a joint piecewise-volatility repricing calibration. Its diagonal forward-difference Jacobian includes sigma’s response to volatility inputs, but not its response to curve quotes or moving calibration strikes. Direct curve risk through valuation still exists. Cube calibration permits at most one underlying tenor per expiry and must remain in-domain. Sensitivity reports therefore include partial calibration risk with these inputs held fixed.

The model chapter distinguishes these closed forms from actual numerical path generation and its zero-parameter limits.