Two-Curve Hull-White Swaption Pricing: Discounting and Forwarding Curves
Summary
The document lays out a question about Monte Carlo pricing of a swaption under an annuity measure in a Hull-White model when discounting and forwarding use separate curves. Its payoff depends on a forward swap rate, constructed from projected floating coupons and discount-curve zero-coupon prices, as well as a fixed-leg annuity. The floating rates are defined using forwarding-curve theoretical bonds, while the model is described as diffusing a short rate.
The central issue is whether the two sets of zero-coupon prices should both be generated from that short rate or treated differently. The text provides the formulas and identifies EURIBOR-linked swaptions as the practical context, but it contains no answer or pricing implementation. It therefore frames the distinction between discount and projection curves without establishing how a one- or multifactor Hull-White model should represent basis risk, calibrate both curves, or simulate the required quantities. Those modeling choices remain unresolved in the document.
Key ideas
- A two-curve swaption setup uses discount factors for cash-flow valuation and forwarding rates for floating coupons.
- The forward swap rate is expressed using projected floating cash flows divided by the fixed-leg annuity.
- The payoff is valued under an annuity measure in the stated Monte Carlo formulation.
- The question asks how discounting and forwarding zero-coupon prices relate to Hull-White short-rate dynamics.
- The document poses this modeling problem but does not supply a solution.
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Full text
# Pricing a swaption in a Hull-White model with two curves
# Pricing a swaption in a Hull-White model with two curves
Let's say the forward swap rate $s_t$ is equal to $$s_t = \frac{\sum_{j=1}^N \delta_j^{\textrm{float}} P_{t,T_j^{\textrm{float}}}^{\textrm{disc}} L_t^{[T_{j-1}^{\textrm{float}},T_j^{\textrm{float}}]}}{A_t}$$ where the annuity $A_t$ is defined by : $$A_t = \sum_{i=1}^M \delta_j^{\textrm{fixed}} P_{t,T_i^{\textrm{fixed}}}^{\textrm{disc}}$$ and that my swaption's payoff at time $T = T_0^{\textrm{fixed}} = T_0^{\textrm{float}}$ is $A_T \left( s_T - K \right)_{+}$ and finally that I want to price it with Monte-Carlo in a Hull-White model with one or several factors, so that I want to compute
$$\pi_0 = A_0 {\mathbf{E}}^{\mathbf{Q}^A} \left[ \left(s_T - K\right)_{+} \right]$$
where $\mathbf{Q}^A$ is the annuity measure.
The Hull-White model diffuses only the short rate $r_t$ while my swaption's payoff needs two types of zero-coupons :
- the "discounting" forward zero-coupons $P_{T,T_i^{\textrm{fixed}}}^{\textrm{disc}}$ and $P_{T,T_j^{\textrm{float}}}^{\textrm{disc}}$ used to discount in the future cash-flows from a more distant future
- the "forwarding" "theoretical" zero-coupons $P_{T,T_j^{\textrm{float}}}^{\textrm{fwd}}$ defined by $L_t^{[T_{j-1}^{\textrm{float}},T_j^{\textrm{float}}]} \equiv \frac{P_{T,T_{j-1}^{\textrm{float}}}^{\textrm{fwd}} - P_{T,T_j^{\textrm{float}}}^{\textrm{fwd}}}{\delta_j^{\textrm{float}} P_{T,T_j^{\textrm{float}}}^{\textrm{fwd}}}$ used to compute the forward "LIBOR" rates (I am indeed practically interested in ICE swaptions "on" EURIBOR rates)
Are both types of zero-coupons computed through the same diffused short rate $r_t$ or not ? What is done exactly ?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.