Building Discrete Forward Inputs for Monotone Convex Curve Interpolation
Summary
The question concerns preparing inputs for Hagan and West’s monotone convex interpolation when constructing a six-month EURIBOR forward curve with an ESTR discount curve. Short-end FRA quotes directly provide discrete forward rates, but longer-maturity par swap quotes constrain weighted sums of several six-month forwards. Subtracting the two-year swap relationship from the three-year relationship leaves a combined constraint on the next two forward periods, rather than identifying each period separately.
The author asks how such swap information can be rearranged into discrete forwards for whole intervals, as the interpolation method requires. The document provides the setup and algebraic difficulty but no answer, derivation, or numerical example. It therefore highlights an input-construction issue without specifying a method for resolving the underdetermined individual forwards; the equations also depend on the stated accrual factors and discount factors.
Key ideas
- FRA quotes can supply discrete forward rates over their quoted accrual periods.
- Par swap quotes constrain discounted sums of forward cash flows across multiple periods.
- Subtracting adjacent maturity swap equations can isolate a combined constraint on newly added periods.
- The document poses, but does not answer, how to convert that combined constraint into interval-level interpolation inputs.
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Full text
# Apply monotone convex interpolation to swap rate input data
# Apply monotone convex interpolation to swap rate input data
I'm trying to apply Hagan & West's monotone convex interpolation to a 6m EURIBOR (forward) curve using ESTR (already bootstrapped) for discounting.
In their paper Hagan & West use discrete forward rates $f_i^d$ belonging to the interval $[\tau_{i-1},\tau_i]$ as input. They then try to identify and instantaneous forward rate function f that satisfies $\frac{1}{\tau_i-\tau_{i-1}} \int_{\tau_{i-1}}^{\tau_i} f(t) dt = f_i^d$ plus some other constraints.
The problem I'm struggling now with is how to get the discrete forwards from my input data, which are FRA rates in the short end and fix-float-swap rates form 2y onwards. In their paper the authors state that "we have (or can rearrange our inputs so that we have) discrete forwards for entire intervals". The FRA rates directly give me discrete forward rates. For the swaps it's a bit more complicated. For a m-year swap to be at par the following condition has to satisfied $$\sum_{j=1}^m S_{my} \cdot dcf(\tau_{j-1},\tau_j) \cdot DF_{ESTR}(\tau_j) = \sum_{i=1}^n f_i^d \cdot dcf(\tau_{i-1},\tau_i) \cdot DF_{ESTR}(\tau_i)$$ whith $\tau_j \in [0,1y,2y,...]$ and $\tau_i \in [0,6m,1y,...]$. Now the 3y formula can be rewritten using the 2y one as $$f_5^d \cdot dcf(\tau_4,\tau_5) \cdot DF_{ESTR}(\tau_5) + f_6^d \cdot dcf(\tau_5,\tau_6) \cdot DF_{ESTR}(\tau_6) = \underbrace{\left( \sum_{j=1}^3 S_{3y} \cdot dcf(\tau_{j-1},\tau_j) \cdot DF_{ESTR}(\tau_j) \right) - \left( \sum_{i=1}^4 S_{2y} \cdot dcf(\tau_{i-1},\tau_i) \cdot DF_{ESTR}(\tau_i) \right)}_{=:A}$$ Or in integral notation as $$ \frac{1}{\tau_5-\tau_4} \int_{\tau_4}^{\tau_5} f(t) dt \cdot dcf(\tau_4,\tau_5) \cdot DF_{ESTR}(\tau_5) + \frac{1}{\tau_6-\tau_5} \int_{\tau_5}^{\tau_6} f(t) dt \cdot dcf(\tau_5,\tau_6) \cdot DF_{ESTR}(\tau_6) = A$$ This means I only have combined information about $f_5^d$ and $f_6^d$.
Can anybody give me a hint on how to continue? The comment of the authors imply that I can rewrite this as $\frac{1}{\tau_6-\tau_4} \int_{\tau_4}^{\tau_6} f(t) dt = B$. But I don't see how ...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.