Global Solver Methods for Multi-Curve Interest Rate Construction
Summary
The document explains why some interest rate curves must be calibrated together: instruments on one curve may depend on another, preventing a sequential bootstrap. Examples include linked short and long tenor curves and multiple benchmark curves used in markets transitioning away from LIBOR. A multidimensional Newton solver is identified as a possible approach, solving simultaneously for curve parameters or discount factors that reprice the instruments.
The discussion raises practical implementation questions rather than answering them. It asks whether larger instrument sets can make exact calibration impossible, how interpolation choices affect convergence and the stability of Greeks and Jacobians, and whether institutions should solve by currency or across many currencies at once. The examples motivate the problem, but the document supplies no benchmark results, solver configuration, or practitioner guidance on these tradeoffs.
Key ideas
- Curves with instruments that depend on one another may require simultaneous calibration.
- A multidimensional Newton method can solve for linked curve values together.
- Exact repricing may become difficult as the calibration set grows, but the document leaves this unresolved.
- Interpolation choices can affect solver behavior and the stability of risk sensitivities.
- Large institutions may need to partition calibration problems, though no specific design is recommended.
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# Implementation of solvers for curve construction # Implementation of solvers for curve construction I'd be really interested to hear people's experiences of implementing global solvers for curve construction, especially with regard to how robust the approach is in practice, numerical performance, stability of the resulting Greeks and Jacobian. The topic is well laid out here, with some good examples. I'm interested in other practitioner's views on the approach, any pitfalls of this method and things to be careful of. By way of background, as the book explains very clearly, the need for global curve construction arises when we build a series of curves that depend on each other and therefore cannot be constructed using a sequential bootstrap. A good example given in the above book is the AUD curve, whereby IR swaps are quarterly upto 3y and semiannual thereafter, with quarterly vs semi-annual basis swaps available across the entire maturity spectrum, thereby requiring 3M and 6M forward curves to be build simultaneously. Another example is in the upcoming demise of LIBOR, where for the case of USD market participants, some combination of LIBOR/SOFR, LIBOR/FF or FF/SOFR could be used to build the relevant curves. For the global curve construction, the numerical algorithm that appears to be used is a multi-dimensional Newton solver. One query I had was whether people frequently encounter the case that as the number of instruments increases, it may not be possible to find a multi-dimensional root, i.e. to solve for the discount factors that exactly reprice the market quotes. If this occurs, what are the typical work-arounds? Any difference in performance depending on which interpolation scheme is used? Also, for a bank which may have say 100 currency curves to build, what would be a sensible way of setting up the solver? I'm guessing that a global solver with 1000s of instruments to price across all currencies is not the way forward. Is it common to say calibrate individual currencies curves at a time, combining them with the cross-currency basis swaps if basis adjusted curves are required? Thanks!
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.