Building an accurate and robust interest rate curve has considerable implications for a broad range of financial operations, from setting benchmark rates to managing risk, and hinges on the use of liquid market instruments. Instruments, such as interest rate swaps, futures, and government bonds, provide insight into market expectations for future rates to ensure accurate pricing. High trading volumes make these instruments less vulnerable to manipulation, resulting in reliable, anomaly-resistant interest rate curves that reflect actual market conditions.
Single curve bootstrap versus global optimisation
Prior to the financial crisis of 2008, interest rate curve building conventionally followed a relatively straightforward process. However, the financial crisis reshaped market dynamics. The once negligible LIBOR-OIS (Overnight Indexed Swap) spread widened and became volatile. This led to the adoption of dual curve discounting, using LIBOR for forward rate projection and OIS for discounting.
As the paradigm shifted, a simultaneous solver for both curves became necessary, intensifying the problem’s complexity. The challenge arises from a circular dependency between the curves, where instruments on one curve need information from the other for pricing. This is because various combinations of projection expectations and discount factors can yield the same arbitrage-free rate, requiring simultaneous estimation for realistic results. This entanglement rules out sequential curve construction, necessitating initial disentanglement or a global optimiser.
Despite LIBOR phasing out, diverse traded interest rates are referenced in derivatives and securities. Accurately constructing risk-free rate benchmarks like SOFR and Fed Funds in the US often demands a global optimisation approach due to complexity. In some markets, like Australian interest rates with four interconnected benchmark curves, disentanglement isn’t feasible. These markets have required global optimisation for years and will likely continue to do so.
Global optimisation can outperform bootstrapping when constructing a single curve, especially with advanced interpolation methods like spline interpolation. This method operates globally across the entire curve, requiring multi-dimensional solving, underscoring the need for global optimisation. In the past decade, significant research has focused on enhancing the stability and performance of these global optimisation algorithms.
Entangled curves
Constructing inter-dependent or “entangled” curves is a challenge in quantitative finance. Using a multi-dimensional optimiser sometimes can be slow and unstable. This has prompted researchers and practitioners to seek alternative methods to sidestep the need for a global optimiser while ensuring accurate and reliable results.
An alternative method focuses on addressing cyclical dependencies, primarily through interest rate swaps, particularly basis swaps. In certain conditions, it’s possible to construct synthetic fixed-to-floating swaps that replicate market data in basis swaps. For instance, a 10-year fixed-to-float swap for an IBOR rate and a 10-year basis swap between IBOR and OIS rates can be combined to create a synthetic 10-year OIS fixed-to-floating swap. This simplifies the entangled curve challenge into a single curve bootstrap problem.
However, this approach has limitations. The required market instruments may not always be sufficiently liquid. Liquidity varies across markets and time periods. Thus, performant global optimisations are essential in some markets. Quantifi has demonstrated that it is possible to implement stable and fast global optimisation algorithms, addressing performance issues.
Calibrating SOFR curves
Following the discontinuation of LIBOR in June 2023, attention shifted to USD’s SOFR and other global Risk-Free Rates (RFRs). Daily SOFR rates are now central in interest rate calculations, particularly for discounting purposes. Creating the SOFR curve mirrors the methodology used for the Fed Funds and LIBOR curves—bootstrapping with highly liquid market quotes, including overnight rates, futures, and swaps.
Since October 2018, CME has cleared SOFR swaps, notably SOFR OIS swaps (SOFR versus fixed rate) and basis swaps against EFFR (Effective Federal Funds Rate). The graphs below illustrate SOFR OIS swap rates and EFFR-SOFR basis. If choosing the SOFR curve for discounting, SOFR rates can be exclusively derived from SOFR quotes, while basis swaps can construct the EFFR curve if needed.
Calibrating the SOFR curve is more complex compared to EFFR and LIBOR. This is due to daily averaging, retrospective nature of payments, geometric compounding, and especially due to necessity of including historical and projected segments in calibration to futures rates, unique to SOFR.
To illustrate this last point, let’s consider the current 3m SOFR futures contract. As of August 2023, this contract is denoted as SR3M3, which started on June 22, and will settle on September 20. The market quote reflects SOFR rates compounded from June to the current date, along with projected rates from the present to the settlement date.
Accounting for the convexity adjustment of forward-future rates is another challenge specific to SOFR rates. This adjustment reflects the distinction between the future rate, calculated under the risk neutral measure, and the forward rate, calculated under the forward measure. Evaluating convexity adjustment requires rate volatilities, e.g. obtained from implied volatilities of SOFR Futures options quoted on CME.
Tackling curve complexity
To navigate the complexity of rate curves and products, financial institutions must either build their own analytics and risk system or leverage third-party solutions. While the specifics of these solutions can vary considerably, they all must incorporate essential features to effectively address these challenges. The key features include:
- Comprehensive interest rate coverage across products such as repos, bonds, futures, forwards, IR and XCCY swaps, caps/floors, swaptions, curve options, scripted IR payoffs and their numerous variations.
- Flexible curve construction for a wide range of market instruments, often with overlapping quotes that require joint fit or disentanglement, as well as modelling central bank dates etc.
- Live risk: particularly important for buy side is the ability to observe their P&L and basic rate sensitivity in live, dynamic updates.
- Ability to calculate sensitivities per curve, per tenor, and per instrument used for the calibration, alongside the capacity to derive sensitivities to zero rates and reproject to instruments beyond those employed in the calibration process.
- Flexible interfaces to trading and risk management tools, such as Excel spreadsheets or applications written in Python or Java, is very important for traders engaged in analysis and structuring activities.
Quantifi has developed a comprehensive solution to address the multifaceted challenges of interest rate curve construction which meets the institutional requirements mentioned above.
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