Modeling the Post-LIBOR Interest Rate Market - Calibration of a Time-Dependent Hull-White Model to SOFR Caps

Abstract

Following the discontinuation of LIBOR, the interest rate market has transitioned to backward-looking benchmarks such as SOFR. Unlike traditional forward-looking rates like LIBOR, SOFR is set in arrears, which affects the pricing methods of derivatives linked to these benchmarks. This thesis studies the impact of this transition on the pricing and calibration of caps within the Hull-White framework. It begins with a derivation of a closed form pricing formula for caplets under a piecewise constant volatility structured Hull-White model. It continues to explore different calibration strategies, such as global and sequential approaches, as well as methods for choosing the mean reversion parameter. Among the calibration strategies evaluated, the sequential approach was found to display the most robustness, computational efficiency, and calibration accuracy. The study also highlights the advantages of using a piecewise constant volatility structure, which improves the model's ability to fit market data.

Popular Abstract

In the aftermath of the LIBOR-scandal, which resulted in senior financial officials being jailed and banks facing substantial fines, markets are navigating the transition to the use of SOFR. As interest rate derivatives constitutes a $579 trillion market, pricing of such derivatives is of the utmost importance. This thesis studies the impact of this transition on the pricing of SOFR Caps.

Interest rate derivatives are financial instruments that derive their value from a reference interest rate and are widely used by financial institutions to hedge risk and speculate on interest rate movements. Historically, these derivatives have been tied to the London Inter-Bank Offered Rate (LIBOR), which in 2012 made out $350 trillion of the interest rate derivative market. However, as a few banks effectively set the rate in a market they themselves were participating in, the skewed incentives led several banks to engage in fraudulent manipulation of the benchmark. This prompted regulators to take action to discontinue LIBOR, with the final synthetic rate being published in 2024.

Instead, regulators opted for more transparent, transaction based benchmark rates like the Secured Overnight Financing Rate (SOFR). These rates differ from LIBOR, as they are based on real transactions and thus are set in arrears instead of in advance which LIBOR is. This has major implications for the pricing of interest rate derivatives, and therefore, the market has had to create new methodologies for accurate valuation.

This paper studies the impact on the valuation of a specific type of interest derivative, namely a cap. A cap is a financial instrument which constitute of multiple caplets. Each caplet is effectively an insurance that the buyer will pay a maximum interest rate over the caplet's period. Thus, an interest rate cap is meant to cap the maximum interest one can pay and thus hedges for unfavorable interest rate movements.

One of the most central ideas in mathematical finance is the notion of arbitrage, i.e. risk-less profit. Interest rate derivatives constitute an enormous market, and it thus becomes of essence to price these in a consistent manner to avoid arbitrage. In order to achieve this consistency, we need a underlying model able to reproduce all market prices. In this project we will try to calibrate one such model, namely the short rate model.

With the use of Overnight Index Swap rates, we will first construct a yield curve, which is necessary to relate money to different time points. Since we want to expand the grid for which we have data, we will use a combination of bootstrapping and interpolation techniques to obtain a continuous yield curve. Furthermore we develop and introduce a pricing formula for the caps, which we use to calibrate the short rate model to market SOFR Cap prices.

After testing multiple calibration methods, we find that a sequential calibration approach yields the most robust and computational efficient results. This calibration method leverages the fact that by matching a parameter structure with the tenors available in the market, we can optimize the parameters one at a time, sequentially.

• author: Stoltz, Rasmus and Sönne, Fabian
• supervisor: Magnus Wiktorsson
• organization: Mathematical Statistics
• course: FMSM01 20251
• year: 2025
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• keywords: SOFR, Hull-White model, caplet pricing, model calibration, sequential calibration, yield curve construction, LIBOR transition
• publication/series: Master's Thesis in Mathematical Sciences
• report number: LUTFMS-3525-2025
• ISSN: 1404-6342
• other publication id: 2025:E66
• language: English
• id: 9201629
• date added to LUP: 2025-06-18 08:30:43
• date last changed: 2025-06-18 08:30:43