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Statistical modeling of SOFR term structure

Teemu Pennanen, Waleed Taoum

Abstract

SOFR derivatives market remains illiquid and incomplete so it is not amenable to classical risk-neutral term structure models which are based on the assumption of perfect liquidity and completeness. This paper develops a statistical SOFR term structure model that is well-suited for risk management and derivatives pricing within the incomplete markets paradigm. The model incorporates relevant macroeconomic factors that drive central bank policy rates which, in turn, cause jumps often observed in the SOFR rates. The model is easy to calibrate to historical data, current market quotes, and the user's views concerning the future development of the relevant macroeconomic factors. The model is well suited for large-scale simulations often required in risk management, portfolio optimization and indifference pricing of interest rate derivatives.

Statistical modeling of SOFR term structure

Abstract

SOFR derivatives market remains illiquid and incomplete so it is not amenable to classical risk-neutral term structure models which are based on the assumption of perfect liquidity and completeness. This paper develops a statistical SOFR term structure model that is well-suited for risk management and derivatives pricing within the incomplete markets paradigm. The model incorporates relevant macroeconomic factors that drive central bank policy rates which, in turn, cause jumps often observed in the SOFR rates. The model is easy to calibrate to historical data, current market quotes, and the user's views concerning the future development of the relevant macroeconomic factors. The model is well suited for large-scale simulations often required in risk management, portfolio optimization and indifference pricing of interest rate derivatives.

Paper Structure

This paper contains 19 sections, 3 theorems, 86 equations, 13 figures, 13 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. The SOFR term structure
  3. SOFR averages and futures
  4. The overnight forward curve
  5. Parameterization of the forward curve
  6. Statistical modelling of the forward curve
  7. SOFR and policy rates
  8. The data and the transformations
  9. The macroeconomic model
  10. The forward curve model
  11. Incorporation of user's views
  12. A simulation study
  13. Simulation of the macroeconomy and the forward curve
  14. Simulating SOFR derivatives payouts
  15. Indifference pricing of SOFR derivatives
  16. ...and 4 more sections

Key Result

Theorem 5

A price process $s$ is arbitrage-free if and only if the discounted price process $S$ satisfies almost surely for every $t=0,\ldots,T-1$.

Figures (13)

  • Figure 1: Estimated forward curve 12 months ahead on 16 August 2023 with a cut-off date of 1 year using piecewise constant basis functions from Example \ref{['ex:cmepar']}. The tenors correspond to the FOMC meeting dates after 16 August 2023. The model-based forward rates are computed from the forward curve $F(t)$ using \ref{['eq:consistency']}. The futures rate quotes are computed using Remark \ref{['rem:imm']} and are plotted in the middle of the reference quarter ( data source: Bloomberg Finance L.P.)
  • Figure 2: Estimated forward curve five years ahead on 16 August 2023 using continuous basis functions from Example \ref{['ex:contpar']}. The model-based forward rates are computed from the forward curve $F(t)$ using \ref{['eq:consistency']}. The rates over the first 12 months differ from those in Figure \ref{['fig:termCMEstep']} because of the different tenors and basis functions. The futures rate quotes are computed using Remark \ref{['rem:imm']} and are plotted in the middle of the reference quarter ( data source: Bloomberg Finance L.P.)
  • Figure 3: Daily values of SOFR, ON RRP, and the limits of federal funds target range $[L, H]$ ( source: FREDSOFRDFEDTARUDFEDTARLRRPONTSYAWARD).
  • Figure 4: Historical daily values of the term structure risk factors $\xi$, the components of which are values of the overnight forward curve at $(0,1m,3m,6m,1y,2y,3y, 4y,5y)$; see Example \ref{['ex:contpar']}. ( data source: Bloomberg Finance L.P.)
  • Figure 5: Historical values of $x^1:=\ln(\xi^1-L)$. Here, $\xi^1$ is the spot rate $F(0)$ and $L$ is the lower limit of the FED target range. The SOFR moves with multiples of 1 basis point, which causes the discrete jumps in the time series.
  • ...and 8 more figures

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Example 3: Piecewise constant parametrization
  • Example 4: Continuous parameterization
  • Theorem 5: Jacod and Shiryaev
  • Theorem 6
  • proof
  • Theorem 7
  • proof