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Short-rate models with stochastic discontinuities: a PDE approach

Alessandro Calvia, Marzia De Donno, Chiara Guardasoni, Simona Sanfelici

TL;DR

The paper develops a PDE-based framework for pricing interest rate derivatives under short-rate dynamics that feature stochastic discontinuities at fixed times and a discontinuous numéraire at roll-over dates. By exploiting a Markovian representation and no-arbitrage, it derives a system of backward PDEs with jump boundary conditions and proves existence, uniqueness, and a Feynman-Kač representation for the solution, including exponential-growth controls. In the affine setting, it obtains closed-form ZCB prices and, for extended Vasicek with Gaussian jumps, closed-form call prices on ZCB, while for general cases it proposes two robust numerical schemes: a semi-analytic Green’s-function method and a finite-difference approach. Numerical results validate the methods against known solutions and demonstrate the framework’s flexibility to handle jumps in both the short-rate and the numéraire, with practical implications for pricing and risk management of RFR-based derivatives. The work thus provides a tractable, adaptable tool for post-LIBOR term structure modeling and derivative pricing in environments with regulatory-driven discontinuities.

Abstract

With the reform of interest rate benchmarks, interbank offered rates (IBORs) like LIBOR have been replaced by risk-free rates (RFRs), such as the Secured Overnight Financing Rate (SOFR) in the U.S. and the Euro Short-Term Rate (\euro STR) in Europe. These rates exhibit characteristics like jumps and spikes that correspond to specific market events, driven by regulatory and liquidity constraints. To capture these characteristics, this paper considers a general short-rate model that incorporates discontinuities at fixed times with random sizes. Within this framework, we introduce a PDE-based approach for pricing interest rate derivatives and establish, under suitable assumptions, a Feynman-Kač representation for the solution. For affine models, we derive (quasi) closed-form solutions, while for the general case, we develop numerical methods to solve the resulting PDEs.

Short-rate models with stochastic discontinuities: a PDE approach

TL;DR

The paper develops a PDE-based framework for pricing interest rate derivatives under short-rate dynamics that feature stochastic discontinuities at fixed times and a discontinuous numéraire at roll-over dates. By exploiting a Markovian representation and no-arbitrage, it derives a system of backward PDEs with jump boundary conditions and proves existence, uniqueness, and a Feynman-Kač representation for the solution, including exponential-growth controls. In the affine setting, it obtains closed-form ZCB prices and, for extended Vasicek with Gaussian jumps, closed-form call prices on ZCB, while for general cases it proposes two robust numerical schemes: a semi-analytic Green’s-function method and a finite-difference approach. Numerical results validate the methods against known solutions and demonstrate the framework’s flexibility to handle jumps in both the short-rate and the numéraire, with practical implications for pricing and risk management of RFR-based derivatives. The work thus provides a tractable, adaptable tool for post-LIBOR term structure modeling and derivative pricing in environments with regulatory-driven discontinuities.

Abstract

With the reform of interest rate benchmarks, interbank offered rates (IBORs) like LIBOR have been replaced by risk-free rates (RFRs), such as the Secured Overnight Financing Rate (SOFR) in the U.S. and the Euro Short-Term Rate (\euro STR) in Europe. These rates exhibit characteristics like jumps and spikes that correspond to specific market events, driven by regulatory and liquidity constraints. To capture these characteristics, this paper considers a general short-rate model that incorporates discontinuities at fixed times with random sizes. Within this framework, we introduce a PDE-based approach for pricing interest rate derivatives and establish, under suitable assumptions, a Feynman-Kač representation for the solution. For affine models, we derive (quasi) closed-form solutions, while for the general case, we develop numerical methods to solve the resulting PDEs.

Paper Structure

This paper contains 20 sections, 7 theorems, 131 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The model
  3. Pricing of derivatives: a PDE formulation
  4. Existence, uniqueness and regularity results
  5. Affine models
  6. Price of a zero-coupon bond
  7. Price of a call option in the extended Vasicek model with gaussian jumps
  8. Numerical results
  9. Distribution of the risk-free rate jumps
  10. Semi-analytic method
  11. Finite difference method
  12. Zero-coupon bond
  13. Call option on $P_T(t, x)$
  14. Conclusions
  15. Proofs of Section \ref{['affinesection']}
  16. ...and 5 more sections

Key Result

Theorem 3.3

Suppose that $f \in {\mathcal{C}} ([0,T]\setminus {\mathcal{R}} \times \mathbb{R}) \cap {\mathcal{C}} ^{1,2}((0,T) \setminus {\mathcal{R}} \times \mathbb{R})$ and that, for any $x \in \mathbb{R}$, the map $t \mapsto f(t,x)$ is right-continuous. Assume, moreover, that $f$ is such that the process $ is a local martingale. Then, under Assumption hp:PDE, $f$ satisfies the backward Partial Differenti

Figures (10)

  • Figure 1: Hull-White spot interest rates with stochastic discontinuities at $s_j=0.3, 0.5, 0.6, 0.8, 0.9$.
  • Figure 2: ZCB price at $t=0$ and relative error of the Semi-analitic method in the Vasicek model, when one jump on the interest rate occurs at time $t=0.5$, with Gaussian distribution ${\mathcal{N}} (0.09,0.25)$. Parameter values: $\alpha = 0.075$, $\beta = -0.3$, $\sigma =0.1$, $T=1$.
  • Figure 3: ZCB price. Parameter values: $\alpha = 0.075$, $\beta = -0.3$, $\sigma =0.1$, $T =1$.
  • Figure 4: ZCB Gaussian CASE 2. Parameter values: $\alpha = 0.075$, $\beta = -0.3$, $\sigma =0.1$, $T =1$, $\mu=a/b$.
  • Figure 5: ZCB Gaussian CASE 3. Parameter values: $\alpha = 0.075$, $\beta = -0.3$, $\sigma =0.1$, $T =1$.
  • ...and 5 more figures

Theorems & Definitions (20)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • ...and 10 more