[Paper Review] Term structure modeling for multiple curves with stochastic discontinuities

Claudio Fontana, Zorana Grbac+2 more|arXiv (Cornell University)|Oct 23, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance42 references3 citations

TL;DR

This paper develops a general HJM-type term structure model for multiple yield curves that explicitly incorporates stochastic discontinuities—jumps occurring at predetermined dates, such as ECB policy meetings. It establishes a fundamental theorem of asset pricing under NAFLVR and introduces a tractable class of affine semimartingale models that generalize existing approaches by relaxing the assumption of stochastic continuity.

ABSTRACT

We develop a general term structure framework taking stochastic discontinuities explicitly into account. Stochastic discontinuities are a key feature in interest rate markets, as for example the jumps of the term structures in correspondence to monetary policy meetings of the ECB show. We provide a general analysis of multiple curve markets under minimal assumptions in an extended HJM framework and provide a fundamental theorem of asset pricing based on NAFLVR. The approach with stochastic discontinuities permits to embed market models directly, unifying seemingly different modeling philosophies. We also develop a tractable class of models, based on affine semimartingales, going beyond the requirement of stochastic continuity.

Motivation & Objective

• To address the lack of modeling frameworks that explicitly account for stochastic discontinuities in interest rate markets, particularly jumps tied to scheduled events like ECB policy meetings.
• To unify existing multiple curve modeling approaches—especially market models and HJM-type frameworks—within a single, general term structure framework.
• To establish a rigorous foundation for no-arbitrage pricing in multiple curve markets by proving the equivalence between NAFLVR and the existence of an equivalent separating measure.
• To extend the classical HJM approach to allow for discontinuous forward rates and to develop a tractable class of models based on affine semimartingales beyond the continuity requirement.

Proposed method

• Extends the Heath-Jarrow-Morton (HJM) framework to multiple curves by modeling forward rates for both OIS and Ibor rates, allowing for stochastic discontinuities at predetermined dates.
• Derives necessary and sufficient drift conditions for risk-neutral measures under a general numéraire, based on the theory of semimartingales and local martingales.
• Introduces a stochastic discontinuity structure via a purely atomic random measure, enabling the modeling of jumps in forward rates at known dates such as central bank meetings.
• Applies the theory of large financial markets to prove a fundamental theorem of asset pricing (FTAP) under NAFLVR, extending prior results to infinite time horizons and multiple curves.
• Proposes a class of affine semimartingale models that generalize affine processes by allowing for jumps and discontinuities, while preserving tractability.
• Uses stochastic exponential representations and semimartingale decompositions to characterize the dynamics of FRA and OIS bond prices under the risk-neutral measure.

Experimental results

Research questions

• RQ1 How can a general HJM-type framework be extended to model multiple yield curves with stochastic discontinuities?
• RQ2 What are the necessary and sufficient conditions for the existence of a risk-neutral measure in a multiple curve market with discontinuous forward rates?
• RQ3 How can market models be embedded directly within a general HJM framework that allows for jumps at predetermined dates?
• RQ4 What is the relationship between the no-arbitrage condition NAFLVR and the existence of an equivalent separating measure in a multiple curve setting?
• RQ5 Can affine semimartingale models be constructed to allow for stochastic discontinuities while preserving analytical tractability?

Key findings

• The paper establishes a general HJM framework for multiple curves that incorporates stochastic discontinuities, providing necessary and sufficient conditions for the existence of a risk-neutral measure under any numéraire.
• It proves a fundamental theorem of asset pricing (FTAP) in the context of multiple curve markets, showing that NAFLVR is equivalent to the existence of an equivalent separating measure, extending results from Cuchiero et al. (2016) to infinite time horizons and multiple curves.
• The authors demonstrate that market models can be directly embedded within the proposed HJM framework by specifying the drift and jump components of forward rates to match the dynamics of observed Libor rates.
• A new class of models based on affine semimartingales is introduced, which generalizes classical affine processes by allowing for discontinuities, thereby going beyond the assumption of stochastic continuity.
• The specification of forward rates is achieved by matching the continuous local martingale parts and the jump parts (totally inaccessible jumps) of the FRA and OIS bond price processes, ensuring consistency with the no-arbitrage condition.
• The paper shows that the process $ (1+ u L( au,T, u))P( au,T+ u)/X^0 $ is a local martingale under the risk-neutral measure, which is crucial for deriving the drift conditions and ensuring no-arbitrage.