Real-world models for multiple term structures: a unifying HJM semimartingale framework

Paper: arXiv 2411.01983

Authors: Claudio Fontana, Eckhard Platen, Stefan Tappe

Abstract

We develop a unified framework for modeling multiple term structures arising in financial, insurance, and energy markets, adopting an extended Heath-Jarrow-Morton (HJM) approach under the real-world probability. We study market viability and characterize the set of local martingale deflators. We conduct an analysis of the associated stochastic partial differential equation (SPDE), addressing existence and uniqueness of solutions, invariance properties and existence of affine realizations.

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous, highly complex mathematical unified HJM framework for multiple term structures using SPDEs under real-world probability. However, it is entirely theoretical with zero empirical validation or backtesting, placing it squarely in the Lab Rats category.

Research Flowchart

  flowchart TD
    A[Research Goal: Unified HJM Framework for<br/>Multiple Term Structures under Real-World Probability] --> B[Inputs: Multi-Market Data &<br/>Semimartingale Driving Processes]
    B --> C[Market Viability Analysis &<br/>Local Martingale Deflators Characterization]
    C --> D[SPDE Modeling & Analysis]
    D --> E[Computational Processes:<br/>Solving SPDEs, Invariance Tests &<br/>Affine Realizations]
    E --> F[Outcomes: Proven Existence/Uniqueness of Solutions<br/>& Tractable Multi-Market Term Structure Models]