Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Claudio Fontana, Eckhard Platen, Stefan Tappe

TL;DR

The paper addresses arbitrage-free modeling of multiple coexisting term structures under the real-world probability. It develops a unified real-world HJM semimartingale framework and provides a complete characterization of local martingale deflators (LMDs), yielding market viability without relying on a risk-neutral measure. A novel SPDE theory for SPDEs with random locally Lipschitz coefficients is proven, establishing existence, uniqueness, monotonicity (via invariance of convex cones), and affine realizations for the real-world HJMM forward-rate dynamics. The MMM example demonstrates viability without a risk-neutral measure and illustrates the practical impact for pricing and risk management in a multi-curve setting.

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.

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

TL;DR

The paper addresses arbitrage-free modeling of multiple coexisting term structures under the real-world probability. It develops a unified real-world HJM semimartingale framework and provides a complete characterization of local martingale deflators (LMDs), yielding market viability without relying on a risk-neutral measure. A novel SPDE theory for SPDEs with random locally Lipschitz coefficients is proven, establishing existence, uniqueness, monotonicity (via invariance of convex cones), and affine realizations for the real-world HJMM forward-rate dynamics. The MMM example demonstrates viability without a risk-neutral measure and illustrates the practical impact for pricing and risk management in a multi-curve setting.

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.

Paper Structure

This paper contains 25 sections, 36 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. Foreign exchange markets
  4. Interbank interest rates
  5. Credit-risky bonds
  6. Longevity bonds
  7. Energy forward contracts
  8. Market viability with multiple term structures
  9. Abstract market setup
  10. Market viability
  11. A real-world HJM semimartingale framework
  12. Probabilistic setup
  13. Characterization of LMDs
  14. Monotonicity of term structures
  15. The risk-neutral setup
  16. ...and 10 more sections

Key Result

Theorem 2.5

For the financial market of Definition def:market, NUPBR holds if and only if there exists an element $\widehat{X}\in\mathcal{X}_1$ with $\widehat{X}>-1$ such that $1/(1+\widehat{X})$ is a supermartingale deflator. Moreover, every local martingale deflator is a supermartingale deflator.

Theorems & Definitions (86)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • ...and 76 more