Cross-Currency Heath-Jarrow-Morton Framework in the Multiple-Curve Setting

ArXiv ID: 2312.13057 “View on arXiv”

Authors: Unknown

Abstract

We provide a general HJM framework for forward contracts written on abstract market indices with arbitrary fixing and payment adjustments, and featuring collateralization in any currency denominations. In view of this, we first provide a thorough study of cross-currency markets in the presence of collateral and incompleteness. Then we give a general treatment of collateral dislocations by describing the instantaneous cross-currency basis spreads by means of HJM models, for which we derive appropriate drift conditions. The framework obtained allows us to simultaneously cover forward-looking risky IBOR rates, such as EURIBOR, and backward-looking rates based on overnight rates, such as SOFR. Due to the discrepancies in market conventions of different currency areas created by the benchmark transition, this is pivotal for describing portfolios of interest-rate products that are denominated in multiple currencies. As an example of contract simultaneously depending on all the risk factors that we describe within our framework, we treat cross-currency swaps using our proposed abstract indices.

Keywords: Heath-Jarrow-Morton (HJM) framework, Cross-currency swaps, Collateralization, SOFR/EURIBOR transition, Interest rate derivatives, Interest Rates / FX

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, featuring extensive stochastic calculus, HJM frameworks, drift conditions, and semimartingale dynamics, indicating very high mathematical complexity. It focuses on model formulation and theoretical extensions (e.g., to semimartingales and abstract indices) without presenting any backtesting results, empirical data analysis, or implementation details, resulting in low empirical rigor.

  flowchart TD
    Goal["Research Goal<br>Create a generalized Cross-Currency HJM framework<br>handling collateral & multiple benchmarks"]
    
    Method["Key Methodology<br>1. Study cross-currency markets with collateral/incompleteness<br>2. Model instantaneous basis spreads via HJM<br>3. Derive drift conditions"]
    
    Inputs["Data & Inputs<br>- Abstract market indices (IBOR/Overnight)<br>- Arbitrage-free stochastic processes<br>- Collateral/cash flow adjustments<br>- Currency basis spreads"]
    
    Process["Computational Process<br>Solve HJM drift constraints under collateralization<br>Simulate cross-currency swaps<br>Simultaneously model SOFR & EURIBOR transition"]
    
    Outcomes["Key Findings/Outcomes<br>1. Unified HJM framework for multi-currency collateralized markets<br>2. Explicit drift conditions for basis spreads<br>3. Simultaneous handling of risky IBOR & backward-looking rates<br>4. Application to cross-currency swaps"]
    
    Goal --> Method
    Method --> Inputs
    Inputs --> Process
    Process --> Outcomes