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

Alessandro Gnoatto, Silvia Lavagnini

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.

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

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.
Paper Structure (36 sections, 18 theorems, 151 equations, 3 figures)

This paper contains 36 sections, 18 theorems, 151 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Violations of the covered interest rate parity
  3. IBOR-OIS spread
  4. The LIBOR discontinuation is not the end of IBORs
  5. Summary of the requirements
  6. Cross-currency HJM framework
  7. HJM framework for collateral discount curves
  8. HJM framework for cross-currency basis curves
  9. HJM framework for foreign-collateral discount curves
  10. Foreign exchange rate models and changes of measure
  11. Forwards of indices
  12. Some examples
  13. Backward-looking rate
  14. Forward-looking rate
  15. Backward-looking in-arrears forward rate
  16. ...and 21 more sections

Key Result

Lemma 2.3

For any $\beta\in \mathcal{U}^{\mathbb{P}, X}$, outside some $d\mathbb{P}\otimes dt$-nullset, it holds that Moreover, the gradient $\nabla_{\beta_t} \Psi_t^{\mathbb{P}, X}(\beta_t)$ of $\Psi_t^{\mathbb{P}, X}(\beta_t)$ in the direction of $\beta_t$ is the $\mathbb{R}^{{d}_{X}}$-valued vector given by We shall use the shorthand $\nabla\Psi_t^{\mathbb{P}, X}(\beta_t) := \nabla_{\beta_t} \Psi_t^{\m

Figures (3)

  • Figure 1: Cross-currency basis swap spread time series for the pair EUR-USD (left panel) and for the pair GBP-USD (right panel). Each curve corresponds to a different maturity.
  • Figure 2: Illustration of an abstract index. The period starts at $T^s$ and ends at $T^e$. The fixing and payment times are adjusted versions of $T^s$ and $T^e$. In this case, we set $T^s < T^f<T^e$ and $T^e=T^p$.
  • Figure 3: This graph summarizes (a part of) the relations between the different pricing measures. Each node in the graph is a pricing measure and each edge represents the link between two probability measures via a suitable Radon-Nikodym derivative. For the sake of readability, we only plot the Radon-Nikodym derivatives with respect to the spot measures. However, each node of the graph can be linked to all the other nodes. For illustrative purposes, we represent these relations only for the forward measure $\mathbb{Q}^{T,k_0-1,k_0-1}$ (blue arrows).

Theorems & Definitions (69)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4: NAFLVR
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • proof
  • ...and 59 more