Paper: arXiv 2212.12318

Authors: Marco Di Francesco, Kevin Kamm

Abstract

In this paper, we improve the performance of the large basket approximation developed by Reisinger et al. to calibrate Collateralized Debt Obligations (CDO) to iTraxx market data. The iTraxx tranches and index are computed using a basket of size $K= 125$. In the context of the large basket approximation, it is assumed that this is sufficiently large to approximate it by a limit SPDE describing the portfolio loss of a basket with size $K\rightarrow \infty$. For the resulting SPDE, we show four different numerical methods and demonstrate how the Magnus expansion can be applied to efficiently solve the large basket SPDE with high accuracy. Moreover, we will calibrate a structural model to the available market data. For this, it is important to efficiently infer the so-called initial distances to default from the Credit Default Swap (CDS) quotes of the constituents of the iTraxx for the large basket approximation. We will show how Deep Learning techniques can help us to improve the performance of this step significantly. We will see in the end a good fit to the market data and develop a highly parallelizable numerical scheme using GPU and multithreading techniques.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a sophisticated mathematical framework combining SPDE limits, Magnus expansions, and deep learning for CDO calibration. While the theoretical physics/math formulation is rigorous and empirically tested against iTraxx data, reproducibility is slightly hindered by the lack of explicit code repositories.

Research Flowchart

  flowchart TD
    A[Goal: Fast & Accurate CDO Calibration via Large Basket SPDE] --> B[Input: Market Data<br/>125 iTraxx CDS & Tranche Quotes]
    B --> C[Method 1: Deep Learning<br/>Infer Initial Distances to Default from CDS]
    C --> D[Method 2: Large Basket SPDE<br/>Model Limit Portfolio Loss K → ∞]
    D --> E[Method 3: Magnus Expansion<br/>Solve SPDE Efficiently with High Accuracy]
    E --> F[Computational Execution<br/>Parallel Schemes on GPU & Multithreading]
    F --> G[Outcome: High-Accuracy Market Fit<br/>& Significantly Improved Calibration Speed]