Fixed Income

A cost of capital approach to determining the LGD discount rate ArXiv ID: 2503.23992 “View on arXiv” Authors: Unknown Abstract Loss Given Default (LGD) is a key risk parameter in determining a bank’s regulatory capital. During LGD-estimation, realised recovery cash flows are to be discounted at an appropriate rate. Regulatory guidance mandates that this rate should allow for the time value of money, as well as include a risk premium that reflects the “undiversifiable risk” within these recoveries. Having extensively reviewed earlier methods of determining this rate, we propose a new approach that is inspired by the cost of capital approach from the Solvency II regulatory regime. Our method involves estimating a market-consistent price for a portfolio of defaulted loans, from which an associated discount rate may be inferred. We apply this method to mortgage and personal loans data from a large South African bank. The results reveal the main drivers of the discount rate to be the mean and variance of these recoveries, as well as the bank’s cost of capital in excess of the risk-free rate. Our method therefore produces a discount rate that reflects both the undiversifiable risk of recovery recoveries and the time value of money, thereby satisfying regulatory requirements. This work can subsequently enhance the LGD-component within the modelling of both regulatory and economic capital. ...

Defaultable bond liquidity spread estimation: an option-based approach ArXiv ID: 2501.11427 “View on arXiv” Authors: Unknown Abstract This paper extends an option-theoretic approach to estimate liquidity spreads for corporate bonds. Inspired by Longstaff’s equity market framework and subsequent work by Koziol and Sauerbier on risk-free zero-coupon bonds, the model views liquidity as a look-back option. The model accounts for the interplay of risk-free rate volatility and credit risk. A numerical analysis highlights the impact of these factors on the liquidity spread, particularly for bonds with different maturities and credit ratings. The methodology is applied to estimate the liquidity spread for unquoted bonds, with a specific case study on the Republic of Italy’s debt, leveraging market data to calibrate model parameters and classify liquid versus illiquid emissions. This approach provides a robust tool for pricing illiquid bonds, emphasizing the importance of marketability in debt security valuation. ...

Direct Inversion for the Squared Bessel Process and Applications ArXiv ID: 2412.16655 “View on arXiv” Authors: Unknown Abstract In this paper we derive a new direct inversion method to simulate squared Bessel processes. Since the transition probability of these processes can be represented by a non-central chi-square distribution, we construct an efficient and accurate algorithm to simulate non-central chi-square variables. In this method, the dimension of the squared Bessel process, equivalently the degrees of freedom of the chi-square distribution, is treated as a variable. We therefore use a two-dimensional Chebyshev expansion to approximate the inverse function of the central chi-square distribution with one variable being the degrees of freedom. The method is accurate and efficient for any value of degrees of freedom including the computationally challenging case of small values. One advantage of the method is that noncentral chi-square samples can be generated for a whole range of values of degrees of freedom using the same Chebyshev coefficients. The squared Bessel process is a building block for the well-known Cox-Ingersoll-Ross (CIR) processes, which can be generated from squared Bessel processes through time change and linear transformation. Our direct inversion method thus allows the efficient and accurate simulation of these processes, which are used as models in a wide variety of applications. ...

Decoding OTC Government Bond Market Liquidity: An ABM Model for Market Dynamics ArXiv ID: 2501.16331 “View on arXiv” Authors: Unknown Abstract The over-the-counter (OTC) government bond markets are characterised by their bilateral trading structures, which pose unique challenges to understanding and ensuring market stability and liquidity. In this paper, we develop a bespoke ABM that simulates market-maker interactions within a stylised government bond market. The model focuses on the dynamics of liquidity and stability in the secondary trading of government bonds, particularly in concentrated markets like those found in Australia and the UK. Through this simulation, we test key hypotheses around improving market stability, focusing on the effects of agent diversity, business costs, and client base size. We demonstrate that greater agent diversity enhances market liquidity and that reducing the costs of market-making can improve overall market stability. The model offers insights into computational finance by simulating trading without price transparency, highlighting how micro-structural elements can affect macro-level market outcomes. This research contributes to the evolving field of computational finance by employing computational intelligence techniques to better understand the fundamental mechanics of government bond markets, providing actionable insights for both academics and practitioners. ...

Deep Hedging Bermudan Swaptions ArXiv ID: 2411.10079 “View on arXiv” Authors: Unknown Abstract Abstract This paper proposes a novel approach to Bermudan swaption hedging by applying the deep hedging framework to address limitations of traditional arbitrage-free methods. Conventional methods assume ideal conditions, such as zero transaction costs, perfect liquidity, and continuous-time hedging, which often differ from real market environments. This discrepancy can lead to residual profit and loss (P&L), resulting in two primary issues. First, residual P&L may prevent achieving the initial model price, especially with improper parameter settings, potentially causing a negative P&L trend and significant financial impacts. Second, controlling the distribution of residual P&L to mitigate downside risk is challenging, as hedged positions may become curve gamma-short, making them vulnerable to large interest rate movements. The deep hedging approach enables flexible selection of convex risk measures and hedge strategies, allowing for improved residual P&L management. This study also addresses challenges in applying the deep hedging approach to Bermudan swaptions, such as efficient arbitrage-free market scenario generation and managing early exercise conditions. Additionally, we introduce a unique “Option Spread Hedge” strategy, which allows for robust hedging and provides intuitive interpretability. Numerical analysis results demonstrate the effectiveness of our approach. ...

Zero-Coupon Treasury Rates and Returns using the Volatility Index ArXiv ID: 2411.03699 “View on arXiv” Authors: Unknown Abstract We study a multivariate autoregressive stochastic volatility model for the first 3 principal components (level, slope, curvature) of 10 series of zero-coupon Treasury bond rates with maturities from 1 to 10 years. We fit this model using monthly data from 1990. Unlike classic models with hidden stochastic volatility, here it is observed as VIX: the volatility index for the S&P 500 stock market index. Surprisingly, this stock index volatility works for Treasury bonds, too. Next, we prove long-term stability and the Law of Large Numbers. We express total returns of zero-coupon bonds using these principal components. We prove the Law of Large Numbers for these returns. All results are done for discrete and continuous time. ...

How does liquidity shape the yield curve? ArXiv ID: 2409.12282 “View on arXiv” Authors: Unknown Abstract The phenomenology of the forward rate curve (FRC) can be accurately understood by the fluctuations of a stiff elastic string (Le Coz and Bouchaud, 2024). By relating the exogenous shocks driving such fluctuations to the surprises in the order flows, we elevate the model from purely describing price variations to a microstructural model that incorporates the joint dynamics of prices and order flows, accounting for both impact and cross-impact effects. Remarkably, this framework allows for at least the same explanatory power as existing cross-impact models, while using significantly fewer parameters. In addition, our model generates liquidity-dependent correlations between the forward rate of one tenor and the order flow of another, consistent with recent empirical findings. We show that the model also account for the non-martingale behavior of prices at short timescales. ...

On Deep Learning for computing the Dynamic Initial Margin and Margin Value Adjustment ArXiv ID: 2407.16435 “View on arXiv” Authors: Unknown Abstract The present work addresses the challenge of training neural networks for Dynamic Initial Margin (DIM) computation in counterparty credit risk, a task traditionally burdened by the high costs associated with generating training datasets through nested Monte Carlo (MC) simulations. By condensing the initial market state variables into an input vector, determined through an interest rate model and a parsimonious parameterization of the current interest rate term structure, we construct a training dataset where labels are noisy but unbiased DIM samples derived from single MC paths. A multi-output neural network structure is employed to handle DIM as a time-dependent function, facilitating training across a mesh of monitoring times. The methodology offers significant advantages: it reduces the dataset generation cost to a single MC execution and parameterizes the neural network by initial market state variables, obviating the need for repeated training. Experimental results demonstrate the approach’s convergence properties and robustness across different interest rate models (Vasicek and Hull-White) and portfolio complexities, validating its general applicability and efficiency in more realistic scenarios. ...

The Credit Markets Go Dark ArXiv ID: ssrn-4879742 “View on arXiv” Authors: Unknown Abstract Keywords: Capital Structure, Corporate Debt, Equity Ownership, Fixed Income, Fixed Income Complexity vs Empirical Score Math Complexity: 1.0/10 Empirical Rigor: 2.0/10 Quadrant: Philosophers Why: The paper is a legal and economic analysis discussing trends in corporate debt ownership and the rise of private credit, relying on narrative data and industry observations without complex mathematical modeling or backtested implementations. flowchart TD A["Research Goal: Analyze diverging trends in<br>equity vs. corporate debt ownership"] --> B["Key Methodology: Empirical & Theoretical<br>Analysis of Institutional Holdings"] B --> C["Data Input: Decades of<br>Equity & Debt Ownership Data"] C --> D["Computational Process:<br>Quantitative Comparison & Trend Analysis"] D --> E["Key Finding: Equity ownership<br>is widely dispersed (institutional rise)"] D --> F["Key Finding: Corporate debt ownership<br>concentrated in opaque 'shadow banking'"] E --> G["Outcome: Credit markets 'go dark'<br>with transparency and liquidity"] F --> G

Dynamically Consistent Analysis of Realized Covariations in Term Structure Models ArXiv ID: 2406.19412 “View on arXiv” Authors: Unknown Abstract In this article we show how to analyze the covariation of bond prices nonparametrically and robustly, staying consistent with a general no-arbitrage setting. This is, in particular, motivated by the problem of identifying the number of statistically relevant factors in the bond market under minimal conditions. We apply this method in an empirical study which suggests that a high number of factors is needed to describe the term structure evolution and that the term structure of volatility varies over time. ...