Efficient Monte Carlo Valuation of Corporate Bonds in Financial Networks

TL;DR

This work proposes a novel approach -- Bi-Level Importance Sampling with Splitting -- and characterize individual bank defaults by decoupling them from the network's complex fixed-point dynamics, which enables a two-stage estimation process that directly generates samples from the banks'default events.

Abstract

Valuing corporate bonds in systemic economies is challenging due to intricate webs of inter-institutional exposures. When a bank defaults, cascading losses propagate through the network, with payments determined by a system of fixed-point equations lacking closed-form solutions. Standard Monte Carlo methods cannot capture rare yet critical default events, while existing rare-event simulation techniques fail to account for higher-order network effects and scale poorly with network size. To overcome these challenges, we propose a novel approach -- Bi-Level Importance Sampling with Splitting -- and characterize individual bank defaults by decoupling them from the network's complex fixed-point dynamics. This separation enables a two-stage estimation process that directly generates samples from the banks' default events. We demonstrate theoretically that the method is both scalable and asymptotically optimal, and validate its effectiveness through numerical studies on empirically observed networks.

Figures (9)

• Figure 1: A graphical illustration of the default event $\{{\bf s}\in\mathbb{R}^2_+:p_2({\bf s})<\bar{p}_2\}$ in a 2-bank system without fire sales. The shaded area represents the set of liquid asset values that cause bank 2 to default. The thick solid line depicts the value of $v_2(s_1)$ for different $s_1$.
• Figure 2: Performance comparison between MC, CIS, and BLISS for the toy example in Section \ref{['subsec:numeric_1']} in terms of relative error (top panels) and running time (bottom panels) when network size changes. The first and second columns correspond to the cases of the complete and ring networks, respectively.
• Figure 3: The recovered interbank network structures from the 2018 EBA stress test data. For each structure, nodes represent individual banks, and edges stand for their interbank exposures. The red-colored nodes correspond to the core banks (banks 1 to 10).
• Figure 4: Performance comparison between MC, ILIS, and BLISS for the real-world example in Section \ref{['subsec:numeric_2']} with the core-periphery network when asset value changes. The top panels compare standard errors of the three methods, and the bottom panels exhibit their efficiency ratios. The left panels correspond to the case where liquid assets are correlated across different banks, while the right panels describe the case of uncorrelated assets.
• Figure 5: Performance comparison between MC, ILIS, and BLISS for the real-world example in Section \ref{['subsec:numeric_2']} with the core-periphery network when volatility changes. The top panels compare standard errors of the three methods, and the bottom panels exhibit their efficiency ratios. The left panels correspond to the case where liquid assets are correlated across different banks, while the right panels describe the case of uncorrelated assets.

Theorems & Definitions (13)

• Proposition 1: Default Condition Reformulation
• Definition 1
• Theorem 1: Asymptotic Optimality of $\Gamma_m^{\tt A}$
• Proposition 2: Effectiveness of Optimizing ${\boldsymbol\mu}_{-n,m}^{\tt A}$
• Theorem 2: Asymptotic Optimality of $\Gamma_m^{\tt B}$
• Proposition 3: Effectiveness of Optimizing ${\boldsymbol\mu}_{-n,m}^{\tt B}$
• proof : Proof of Proposition \ref{['prop:fictitious']}
• proof : Proof of Theorem \ref{['thm:asympOpt_A']}
• proof : Proof of Proposition \ref{['prop:superiority']}
• proof : Proof of Theorem \ref{['thm:asympOpt_B']}