Merton model and Poisson process with Log Normal intensity function

TL;DR

This paper studies a Merton default model with temporal correlation by showing that, in a double limit, default counts converge to a Poisson process with a log-normal intensity $\lambda(y)=\lambda_0 e^{\alpha \hat{y}}$ driven by a latent factor $y_t$. It analyzes two temporal-correlation decays—exponential and power-law—and demonstrates a power-law–driven super-normal phase transition, with the transition organizing the diffusion behavior: normal for $\gamma>1$ and anomalous for $\gamma\le 1$. The authors develop both maximum-likelihood and Bayesian estimation procedures on Moody's and S&P default histories, finding that power-law memory yields better long-horizon predictive performance, while exponential memory is competitive for shorter horizons. The work clarifies connections to Hawkes and SE-NBD processes and provides a practical framework for modeling long-memory default phenomena in default portfolios, with implications for credit risk management and risk forecasting.

Abstract

This study considers the Merton model with temporal correlation. We show the Merton model becomes Poisson process with the log-normal distributed intensity function in the limit. We discuss the relation between this model and Hawkes process. In this model we confirm the super-normal transition when the temporal correlation is power case. The phase transition is same as seen before the limit. We apply this model to the default portfolios and find that the power decay model provides better generalization performance for the long term data.

Figures (5)

• Figure 1: The relation among the variables of (a) this process and (b) SE-NBD and Hawkes process. In (a) $y_t$ is the hidden process. We can observe only $k_t$. In (a) and (b) $\lambda_t$ is the expected value of Poisson process.
• Figure 2: Plot of $V (\lambda_T)/V(\lambda(y))$ (a) exponential decay and (b) power decay. We calculate $\theta\in \{0.8, 0.9,0.99, 0.999\}$ for (a) and $\gamma\in \{0.1, 0.5,1.0, 2.0\}$ for (b). In (a), $V (\lambda_T)/V(\lambda(y))$ converges as the the normal distribution. In (b), it converges slower than the normal distribution when $\gamma\leq 1$ and converges as the normal distribution distribution when $\gamma>1$.
• Figure 3: Plots of $\log_2 (Z(\lambda_T)/Z(\lambda_{2T}))$ vs. $\gamma$. $\alpha =1.0(\rm{solid}), 0.5(\rm{broken})$. We can confirm the super-normal phase transition st $\gamma=1$.
• Figure 4: Annual default counts per 3000 obligors for Moody’s-rated firms from 1920 to 2023.
• Figure 5: Sample autocorrelation functions (ACF) of the inferred latent macroeconomic factor $y_t$ for nine rating segments, with fitted exponential and power-law decay models. Each panel corresponds to a different data subset (Moody’s or S&P, across All, Speculative Grade, and Investment Grade). Solid black points represent empirical ACF values. Dashed gray lines with circles show exponential decay fits, and dotted gray lines with boxes show power-law decay fits. The shape of the ACF helps determine whether short- or long-range dependence is present in each case.