Parameter estimation of default portfolios using the Merton model and Phase transition

Masato Hisakado; Shintaro Mori

doi:10.1016/j.physa.2020.125435

Home/Archive/arXiv

arXiv · 2005.07967

Parameter estimation of default portfolios using the Merton model and Phase transition

Masato Hisakado·Shintaro Mori

View original source ↗Read source document ↗Export metadata ↓

Abstract

We discuss the parameter estimation of the probability of default (PD), the correlation between the obligors, and a phase transition. In our previous work, we studied the problem using the beta-binomial distribution. A non-equilibrium phase transition with an order parameter occurs when the temporal correlation decays by power law. In this article, we adopt the Merton model, which uses an asset correlation as the default correlation, and find that a phase transition occurs when the temporal correlation decays by power law. When the power index is less than one, the PD estimator converges slowly. Thus, it is difficult to estimate PD with limited historical data. Conversely, when the power index is greater than one, the convergence speed is inversely proportional to the number of samples. We investigate the empirical default data history of several rating agencies. The estimated power index is in the slow convergence range when we use long history data. This suggests that PD could have a long memory and that it is difficult to estimate parameters due to slow convergence.

Explore related subjects

q-fin.RM cond-mat.stat-mech

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Masato Hisakado, Shintaro Mori. 2020-05-16. Parameter estimation of default portfolios using the Merton model and Phase transition. https://doi.org/10.1016/j.physa.2020.125435

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Dynamic reinsurance via martingale transport

We formulate a dynamic reinsurance problem in which the insurer seeks to satisfy prescribed terminal moment or risk-based constraints while minimizing the $L^2$-norm of the ceded risk. As a tool for this analysis, we first use techniques from martingale optimal transport to study the auxiliary problem in which the insurer matches a given terminal distribution of the surplus process. We show that, under suitable assumptions, this auxiliary problem admits a tractable solution analogous to the Bass martingale. We then relax this condition by only requiring certain moment or risk-based constraints.

q-fin.RM↗

arXiv2026

Risk Measures under Paired-Ambiguity: A Deep Learning Reflected BSDE Framework

We study optimal stopping under dynamic risk measures with simultaneous ambiguity in the probability model and the discount rate. We introduce a paired ambiguity framework combining Girsanov model uncertainty with cash subadditive risk evaluation and characterize the stopping value by an upper reflected backward stochastic differential equation (BSDE). We establish structural properties of the resulting stopping operator and study quadratic drivers associated with entropic risk measures, obtaining explicit stopping rules in several benchmark cases. We then develop a deep learning scheme for the reflected quadratic BSDE. The convergence analysis uses discrete reflection and truncation to reduce the quadratic problem to a globally Lipschitz system and combines reflected BSDE discretization estimates with neural network approximation errors. Numerical experiments for American options illustrate the effects of discount rate and entropic ambiguity on stopping values and exercise decisions.

q-fin.RM↗

arXiv2026

When Is the Gini Loading More Prudent? Tail Structure and the Ordering of the Standard Deviation and the Gini Mean Difference

The standard deviation (SD) and the Gini mean difference (GMD) are the two canonical measures of variability used to load premiums, set risk margins and allocate capital, yet no universal ordering between them exists. We show that the comparison is \emph{equivalent} to asking whether the coefficient of variation of the spacing $|X-X'|$ generated by two independent copies of the risk exceeds unity, so that the exponential law -- whose spacing is again exponential -- is the universal knife-edge separating the two regimes. Reading the GMD as twice the maxiance, that is, as a second-order \emph{dual} moment in the sense of Yaari's dual theory, the problem becomes an explicit comparison of primal and dual second-order variability. We derive a closed-form representation of the mean excess function of the spacing in terms of the hazard and reverse hazard rates of $X$, and use it to prove that heavy-tailed behavior -- a decreasing hazard rate or an increasing reverse hazard rate -- yields SD dominance, whereas two-sided light tails yield GMD dominance; within the monotone aging classes, equality characterizes the exponential law. Both regimes are stable under truncation, convolution and mixing, which makes them operational in collective risk and frailty models. We classify the severity, lifetime and frequency distributions of actuarial practice accordingly, quantify the consequences for SD- and Gini-loaded premium principles and for Gini-type tail risk measures, and show that the sign of $\mathrm{SD}-\mathrm{GMD}$ across thresholds furnishes a simple diagnostic for tail aging.

q-fin.RM↗