In this talk, we introduce a novel methodology to model rating transitions with a stochastic process. To introduce stochastic processes, whose values are valid rating matrices, we noticed the geometric properties of stochastic matrices and its link to matrix Lie groups. We give a gentle introduction to this topic and demonstrate how Itô-SDEs in R will generate the desired model for rating transitions. To calibrate the rating model to historical data, we use a Deep-Neural-Network (DNN) called TimeGAN to learn the features of a time series of historical rating matrices. Then, we use this DNN to generate synthetic rating transition matrices. Afterwards, we fit the moments of the generated rating matrices and the rating process at specific time points, which results in a good fit. After calibration, we discuss the quality of the calibrated rating transition process by examining some properties that a time series of rating matrices should satisfy, and we will see that this geometric approach works very well.
This presentation is based on a joint work with Kevin Kamm who is a PhD student at the University of Bologna.
|About the speaker||
Ms. Michelle Muniz is a PhD student at the Bergische Universität Wuppertal (BUW), where she has received both the Bachelor’s and Master’s degree in mathematics. In her Bachelor’s and Master’s thesis she worked on geometric numerical methods for matrix ordinary differential equations (ODEs) on Lie groups with application in Lattice quantum chromodynamics (QCD). Trying to solve the problem of constructing numerical methods of high convergence order but at the same time having to preserve all the geometric requirements got her intrigued, such that she decided to keep on working on geometric numerical integration methods for the PhD thesis. After studying financial economics, she then focuses her research on numerical methods for stochastic differential equations (SDEs) on Lie groups with application in financial mathematics.