# Mathematical models for the regulation of the banking system

• Pui Wah Choi

Student thesis: Phd

### Abstract

This thesis looks into mathematical models with the aim of providing insight into two key areas that are commonly discussed in the fluctuating conditions of the financial markets; how can we simulate a model to fit into real world conditions in order to provide useful results that show the potential effects of a shock in the markets? Can we provide results that are based on real bank financial data to the regulator such that the regulator is able to make the most informed decisions about changes to the banking regulations?\\ \noindent We begin by extending the framework of Barucci and Del Viva \cite{barucci2013dynamic} who make use of Contingent Convertible (CoCo) bonds and assess the optimal capital structure of a firm under a geometric Brownian motion.They determine the optimal firm values for when to convert the CoCo bonds into equity and when to go bankrupt. Our model consists of a jump-diffusion process for the firm value simulating large shocks. We employ numerical techniques to solve the complex OIDE since many issues can arise from the inclusions of jumps as we will demonstrate. Finite-difference methods combined with non-uniform grid spacing and a logarithmic transformation were required in order to provide stable and accurate results. We further extended the model by detailing asymptotic expansions of the OIDE to provide a less computationally expensive method that yields results of similar accuracy.\\ \noindent Next we examined the framework of Evatt et al. \cite{evatt2014optimal} who provide a model that takes into account of the objectives of both the bank and the regulator. We simplified the problem to assess the effects of capital requirements on the bank and we included the valuation of the debt holder's claim on the bank. The simplified numerical scheme provides the framework for us to calibrate the parameters in our model to the UK's five largest banks. We hope that by taking the largest banks in the UK, we can provide information that represents the effects on the UK banking sector as a whole that can in turn supply the regulator with key knowledge on how best to adapt the regulations.
Date of Award 1 Aug 2019 English The University of Manchester Peter Duck (Supervisor) & Paul Johnson (Supervisor)

### Keywords

• CoCo bonds
• stochastic
• finite-difference
• real world
• banking
• capital requirements
• optimal control
• jump-diffusion

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