# Risikomodelle

### Liquidity Stress Testing in Limit Order Books

*Master Thesis in "Mathematical Finance", Oxford (2018)*

Limit order books (LOBs) represent the predominant mechanism for asset trading in organized markets. The intricate relationship between order flow and price formation has been attracting attention from scienti˝c communities in economics, mathematics and econophysics throughout the last decades. In this thesis, LOB behaviour is investigated with a focus on stress situations in the market. We conduct a statistical analysis revealing structural differences in order flow between the stressed and unstressed state. An important part of the analysis uses a model-based approach. Here, a marked point process model for order flow is fitted to real-world LOB data. In this way, changes in model parameters are related to structural changes in the order flow. In literature on so-called zero-intelligence models, many components of the marked point process we investigate have been studied for modelling of LOBs before. As an original contribution of this thesis, we introduce new instruments for model fitting and apply them with a specific focus on the detection of structural differences between order flows in stressed and unstressed markets. As a backbone of the fitting method, we derive a formula for the mark co-variance function of a certain class of marked Poisson processes. To the best of our knowledge, this formula cannot be found elsewhere in literature. It represents the core mathematical result of this thesis and in particular applies to the order flow models we are interested in. This allows us to utilize our LOB model for the statistical analysis of real-world LOB data in stressed and unstressed condition.

In econophysics, point process modelling of LOBs aims at explaining complex real-world market phenomena as emergent properties of rather elementary interacting model constituents. In this spirit, we investigate in a simulation study whether our order flow model generates the macroscopic price behaviour observed in real LOBs when parameters are set to stressed and unstressed configurations.

### Why is the slope a good predictor of excess returns ?

*Master Thesis in "Mathematical Finance", Oxford (2017)*

The slope of the yield curve is an important factor in predicting excess bond returns. Systematically borrowing at the short rate and investing long should not make a profit in the long run, according to the Strong Expectations Hypothesis. In reality, however, the unconditional long duration strategy in US Treasuries has yielded profits over the last few decades.

The classical explanation is that investors demand higher yields in compensation for the risk that their estimates of the future short rate are incorrect. However, we present statistical evidence that suggests that the excess returns are unlikely to be fully ascribable to `simple' risk premia, as the observed sign of the excess returns does not correlate with the sign of the CAPM risk premium over the last four decades. The yield curve has sloped upwards for the majority of recent history and this has been taken as prima facie evidence of risk premia. By running representative simulations of the yield curve we find, however, that in one-in- five cases we obtain an even steeper slope than the observed average 50-year slope just by chance alone. This demonstrates that it is possible for the average yield curve to be as steep as it has been without invoking any risk premia.

If excess returns are just a `quirk of fate' however, this does not explain why the slope is a good predictor. If excess returns are not directly and completely due to risk premia, and since it is dicult to believe there are large institutional frictions in the US Treasury market, we turn to behavioural finance explanations instead. Following in the footsteps of well-established research, in particular Shiller (1980), we propose a simple investor overreaction explanation. Our simulations of simple, but representative, models with risk-neutral investors overreacting in different ways to the central bank's interest rate policy all produce similar regression results to those observed in the real data. We demonstrate that small amounts of investor overreaction are sufficient to produce excess returns and cause the slope to become predictive.

We also contribute to the ongoing debate over the return predicting factor proposed by Cochrane and Piazzesi (2005). From our simple model we recover many of the different patterns for the return predicting factor found in the literature and show that minor changes in model parameters radically change the shape of the factor. Though excess returns could be explained by chance and risk premia, and these are contributing factors, we propose that overreaction can convincingly explain why the slope is a good predictor of excess returns.

### Dependence Modeling for Operational Risks

*Master Thesis in "Risk Management and Regulation", Frankfurt (2016)*

### Modelling of Credit Spread Risk - in consideration of Migration Risk

*Master Thesis in "Mathematical Finance", Oxford (2015)*

With the publication of the final version of the Basel Committee's Fundamental review of the trading book (FRTB) in 2015/2016 banks will have to reorganise their measurement of credit risk and implement an integrated migration and spread risk model to calculate the Integrated Credit Spread Risk Capital Charge (CSR). In the course of this work we will present and develop a consistent athematical framework to calculate the CSR by transforming and extending current common market practice models for the measurement of migration risk to additionally account for credit spread risk. Making use of the same input data as current credit portfolio models the presented approach could prove to be an attractive opportunity for banks to implement latest regulatory requirements based on input data already processed and available. In the course of this work we will review the relevant regulatory requirements. We will present a standard credit portfolio model considering migration and default risk which will serve as the basic model throughout this work and will be transformed to additionally account for credit spread risk thereafter. Here, we make use of the pearson distribution family - a distribution family which is able to capture the highly skewed and fat-tailed shapes of credit risk distributions. Furthermore, validation methods - and for the case of unsatisfactory validation results - further model extensions are introduced. We perform several simulation studies to show that the specied approach works as designed and is suitable to model credit spread and migration risk in an integrated way.

### Modelling behavioural effects in stock markets

*Master Thesis in "Mathematical Finance", Oxford (2014)*

In this work behavioural effects on stock market prices are investigated. Starting from fundamental considerations on behavioural effects, the starting point to model these effects is the model developed by Caginalp and Balenovich. It is investigated in which ways the model should be extended to capture important points which are not modelled yet. In stock markets stylized facts, especially scaling laws, are important. Thus it is also investigated, whether and in which ways the model predicts these, and in which way they are modelled.

### Liquidity-at-Risk using Extreme Value Theory - a critical assessment

*Master Thesis in "Mathematical Finance", Oxford (2012)*

The presented thesis focuses on Liquidity-at-Risk (LaR) based on the peak-over-threshold (POT) method. This stress figure is motivated by the fact, that liquidity risk is located in the extreme tails of the distribution of the autonomous account movements. Therefore standard methods for risk assessment fail to capture all aspects of the inherited risk correctly. As the POT-method explicitly focuses on the tail-behavior of a distribution it is better suited to estimate liquidity risk than other approaches, e.g. standard quantile estimator based on a normal distribution. Nevertheless the POT-method has still some major drawbacks, e.g. its limited usability as a control instrument for liquidity risk, which are examined next. It is then concluded that the strength of the LaR using POT is the possibility to calculate the amount of liquid assets, that must be available to prevent insolvency with some confidence level. Thus it helps to fulfill one fundamental requirement of all legal texts. But as soon as more information, e.g. for management decisions, are to be extracted from LaR it becomes awkward.

### An Integrated View on Market and Credit Risk

*Master Thesis in "Mathematical Finance", Oxford (2012)*

In this thesis we present an integrated approach to market and credit risk modeling that offers the possibility to consider interest rate risk, credit spread risk, migration and default risk within a consistent theoretical framework. The contribution of this work is to link the interest rate and credit spread evolution to the migration and default risk by a structural credit portfolio model. In the standard approach to credit risk modeling, the dynamic of rating transitions is not intrinsically linked to the stochastic evolution of credit spreads. Furthermore, conventional migration-mode models have only limited potentialto produce reasonable spread distributions which motivates the need for more elaborate models. In our model, credit state and credit spread evolution are driven by the same underlying process. This incorporates the joint simulation of credit states and interest rates at the risk horizon. The issuer’s spread term structure at the risk horizon is generated by an asset value threshold model. It is derived from the first passage time density for the asset value hitting a lower stochastic barrier. Therefore, an extended structural term structure model is embedded into a multi-factor framework to simulate the joint evolution of issuer credit states. We focus our analysis on changes in market values due to changes in the credit states and the associated issuer specific credit spread curves as well as the risk-free interest rate curves. Thus, the integratedportfolio model incorporates credit risk associated with migration and defaults with stochastic recovery rates, systematic and idiosyncratic credit spread risk as well as stochastic interest rate risk. We perform a simulation study to show that the specified model reproduces the stylized facts of the credit spread term structure and can model migration and default risk properly.