Contributions published at Quantitative Finance (Erich Walter Farkas)

A new expansion method to acquire the risk-neutral probability measure from observed option prices is implemented (Necula et al., 2018). The closed formula to price European options is obtained by modifying the Gram-Charlier Type A series expansion through the "physicists" Hermite polynomials, known as the Gauss-Hermite expansion. Thanks to the modification, the expansion has convergence proprieties for fat-tailed distributions, usually analyzed in financial returns. The expansion coefficients can be calculated, e.g. from probability density function or characteristic function known in closed form, or calibrated from observed option prices. This thesis aims to contribute to the empirical studies about option pricing model based on expansion methods, adapted for the Dow Jones Index as underlying. After a calibration study in both simulated and real-word option prices, it is possible to conclude that the Gauss-Hermite expansion outperforms the other methods both in-sample and out-sample.

In the last years, variable annuities, a life-insurance product, have assumed an increasing importance in the academic research. The purpose of this thesis is to analyse the systemic impact caused by the hedging of variable annuities. In order to provide an idea of the main features of this product, a review concerning variable annuities is proposed in the rst chapter. Moreover, in order to analyse the guarantees risk exposure, the same chapter presents Greeks. The second chapter deals with dynamic Delta hedging, with the aim to show the effectiveness of this strategy. In the last chapter, a review about systemic risk studies is presented, with a specic focus on systemic risk in the insurance sector. In addition, in this last chapter, a theoretical answer to the main question of the thesis is provided.

Applying to VIX options an expansion-based method for retrieving the risk neutral probability measure from option market prices. This method, so-called Gauss-Hermite series expansion, makes use of the "physisists" Hermite polynomials and, quite convenient in finance, converges for fat-tailed distributions. This approach has the additional significant advantage to enable getting the expansion coefficients by calibration to observed options prices. We mention different properties of the VIX index and its derivatives, notably the fact that the underlying for VIX options is the forward VIX. The latter can be proxied by VIX futures. Inspired by the literature regarding VIX options, we slightly adapt the closed-form formula for call option pricing suggested in Necula & al. (2018) and compare its pricing and hedging performance in-sample and out-of-sample with a cubic spline interpolation technique over the period 2013-2017.

The phenomenon of post-earnings-announcement drift in stock market refers to the continual trends of the estimated cumulative abnormal returns for stocks both with good earnings news and bad earnings news. In this study, we replicate post-earnings-announcement drift using two approaches to quantify stock earnings shocks, we use a seasonal autoregression and a simple averaging approach to estimate expected earnings. And in addition to controlling only for firm size risk when estimating abnormal returns, we use Fama-French fivefactor model with an additional momentum factor to account for more risk factors. Based on our replication, we examine whether the phenomenon remains and the magnitude of drift that is left when controlling for more risk factors. And we also examine the features of post-announcement drifts before and after financial crisis. The profitability of monthly stock portfolios based on post-announcement drifts is also studied, we construct a monthly stock portfolio based on the phenomenon and the profit and loss result is compared to other momentum strategies based on stock earnings and price.

Non-maturing deposits (NMD) such as sight and saving accounts make up a significant percentage of a bank's balance sheet. The behavioural optionalities embedded into the NMD drive the uncertainty in the future cash ow arising from client deposit interest income and expenses. This highlights the importance to model the client deposit rate and client volume. The thesis shows that in order to model the client deposit rate and client volume in a regression framework the least absolute shrinkage and selection operator (LASSO) method which is frequently used in machine learning techniques should be considered. The LASSO method leads to a parsimonious model with a high degree of explanatory power and stability with little risk of overfitting. In the context of NMD various regulatory requirements demand from banks to incorporate expert judgement in their risk management processes. Therefore the banks are encouraged to combine expert judgement and quantitative models. Instead of simply combining forecasts this thesis will in addition discuss how quantitative models can be amended to include expert opinion. We conclude that simple regression methods are not outperformed when using methods based on standard artificial intelligence methods such as the Kalman filter. Without a history of expert judgement the Kalman filter contains as much ambiguity as the regression based methods and therefore does not provide more significance. For practitioners this means that regression based techniques are a safe choice when it comes to the incorporation of expert judgement into a quantitative model.

Target Accumulation Redemption Note (TARN) is a structured product with a path dependent payoff. It pays periodic coupons based on the performance of the underlying asset and can be knocked out if the accumulated amount reaches a target. Given the complexity, TARN is typically evaluated by a Monte-Carlo method. In PDE, its features require the accumulated amount to be tracked and therefore introducing an additional dimension. The accumulated amount and the value of TARN jump along this second dimension per coupon event, which requires an interpolation. We explore possible methods to tackle these difficulties in the finite difference method and compare our implementation with a quasi-Monte-Carlo method. The results provide strong support for the finite difference method, especially in the calculation of Greeks, which is the main need for the pricing. Further extensions for better hedging including the Uncertain Local Volatility model and overhedging are introduced at the end.

Das Ziel dieser Bachelorarbeit ist es, den Zusammenhang zwischen Negativzinsen und Optionen wiedergeben zu können. Seit der Finanzkrise verfolgte die Schweizerische Nationalbank eine expansive Geldpolitik. Dies führt dazu, dass die Zinsen sanken. Jedoch wurde der Schweizer Franken mit der Zeit immer stärker, woraufhin die SNB 2011 einen Mindestkurs für den Euro einführte. Ende 2014 verkündete die SNB zudem die Einführung eines Negativzinses. Einen Monat später erfolgte der nächste Medienauftritt des Schweizerische Nationalbankpräsidenten Thomas Jordan mit einer überraschenden Aufhebung des Mindestkurses. Diese Massnahmen haben die Marktteilnehmer überrascht. Daraus hat sich die Themensetzung ergeben, welche einen Überblick über die Folgen dieser Entwicklungen im Derivatemarkt schaffen sollte. Im ersten Teil wird in die Optionstheorie und strukturierte Produkte eingeführt und ein Überblick über das Geschehen vermittelt. Im zweiten Teil erfolgt eine Einführung in die Regressionsanalyse, und die verwendeten historischen Datensätze werden vorgestellt. Im dritten und letzten Teil wird die negative Beziehung des Zinssatzes und der Optionen/strukturierten Produkten besprochen. Zudem wird der Effekt mit der Theorie aus den Wirtschaftswissenschaften erklärt.

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Model risk needs to be recognized and accounted for in addition to market risk. Uncertainty in risk measures estimates may lead to false security in financial markets. We argue that quantile type risk-measures are at least as good as expected shortfall. We demonstrate how a bank can choose among competing models for measuring market risk and account for model risk. Some BCBS capital requirements formula currently in effect leads to excessive capital buffers even on an unstressed basis. We highlight that the loss to society associated with the inefficient minimum capital requirements calculations is economically substantial over time.

Monetary risk measures classify a financial position by the minimal amount of external capital that must be added to the position to make it acceptable. We propose a new concept: intrinsic risk measures. The definition via external capital is avoided and only internal resources appear. An intrinsic risk measure is defined by the smallest percentage of the currently held financial position which has to be sold and reinvested in an eligible asset such that the resulting position becomes acceptable. We show that this approach requires less nominal investment in the eligible asset to reach acceptability. It provides a more direct path from unacceptable positions towards the acceptance set and implements desired properties such as monotonicity and quasi-convexity solely through the structure of the acceptance set. We derive a representation on cones and a dual representation on convex acceptance sets and we detail the connections of intrinsic risk measures to their monetary counterparts.

Private equity historically performed better than public market. Understanding its performance has been of core interest to both to academics and to practitioners. This thesis attempts shed light on return drivers employed by general partners by analyzing proprietary deal-level data provided by an established limited partner. This study finds that return on EBITDA multiple mimics the behavior of overall private equity returns. It shows similar sensitivities to the risk factors as the overall PE performance. Our results are similar to those conducted on portfolio level study by (Franzoni, Nowak, and Phalippou 2011). Furthermore, this study shows that private equity displays similar risk premium to the value stocks.

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This thesis covers a large scale modelling analysis dealing with the prediction of the loss given default for Banking products, with a focus on loans to multinational corporations. The work enhances the available literature on the topic by a comprehensive study of both, the analysis of a large number of different parametric, semi-parametric and non-parametric statistical modelling approaches, and the selection of predictor variables at the same time. Hence, the work merges the two main choices to be made in practice when building a loss given default model. The resulting models are designed to be point-in-time, in the sense that they describe the variation of the loss given default with changing economic conditions. This is especially relevant for downturn conditions under the Basel II regulatory framework, and for the whole economic cycle in the new IFRS 9 accounting standard becoming binding for financial institutions starting 2018. Departing from different combinations of modelling approaches and explanatory variables, almost 500 different model specifications are analyzed. The decision for a final model specification is based on the most important characteristics a loss given default needs to show to be used in practice. Looking at accuracy of the models using mean squared error and mean absolute error, rank-ordering of predicted loss given default, economic-intuition and sensitivity of the model outcomes, as well as the most important model assumptions inherent in different modelling techniques, it is found that the zero- and one-inflated beta regression model is the most promising approach to model the loss given default for loans to multinational companies. Different reasonable choices of macroeconomic variables are found. These are the AAA corporate bond spread, the credit cycle index, the unemployment rate and the annual %-differences in the real GDP. Accounting for the desired sensitivity of the model outcomes, the AAA corporate bond spread is selected as the macro-economic variable for the champion model. There still exist important caveats for the resulting champion model. A practitioner however, can use the results of the model analysis and the caveats in order to develop a suitable point-in-time loss given default model for her or his specific purpose.

In order to introduce the basic concepts and theories needed for understanding the importance of risk sharing in finance, this literature-based thesis starts with the imperfect risk perception and in particular places a focus on the cognitive and behavioural biases that characterize the modern finance, showing the fundamental differences with the Efficient Market Hypothesis. In the core part of the thesis, we analyse some financial instruments that enable the risk sharing in finance and lead to more flexibility in the financial markets. Therefore, the main part of this work comprises the application of optimal risk sharing in finance, showing the feasible utilizations in various fields of the financial industry, such as asset management, investment banking and risk management. The final part of the work will outline the possible involvement of game theory on the optimization of contracts across individuals. To conclude the thesis, we will concentrate on the interdisciplinary correlation among the various fields and discuss some general arguments.

This bachelor thesis shows a frontier between mathematics and finance. Currently, there are several different views and transformations on financial markets and risks, which make understanding, or attempts to understand, the financial environment particularly difficult and complex. This literature-based thesis moves from the Efficient Market Hypothesis (EMH) to the Fractal Market Hypothesis (FMH), using fractal geometry and its application, which represent the tool for this transition. This paper does not perform an in depth examination of fractal geometry, but rather focuses on giving an overview of the main concepts in this area of mathematics and the basics of random walks and Brownian motion. Strengths and the weaknesses of both the EMH and FMH are discussed. The main approaches using fractals are shown, addressing developments and different thoughts proposed by Mandelbrot, the “father” of fractals, as well as other researchers. Finally, current research themes are covered and the future of this field in finance discussed.

This bachelor thesis examines the VIX index, the most followed gauge of expected market volatility, and its stylised facts. The VIX measures the expected volatility in the American stock market by extracting the implied volatility of S&P 500 index options. In a first step, the emergence of volatility as an own asset class will be evaluated and an overview of today’s established products will be provided. Also known as the "fear index", the VIX is based on a sophisticated calculation methodology and has interesting quantitative facts, which will be studied. In conclusion, the stylised facts of volatility-based products will be analysed with an examination of historical data.

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In the context of a continuous-time pure-exchange economy model, the paper develops a novel methodology, based on measure-valued stochastic processes, for analyzing the evolution of heterogeneity in a tractable manner and studying its impact on asset prices. The agents in the economy differ with respect to impatience, risk aversion, beliefs about the growth rate of output, and to the rules for updating beliefs. The heterogeneity itself is described by a single object, a measure, and its dynamics by a measure-valued stochastic process. A key contribution of the paper consists in obtaining a closed form formula for the stock price in the case in which preferences are homogeneous with the risk aversion parameter given by a natural number. We also synthesize and generalize existing results about the equilibrium in heterogeneous pure-exchange complete markets economies and we highlight the importance of the endogenously determined risk tolerance weighted consumption distribution as a key ingredient in driving the equilibrium variables.

The common risk measure classifies a financial position by the minimal amount of external capital that must be added to the position to make it acceptable.We introduce a new concept: intrinsic risk measures, which provide a more direct path from unacceptable positions towards the acceptance set. External capital is avoided and only internal resources are used. An intrinsic risk measure is defined by the smallest percentage of the currently held financial position which has to be sold and reinvested in an eligible asset such that the resulting position becomes acceptable. We show that this approach requires less investment in the eligible asset to arrive at acceptable positions. It evades the problem of infinite values while desired properties such as monotonicity and quasi-convexity are preserved. We derive a representation on cones and a dual representation on convex acceptance sets and we detail the connections of intrinsic risk measures to their monetary counterparts.

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