Contributions published at Quantitative Finance (Erich Walter Farkas)

In this a paper a non-linear macro-stress testing methodology with focus on early warning is developed. The methodology builds on a variant of Random Forests and its proximity measures. It develops a framework in which naturally defined contagion and feedback effects transfer the impact of stressing a relatively small part of the observations on the whole dataset and thus allow to estimate the a stressed future state. It will be shown that contagion can be directly derived from the proximities while iterating the proximity based contagion leads to naturally defined feedback effects. This procedure allows accurate forecast of events under stress and thus allows to forecast the emergence of a potential crisis. The framework also estimates a set of the most influential economic indicators leading to the potential crisis, which can then be used as indications of remediation or prevention of that crisis. Since the methodology is Random Forests based the framework is suitable for big data analysis.

This thesis considers two main subjects divided in four problems in the broad field of mathematical finance. The first chapter treats option pricing followed by three chapters on the application of the machine learning algorithm of Random Forests to finance, specifically to risk capital aggregation, portfolio optimization and macro stress testing. In all four chapters new methodologies to treat the respective subjects are developed. All proposed models are benchmarked against commonly applied methods in the respective fields and found to outperform their peers.

A new method to retrieve the risk-neutral probability measure from observed option prices is developed and a closed form pricing formula for European options is obtained by employing a modified Gram-Charlier series expansion, known as the Gauss-Hermite expansion. This expansion converges for fat-tailed distributions commonly encountered in the study of financial returns. The expansion coefficients can be calibrated from observed option prices and can also be computed, for example, in models with the probability density function or the characteristic function known in closed form. We investigate the properties of the new option pricing model by calibrating it to both real-world and simulated option prices and find that the resulting implied volatility curves provide an accurate approximation for a wide range of strike prices. Based on an extensive empirical study, we conclude that the new approximation method outperforms other methods both in-sample and out-of-sample.

In this paper we develop a one-factor non-affine stochastic volatility option pricing model where the dynamics of the underlying is endogenously determined from micro-foundations. The interaction and herding of the agents trading the underlying asset induce an amplification of the volatility of the asset over the volatility of the fundamentals. Although the model is non-affine, a closed form option pricing formula can still be derived by using a Gauss-Hermite series expansion methodology. The model is calibrated using S&P 500 index options for the period 1996-2013. When its results are compared to some benchmark models we find that the new non-affine one-factor model outperforms the affine one-factor Heston model and it is competitive, especially out-of-sample, with the affine two-factor double Heston model.

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The risk of financial positions is measured by the minimum amount of capital to raise and invest in eligible portfolios of traded assets in order to meet a prescribed acceptability constraint. We investigate nondegeneracy, finiteness and continuity properties of these risk measures with respect to multiple eligible assets. Our finiteness and continuity results highlight the interplay between the acceptance set and the class of eligible portfolios. We present a simple, alternative approach to the dual representation of convex risk measures by directly applying to the acceptance set the external characterization of closed, convex sets. We prove that risk measures are nondegenerate if and only if the pricing functional admits a positive extension which is a supporting functional for the underlying acceptance set, and provide a characterization of when such extensions exist. Finally, we discuss applications to set-valued risk measures, superhedging with shortfall risk, and optimal risk sharing.

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Regulators dedicate much attention to a financial institution’s option to default, i.e. the option that distressed financial institutions have to transfer losses to their creditors. It is generally recognized that the existence of this option provides intermediaries with a powerful incentive to keep firm capital close to the minimal requirement. We argue, however, that undercapitalization harms profitable growth opportunities, i.e. the institution’s franchise value. Indeed, the capitalization of a financial institution will be ultimately driven by the net impact of capital levels on the default option and the franchise value. By considering the impact of the default option, our work complements and extends, within a simple Black-Scholes framework, the model used by Froot and Stein (1998) in the context of banks and by Froot (2007) in the context of insurance.

We discuss risk measures representing the minimum amount of capital a financial institution needs to raise and invest in a pre-specified eligible asset to ensure it is adequately capitalized. Most of the literature has focused on cash-additive risk measures, for which the eligible asset is a risk-free bond, on the grounds that the general case can be reduced to the cash-additive case by a change of numéraire. However, discounting does not work in all financially relevant situations, especially when the eligible asset is a defaultable bond. In this paper, we fill this gap by allowing general eligible assets. We provide a variety of finiteness and continuity results for the corresponding risk measures and apply them to risk measures based on value-at-risk and tail value-at-risk on L p spaces, as well as to shortfall risk measures on Orlicz spaces. We pay special attention to the property of cash subadditivity, which has been recently proposed as an alternative to cash additivity to deal with defaultable bonds. For important examples, we provide characterizations of cash subadditivity and show that when the eligible asset is a defaultable bond, cash subadditivity is the exception rather than the rule. Finally, we consider the situation where the eligible asset is not liquidly traded and the pricing rule is no longer linear. We establish when the resulting risk measures are quasiconvex and show that cash subadditivity is only compatible with continuous pricing rules.

In this article, we generalize the classical Edgeworth series expansion used in the option pricing literature. We obtain a closed-form pricing formula for European options by employing a generalized Hermite expansion for the risk neutral density. The main advantage of the generalized expansion is that it can be applied to heavy-tailed return distributions, a case for which the standard Edgeworth expansions are not suitable. We also show how the expansion coefficients can be inferred directly from market option prices.

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