## 91B30 Risk theory, insurance

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#### Keywords

- Cholesky-Verfahren (1)
- Commodity Index (1)
- Conditional Value-at-Risk (1)
- Financial Engineering (1)
- Finanzmathematik (1)
- Fourier-Transformation (1)
- Handelsstrategien (1)
- Kreditrisiko (1)
- Marktrisiko (1)
- Optimization (1)

This thesis deals with the solution of special problems arising in financial engineering or financial mathematics. The main focus lies on commodity indices. Chapter 1 addresses the important issue for the financial engineering practice of developing well-suited models for certain assets (here: commodity indices). Descriptive analysis of the Dow Jones-UBS commodity index compared to the Standard & Poor 500 stock index provides us with first insights of some features of the corresponding distributions. Statistical tests of normality and mean reversion then helps us in setting up a model for commodity indices. Additionally, chapter 1 encompasses a thorough introduction to commodity investment, history of commodities trading and the most important derivatives, namely futures and European options on futures. Chapter 2 proposes a model for commodity indices and derives fair prices for the most important derivatives in the commodity markets. It is a Heston model supplemented with a stochastic convenience yield. The Heston model belongs to the model class of stochastic volatility models and is currently widely used in stock markets. For the application in the commodity markets the stochastic convenience yield is included in the drift of the instantaneous spot return process. Motivated by the results of chapter 1 it seems reasonable to model the convenience yield by a mean reverting Ornstein-Uhlenbeck process. Since trading desks only apply and consider models with closed form solutions for options I derive such formulas for commodity futures by solving the corresponding partial differential equation. Additionally, semi-closed form formulas for European options on futures are determined. The Cauchy problem with respect to these options is more challenging than the first one. A solution can be provided. Unlike equities, which typically entitle the holder to a continuing stake in a corporation, commodity futures contracts normally specify a certain date for the delivery of the underlying physical commodity. In order to avoid the delivery process and maintain a futures position, nearby contracts must be sold and contracts that have not yet reached the delivery period must be purchased (so called rolling). Optimal trading days for selling and buying futures are determined by applying statistical tests for stochastic dominance. Besides the optimization of the rolling procedure for commodity futures we dedicate ourselves in chapter 3 with the optimization of the weightings of the commodity futures that make up the index. To this end, I apply the Markowitz approach or mean-variance optimization. The mean-variance optimization penalizes up-side and down-side risk equally, whereas most investors do not mind up-side risk. To overcome this, I consider in the next step other risk measures, namely Value-at-Risk and Conditional Value-at-Risk. The Conditional Value-at-Risk is generalized to discontinuous cumulative distribution functions of the loss. For continuous loss distributions, the Conditional Value-at-Risk at a given confidence level is defined as the expected loss exceeding the Value-at-Risk. Loss distributions associated with finite sampling or scenario modeling are, however, discontinuous. Various risk measures involving discontinuous loss distributions shall be introduced and compared. I then apply the theoretical results to the field of portfolio optimization with commodity indices. Furthermore, I uncover graphically the behavior of these risk measures. For this purpose, I consider the risk measures as a function of the confidence level. Based on a special discrete loss distribution, the graphs demonstrate the different properties of these risk measures. The goal of the first section of chapter 4 is to apply the mathematical concept of excursions for the creation of optimal highly automated or algorithmic trading strategies. The idea is to consider the gain of the strategy and the excursion time it takes to realize the gain. In this section I calculate formulas for the Ornstein-Uhlenbeck process. I show that the corresponding formulas can be calculated quite fast since the only function appearing in the formulas is the so called imaginary error function. This function is already implemented in many programs, such as in Maple. My main contribution of this topic is the optimization of the trading strategy for Ornstein-Uhlenbeck processes via the Banach fixed-point theorem. The second section of chapter 4 deals with statistical arbitrage strategies, a long horizon trading opportunity that generates a riskless profit. The results of this section provide an investor with a tool to investigate empirically if some strategies (for example momentum strategies) constitute statistical arbitrage opportunities or not.

The central theme in this thesis concerns the development of enhanced methods and algorithms for appraising market and credit risks and their application within the context of standard and more advanced market models. Generally, methods and algorithms for analysing market risk of complex portfolios involve detailed knowledge of option sensitivities, the so-called "Greeks". Based on an analysis of symmetries in financial market models, relations between option sensitivities are obtained, which can be used for the efficient valuation of the Greeks. Mainly, the relations are derived within the Black Scholes model, however, some relations are also valid for more general models, for instance the Heston model. Portfolios are usually influenced by lots of underlyings, so it is necessary to characterise the dependencies of these basic instruments. It is usual to describe such dependencies by correlation matrices. However, estimations of correlation matrices in practice are disturbed by statistical noise and usually have the problem of rank deficiency due to missing data. A fast algorithm is presented which performs a generalized Cholesky decomposition of a perturbed correlation matrix. In contrast to the standard Cholesky algorithm, an advantage of the generalized method is that it works for semi-positive, rank deficient matrices as well. Moreover, it gives an approximative decomposition when the input matrix is indefinite. A comparison with known algorithms with similar features is performed and it turns out, that the new algorithm can be recommended in situations where computation time is the critical issue. The determination of a profit and loss distribution by Fourier inversion of its characteristic function is a powerful tool, but it can break down when the characteristic function is not integrable. In this thesis, methods for Fourier inversion of non-integrable characteristic functions are studied. In this respect, two theorems are obtained which are based on a suitable approximation of the unknown distribution with known density and characteristic function. Further it will be shown, that straightforward Fast Fourier inversion works, when the according density lives on a bounded interval. The above techniques are of crucial importance to determine the profit and loss distribution (P&L) of large portfolios efficiently. The so-called Delta Gamma normal approach has become industrial standard for the estimation of market risk. It is shown, that the performance of the Delta Gamma normal approach can be improved substantially by application of the developed methods. The same optimization procedure also applies to the Delta Gamma Student model. A standard tool for computing the P&L distribution of a loan portfolio is the CreditRisk+ model. Basically, the CreditRisk+ distribution is a discrete distribution which can be computed from its probability generating function. For this a numerically stable method is presented and as an alternative, a new algorithm based on Fourier inversion is proposed. Finally, an extension of the CreditRisk+ model to market risk is developed, which distribution can be obtained efficiently by the presented Fourier inversion methods as well.