This thesis proposes an optimization formulation to ensure accuracy and stability in the local volatility function calibration. The unknown local volatility function is represented by kernel splines. The proposed optimization formulation minimizes calibration error and an L1 norm of the vector of coefficients for the kernel splines. The L1 norm regularization forces some coefficients to be zero at the termination of optimization. The complexity of local volatility function model is determined by the number of nonzero coefficients. Thus by using a regularization parameter, the proposed formulation balances the calibration accuracy with the model complexity. In the context of the support vector regression for function based on finite observations, this corresponds to balance the generalization error with the number of support vectors. In this thesis we also propose a trust region method to determine the coefficient vector in the proposed optimization formulation. In this algorithm, the main computation of each iteration is reduced to solving a standard trust region subproblem. To deal with the non-differentiable L1 norm in the formulation, a line search technique which allows crossing nondifferentiable hyperplanes is introduced to find the minimum objective value along a direction within a trust region. With the trust region algorithm, we numerically illustrate the ability of proposed approach to reconstruct the local volatility in a synthetic local volatility market. Based on S&P 500 market index option data, we demonstrate that the calibrated local volatility surface is smooth and resembles in shape the observed implied volatility surface. Stability is illustrated by considering calibration using market option data from nearby dates.

The search for deterministic chaos in economic and financial time series has attracted much interest over the past decade. However, clear evidence of chaotic structures is usually prevented by large random components in the time series. In the first part of this paper we show that even if a sophisticated algorithm estimating and testing the positivity of the largest Lyapunov exponent is applied to time series generated by a stochastic dynamical system or a return series of a stock index, the results are difficult to interpret. We conclude that the notion of sensitive dependence on initial conditions as it has been developed for deterministic dynamics, can hardly be transfered into a stochastic context. Therefore, in the second part of the paper our starting point for measuring dependencies for stochastic dynamics is a distributional characterization of the dynamics, e.g. by heteroskedastic models for economic and financial time series. We adopt a sensitivity measure proposed in the literature which is an information-theoretic measure of the distance between probability density functions. This sensitivity measure is well defined for stochastic dynamics, and it can be calculated analytically for the classes of stochastic dynamics with conditional normal distributions of constant and state-dependent variance. In particular, heteroskedastic return series models such as ARCH and GARCH models are investigated. (author's abstract) / Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

We used neural-network based modelling to generalize the linear econometric return models and compare their out-of-sample predictive ability in terms of different performance measures under three density specifications. As error measures we used the likelihood values on the test sets as well as standard volatility measures. The empirical analysis was based on return series of stock indices from different financial markets. The results indicate that for all markets there was found no improvement in the forecast by non-linear models over linear ones, while nongaussian models significantly dominate the gaussian models with respect to most performance measures. The likelihood performance measure mostly favours the linear model with Student-t distribution, but the significance of its superiority differs between the markets. (author's abstract) / Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

This thesis proposes an optimization formulation to ensure accuracy and stability in the local volatility function calibration. The unknown local volatility function is represented by kernel splines. The proposed optimization formulation minimizes calibration error and an L1 norm of the vector of coefficients for the kernel splines. The L1 norm regularization forces some coefficients to be zero at the termination of optimization. The complexity of local volatility function model is determined by the number of nonzero coefficients. Thus by using a regularization parameter, the proposed formulation balances the calibration accuracy with the model complexity. In the context of the support vector regression for function based on finite observations, this corresponds to balance the generalization error with the number of support vectors. In this thesis we also propose a trust region method to determine the coefficient vector in the proposed optimization formulation. In this algorithm, the main computation of each iteration is reduced to solving a standard trust region subproblem. To deal with the non-differentiable L1 norm in the formulation, a line search technique which allows crossing nondifferentiable hyperplanes is introduced to find the minimum objective value along a direction within a trust region. With the trust region algorithm, we numerically illustrate the ability of proposed approach to reconstruct the local volatility in a synthetic local volatility market. Based on S&P 500 market index option data, we demonstrate that the calibrated local volatility surface is smooth and resembles in shape the observed implied volatility surface. Stability is illustrated by considering calibration using market option data from nearby dates.

The Black-Scholes option pricing model has several well recognized deficiencies, one of which is its assumption of a constant and time-homogeneous stock return volatility term. The implied volatility smile has been studied by subsequent researchers and various models have been developed in an attempt to reproduce this phenomenon from within the models. However, few of these models yield closed-form pricing formulas that are easy to implement in practice. In this thesis, we study a Mixture Lognormal model (MLN) for European option pricing, which assumes that future stock prices are conditionally described by a mixture of lognormal distributions. The ability of mixture models in generating volatility smiles as well as delivering pricing improvement over the traditional Black-Scholes framework have been much researched under multi-component mixtures for many derivatives and high-volatility individual stock options. In this thesis, we investigate the performance of the model under the simplest two-component mixture in a market characterized by relative tranquillity and over a relatively stable period for broad-based index options. A careful interpretation is given to the model and the results obtained in the thesis. This di erentiates our study from many previous studies on this subject. Throughout the thesis, we establish the unique advantage of the MLN model, which is having closed-form option pricing formulas equal to the weighted mixture of Black-Scholes option prices. We also propose a robust calibration methodology to fit the model to market data. Extreme market states, in particular the so-called crash-o-phobia effect, are shown to be well captured by the calibrated model, albeit small pricing improvements are made over a relatively stable period of index option market. As a major contribution of this thesis, we extend the MLN model to price exotic options including binary, Asian, and barrier options. Closed-form formulas are derived for binary and continuously monitored barrier options and simulation-based pricing techniques are proposed for Asian and discretely monitored barrier options. Lastly, comparative results are analysed for various strike-maturity combinations, which provides insights into the formulation of hedging and risk management strategies.

The occurence of financial contagion can lead to hazardous results for financial institutions, financial markets as well as for the whole economy. Therefore it can have even serious economic effects on everybody´s life. That is why it is of great interest to deeper understand its characteristics. As classical finance theory seems not to give the best answers to this topic, the young academic field of behavioural finance can deliver new insights. The main purpose of this work is to provide an introduction mainly to professionals in portfolio and risk management and help them to tackle the problem of contagion at an early stage. Therefore not only aspects of behavioural finance are discussed, but the topic contagion is also brought into connection with network analyses and the current regulation process. Our paper can not answer all questions related to contagion, but it can help the reader to better understand its main aspects and enables him to delve deeper into this field.

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In recent years, the field of Fractional Brownian motion, Fractional Gaussian noise and long-range dependent processes has gained growing interest. Fractional Brownian motion is of great interest for example in telecommunications, hydrology and the generation of artificial landscapes. In fact, Fractional Brownian motion is a basic continuous process through which we show that it is neither a semimartingale nor a Markov process. In this work, we will focus on the path properties of Fractional Brownian motion and will try to check the absence of the property of a semimartingale. The concept of volatility will be dealt with in this work as a phenomenon in finance. Moreover, some statistical method like R/S analysis will be presented. By using these statistical tools we examine the volatility of shares and we demonstrate empirically that there are in fact shares which exhibit a fractal structure different from that of Brownian motion.

Starting from basic financial mathematics, we cover the mathematics of pricing swaptions, options on interest rate swaps. We then continue to the topic of obtaining an approximately pure volatility exposure. This exposure to volatility, which in practice enables us to trade volatility according to our perceptions of the market, is obtained by buying or selling swaptions and appropriate amounts of the underlying interest rate swap contract. Taking offsetting positions in the underlying contract is called hedging and is covered in depth. We note that hedging can primarily be done in two ways, and discuss the advantages and disadvantages of each of them. After deriving the value formulas for such a swaption strategy aimed at isolating volatility exposure we end with a discussion on the transition from theory to practice.We find that this way of trading volatility is conceptually simple, but that pre-trade profitability analysis is difficult due to the sometimes poor availability of the sophisticated data needed to simulate such a swaption strategy. Despite the possible limitations in the data necessary to translate this theory into an experimental setup, this thesis serves as a good basis for further research on the profitability of a volatility trading strategy using interest rate swaptions.

The dissertation consists of three essays. Chapter II examines the asymmetric effects of monetary policy on stock prices by using an unobserved components model with Markov-switching. My results show that monetary policy has negative effects on stock prices, which is consistent with the most recent literature. When the transitory component is in the low volatility state, a contractionary monetary policy significantly reduces stock prices. When the transitory component is in the high volatility state, the negative effect of monetary policy becomes larger, but the difference of the monetary policy effects between two states is not significant. Besides, a contractionary monetary policy will lower the probability of stock prices staying in the low volatility state. Monetary policy also reduces the total volatility of stock prices and the volatility of the transitory component of stock prices. Chapter III employs the smooth transition autoregressive (STAR) models to investigate the nonlinear effect of monetary policy on stock returns. The change in the Federal funds rate is used as an endogenous measure of monetary policy and the growth rate of industrial production is also considered in the model. My empirical results show that excess stock returns, the change in the Federal funds rate, and the growth rate of industrial production all can be expressed in the nonlinear STAR models. The estimated coefficients and the impulse response functions show that the effect of monetary policy on excess returns of stock prices is significantly negative and nonlinear. The change in the Federal funds rate has a larger negative effect on excess returns in the extreme low excess returns regime and the effect becomes smaller when the excess returns are greater than the threshold value. In chapter IV, I use a panel data approach to investigate the impact of exchange rate volatility on bilateral exports of the U.S. to the thirteen major trading partners. I further test the possibility of nonlinear effects of exchange rate volatility on exports by using threshold regression methods for non-dynamic panels with individual-specific fixed effects proposed by Hansen (1999). The results indicate that the effect of exchange rate volatility on bilateral exports is nonlinear. When the relative real GDP per capita of the exporting partner is lower than the threshold value, the response of bilateral U.S. exports to exchange rate volatility is positive. But, exchange rate volatility decreases bilateral exports of the U.S. to the exporting partners when their relative real GDP per capita surpass the threshold value.