In this thesis, a new method based on characteristic functions is proposed to estimate the jump component in a finite-activity Levy process, which includes the jump frequency and the jump size distribution. Properties of the estimators are investigated, which show that this method does not require high frequency data. The implementation of the method is discussed, and examples are provided. We also perform a comparison which shows that our method has advantages over an existing threshold method. Finally, two applications are included: one is the classification of the increments of the model, and the other is the testing for a change of jump frequency.

Sharpe ratio is extensively used in performance of portfolio. However, it is based on assumption that return follows normal distribution. In other words, when return in asset is not normal distribution, the Sharpe ratio is not meaningful. This research focuses on Generalized Sharpe ratio with different distribution in eight indexes from 2001/12/31 to 2010/12/31. We try to find a suitable levy process to fit our data. Instead of Normal distribution assumption, we use Jump diffusion, Variance Gamma, Normal Inverse Gaussian, Hyperbolic, Generalized Hyperbolic, as our distribution to solve stylized fact like skewness and kurtosis. Compared the difference between standard Sharpe ratio and Generalized Sharpe ratio, we come to these conclusions: first of all, Generalized Hyperbolic is better levy process to fit our eight indexes. Second, Sharpe ratio under GH levy process has low autocorrelation, and it present that modified Sharpe ratio is more elastic. Third, Generalized Sharpe ratio can uncover the strategy that fund manager manipulate Sharpe ratio. At last, Generalized Sharpe ratio have better predict than standard Sharpe ratio. Keywords: Sharpe ratio, Levy process, GH distribution, portfolio, utility function

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京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21534号 / 理博第4441号 / 新制||理||1638(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 矢野 孝次, 教授 重川 一郎, 教授 泉 正己 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

In this work we are concerned with the characterization of certain classes of stochastic processes via duality formulae. First, we introduce a new formulation of a characterization of processes with independent increments, which is based on an integration by parts formula satisfied by infinitely divisible random vectors. Then we focus on the study of the reciprocal classes of Markov processes. These classes contain all stochastic processes having the same bridges, and thus similar dynamics, as a reference Markov process. We start with a resume of some existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. In the context of pure jump processes we derive the following new results. We will analyze the reciprocal classes of Markov counting processes and characterize them as a group of stochastic processes satisfying a duality formula. This result is applied to time-reversal of counting processes. We are able to extend some of these results to pure jump processes with different jump-sizes, in particular we are able to compare the reciprocal classes of Markov pure jump processes through a functional equation between the jump-intensities.

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We derive at-the-money call-price and implied volatility asymptotic expansions in time to maturity for a selection of exponential Lévy models, restricting our attention to asset-price models whose log returns structure is a Lévy process. We consider two main problems. First, we consider very general Lévy models that are in the domain of attraction of a stable random variable. Under some relatively minor assumptions, we give first-order at-the-money call-price and implied volatility asymptotics. In the case where our Lévy process has Brownian component, we discover new orders of convergence by showing that the rate of convergence can be of the form t¹/ᵃℓ(t) where ℓ is a slowly varying function and $\alpha \in (1,2)$. We also give an example of a Lévy model which exhibits this new type of behavior where ℓ is not asymptotically constant. In the case of a Lévy process with Brownian component, we find that the order of convergence of the call price is √t. Second, we investigate the CGMY process whose call-price asymptotics are known to third order. Previously, measure transformation and technical estimation methods were the only tools available for proving the order of convergence. We give a new method that relies on the Lipton-Lewis formula, guaranteeing that we can estimate the call-price asymptotics using only the characteristic function of the Lévy process. While this method does not provide a less technical approach, it is novel and is promising for obtaining second-order call-price asymptotics for at-the-money options for a more general class of Lévy processes.

This thesis introduces a novel theoretical option pricing ensemble framework to improve the bias and variance of option pricing models, especially those based on Levy Processes. In particular, we present a completely new, yet very general theoretical framework to calibrate and combine several option pricing models using ensemble methods. This framework has four main steps: general option pricing tasks, ensemble generation, ensemble pruning and ensemble integration. The modularity allows for a exible implementation in terms of asset classes, base models, pricing techniques and ensemble architecture.

京都大学 / 0048 / 新制・課程博士 / 博士(経済学) / 甲第20145号 / 経博第543号 / 新制||経||280(附属図書館) / 京都大学大学院経済学研究科経済学専攻 / (主査)教授 江上 雅彦, 教授 原 千秋, 准教授 砂川 伸幸 / 学位規則第4条第1項該当 / Doctor of Economics / Kyoto University / DGAM

This thesis is concerned with the small-time asymptotics and expansions of call option prices, when the log-return processes of the underlying stock prices follow several Levy-based models. To be specific, we derive the time-to-maturity asymptotic behavior for both at-the-money (ATM), out-of-the-money (OTM) and in-the-money (ITM) call-option prices under several jump-diffusion models and stochastic volatility models with Levy jumps. In the OTM and ITM cases, we consider a general stochastic volatility model with independent Levy jumps, while in the ATM case, we consider the pure-jump CGMY model with or without an independent Brownian component. An accurate modeling of the option market and asset prices requires a mixture of a continuous diffusive component and a jump component. In this thesis, we first model the log-return process of a risk asset with a jump diffusion model by combining a stochastic volatility model with an independent pure-jump Levy process. By assuming smoothness conditions on the Levy density away from the origin and a small-time large deviation principle on the stochastic volatility model, we derive the small-time expansions, of arbitrary polynomial order, in time-t, for the tail distribution of the log-return process, and for the call-option price which is not at-the-money. Moreover, our approach allows for a unified treatment of more general payoff functions. As a consequence of our tail expansions, the polynomial expansion in t of the transition density is also obtained under mild conditions. The asymptotic behavior of the ATM call-option prices is more complicated to obtain, and, in general, is given by fractional powers of t, which depends on different choices of the underlying log-return models. Here, we focus on the CGMY model, one of the most popular tempered stable models used in financial modeling. A novel second-order approximation for ATM option prices under the pure-jump CGMY Levy model is derived, and then extended to a model with an additional independent Brownian component. The third-order asymptotic behavior of the ATM option prices as well as the asymptotic behavior of the corresponding Black-Scholes implied volatilities are also addressed.