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74
Lévy Processes in Finance: Theory, Numerics, and Empirical Facts
, 2000
"... Lévy processes are an excellent tool for modelling price processes in mathematical finance. On the one hand, they are very flexible, since for any time increment ∆t any infinitely divisible distribution can be chosen as the increment distribution over periods of time ∆t. On the other hand, they have ..."
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Cited by 77 (2 self)
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Lévy processes are an excellent tool for modelling price processes in mathematical finance. On the one hand, they are very flexible, since for any time increment ∆t any infinitely divisible distribution can be chosen as the increment distribution over periods of time ∆t. On the other hand, they have a simple structure in comparison with general semimartingales. Thus stochastic models based on Lévy processes often allow for analytically or numerically tractable formulas. This is a key factor for practical applications. This thesis is divided into two parts. The first, consisting of Chapters 1, 2, and 3, is devoted to the study of stock price models involving exponential Lévy processes. In the second part, we study term structure models driven by Lévy processes. This part is a continuation of the research that started with the author's diploma thesis Raible (1996) and the article Eberlein and Raible (1999). The content of the chapters is as follows. In Chapter 1, we study a general stock price model where the price of a single stock follows an exponential Lévy process. Chapter 2 is devoted to the study of the Lévy measure of infinitely divisible distributions, in particular of generalized hyperbolic distributions. This yields information about what changes in the distribution of a generalized hyperbolic Lévy motion can be achieved by a locally equivalent change of the underlying probability measure. Implications for
Option Pricing by Transform Methods: Extensions, Unification, and Error Control
 Journal of Computational Finance
"... We extend and unify Fourieranalytic methods for pricing a wide class of options on any underlying state variable whose characteristic function is known. In this general setting, we bound the numerical pricing error of discretized transform computations, such as DFT/FFT. These bounds enable algorith ..."
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Cited by 75 (6 self)
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We extend and unify Fourieranalytic methods for pricing a wide class of options on any underlying state variable whose characteristic function is known. In this general setting, we bound the numerical pricing error of discretized transform computations, such as DFT/FFT. These bounds enable algorithms to select efficient quadrature parameters and to price with guaranteed numerical accuracy.
Optimal stopping and perpetual options for Lévy processes
, 2000
"... Solution to the optimal stopping problem for a L'evy process and reward functions (e x \Gamma K) + and (K \Gamma e x ) + , discounted at a constant rate is given in terms of the distribution of the overall supremum and infimum of the process killed at this rate. Closed forms of this sol ..."
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Cited by 51 (6 self)
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Solution to the optimal stopping problem for a L'evy process and reward functions (e x \Gamma K) + and (K \Gamma e x ) + , discounted at a constant rate is given in terms of the distribution of the overall supremum and infimum of the process killed at this rate. Closed forms of this solutions are obtained under the condition of positive jumps mixedexponentially distributed. Results are interpreted as admissible pricing of perpetual American call and put options on a stock driven by a L'evy process, and a BlackScholes type formula is obtained. Keywords and Phrases: Optimal stopping, L'evy process, mixtures of exponential distributions, American options, Derivative pricing. JEL Classification Number: G12 Mathematics Subject Classification (1991): 60G40, 60J30, 90A09. 1 Introduction and general results 1.1 L'evy processes Let X = fX t g t0 be a real valued stochastic process defined on a stochastic basis(\Omega ; F ; F = (F t ) t0 ; P ) that satisfy the usual conditions. A...
Hyperbolic Processes in Finance
, 2001
"... Distributions that have tails heavier than the normal distribution are ubiquitous in finance. For purposes such as risk management and derivative pricing it is important to use relatively simple models that can capture the heavy tails and other relevant features of financial data. A class of distrib ..."
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Cited by 34 (7 self)
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Distributions that have tails heavier than the normal distribution are ubiquitous in finance. For purposes such as risk management and derivative pricing it is important to use relatively simple models that can capture the heavy tails and other relevant features of financial data. A class of distributions that is very often able to fit
The defaultable Lévy term structure: ratings and restructuring
 Mathematical Finance
, 2003
"... Abstract. We introduce the intensitybased defaultable Lévy term structure model. It generalizes the defaultfree Lévy term structure model by Eberlein and Raible, and the intensitybased defaultable HeathJarrowMorton approach of Bielecki and Rutkowski. Furthermore, we include the concept of mul ..."
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Cited by 24 (6 self)
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Abstract. We introduce the intensitybased defaultable Lévy term structure model. It generalizes the defaultfree Lévy term structure model by Eberlein and Raible, and the intensitybased defaultable HeathJarrowMorton approach of Bielecki and Rutkowski. Furthermore, we include the concept of multiple defaults, based on Schönbucher, within this generalization. Key words: default risk, Lévy processes, term structure of interest rates, migration process, ratings, restructuring, market price of risk 1
Normal modified stable processes
, 2001
"... This paper discusses two classes of distributions, and stochastic processes derived from them: modified stable (MS) laws and normal modified stable (NMS) laws. This extends corresponding results for the generalised inverse Gaussian (GIG) and generalised hyperbolic (GH) or normal generalised inverse ..."
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Cited by 22 (4 self)
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This paper discusses two classes of distributions, and stochastic processes derived from them: modified stable (MS) laws and normal modified stable (NMS) laws. This extends corresponding results for the generalised inverse Gaussian (GIG) and generalised hyperbolic (GH) or normal generalised inverse Gaussian (NGIG) laws. The wider framework thus established provides, in particular, for added flexibility in the modelling of the dynamics of financial time series, of importance especially as regards OU based stochastic volatility models for equities. In the special case of the tempered stable OU process an exact option pricing formula can be found, extending previous results based on the inverse Gaussian and gamma distributions.
2002), Integrated OU processes and nonGaussian OUbased stochastic volatility models, Scandinavian Journal of Statistics, forthcoming
"... In this paper we study the detailed distributional properties of integrated nonGaussian OU (intOU) processes. Both exact and approximate results are given. We emphasise the study of the tail behaviour of the intOU process. Our results have many potential applications in financial economics, for OU ..."
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Cited by 20 (2 self)
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In this paper we study the detailed distributional properties of integrated nonGaussian OU (intOU) processes. Both exact and approximate results are given. We emphasise the study of the tail behaviour of the intOU process. Our results have many potential applications in financial economics, for OU processes are used as models of instantaneous volatility in stochastic volatility (SV) models. In this case an intOU process can be regarded as a model of integrated volatility. Hence the tail behaviour of the intOU process will determine the tail behaviour of returns generated by SV models.
Pricing Asian options in a semimartingale model
 Quant. Finan
, 2006
"... Abstract. In this article we study arithmetic Asian options when the underlying stock is driven by special semimartingale processes. We show that the inherently path dependent problem of pricing Asian options can be transformed into a problem without path dependency in the payoff function. We also s ..."
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Cited by 16 (1 self)
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Abstract. In this article we study arithmetic Asian options when the underlying stock is driven by special semimartingale processes. We show that the inherently path dependent problem of pricing Asian options can be transformed into a problem without path dependency in the payoff function. We also show that the price satisfies a simpler integrodifferential equation in the case the stock price is driven by a process with independent increments, Lévy process being a special case.
Exotic Options under Lévy Models: An Overview
, 2004
"... In this paper we overview the pricing of several socalled exotic options in the nowdays quite popular exponential Lévy models. ..."
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Cited by 16 (0 self)
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In this paper we overview the pricing of several socalled exotic options in the nowdays quite popular exponential Lévy models.
On the duality principle in option pricing: semimartingale setting. Finance Stoch
, 2008
"... Abstract. The purpose of this paper is to develop the appropriate mathematical tools for the study of the duality principle in option pricing for models where prices are described by general exponential semimartingales. Particular cases of these models are the ones which are driven by Brownian mot ..."
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Cited by 15 (5 self)
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Abstract. The purpose of this paper is to develop the appropriate mathematical tools for the study of the duality principle in option pricing for models where prices are described by general exponential semimartingales. Particular cases of these models are the ones which are driven by Brownian motions and by Lévy processes, which have been considered in several papers. Generally speaking the duality principle states that the calculation of the price of a call option for a model with price process S = eH (w.r.t. the measure P) is equivalent to the calculation of the price of a put option for a suitable dual model S ′ = eH (w.r.t. a dual measure P ′). More sophisticated duality results are derived for a broad spectrum of exotic options. From the paper it is clear that appealing to general exponential semimartingale models leads to a deeper insight into the essence of the duality principle. 1.