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14
Empirical properties of asset returns: stylized facts and statistical issues
 Quantitative Finance
, 2001
"... We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then des ..."
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Cited by 149 (2 self)
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We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.
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 32 (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
Static Hedging of Asian Options under Lévy Models: The Comonotonicity Approach
 Research Report, Department of Applied Economics, Kuleuven , OR 0365
, 2003
"... In this paper we present a simple static superhedging strategy for the payo# of an arithmetic Asian option in terms of a portfolio of European options. Moreover, it is shown that the obtained hedge is optimal in some sense. The strategy is based on stoploss transforms and is applicable under g ..."
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Cited by 14 (10 self)
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In this paper we present a simple static superhedging strategy for the payo# of an arithmetic Asian option in terms of a portfolio of European options. Moreover, it is shown that the obtained hedge is optimal in some sense. The strategy is based on stoploss transforms and is applicable under general stock price models. We focus on some popular Levy models.
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 6 (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.
Empirical distributions of logreturns: Between the stretched exponential and the power law? Quantitative Finance
"... A large consensus now seems to take for granted that the distributions of empirical returns of financial time series are regularly varying, with a tail exponent b close to 3. First, we show by synthetic tests performed on time series with time dependence in the volatility with both Pareto and Stretc ..."
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Cited by 2 (2 self)
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A large consensus now seems to take for granted that the distributions of empirical returns of financial time series are regularly varying, with a tail exponent b close to 3. First, we show by synthetic tests performed on time series with time dependence in the volatility with both Pareto and StretchedExponential distributions that for sample of moderate size, the standard generalized extreme value (GEV) estimator is quite inefficient due to the possibly slow convergence toward the asymptotic theoretical distribution and the existence of biases in presence of dependence between data. Thus it cannot distinguish reliably between rapidly and regularly varying classes of distributions. The Generalized Pareto distribution (GPD) estimator works better, but still lacks power in the presence of strong dependence. Then, we use a parametric representation of the tail of the distributions of returns of 100 years of daily return of the Dow Jones Industrial Average and over 1 years of 5minutes returns of the Nasdaq Composite index, encompassing both a regularly varying distribution in one limit of the parameters and rapidly varying distributions of the class of the StretchedExponential (SE) and LogWeibull distributions in other limits. Using the method of nested hypothesis testing (Wilks ’ theorem),
Extending TimeChanged Lévy Asset Models Through Multivariate Subordinators, Working Paper Università degli Studi di
, 2007
"... • (Clark (1973)) Time changed Brownian motions to model the departure of financial returns from normality. Business time is information driven: arrival of information increase activity with respect to the uniform progress of calendar time, lack of news slows it down. Ané and Geman (2000) show that ” ..."
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Cited by 2 (1 self)
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• (Clark (1973)) Time changed Brownian motions to model the departure of financial returns from normality. Business time is information driven: arrival of information increase activity with respect to the uniform progress of calendar time, lack of news slows it down. Ané and Geman (2000) show that ”the clock that allows one to recover normality for asset returns is indeed defined by the number of trade”. • The traditional multidimensional extension assumes a common business time for each stock: seems not to be very realistic in the stock market setting. Change of time has market activity as a proxy, a more realistic assumption, as confirmed by the empirical analysis in Harris(1986), is that each return has its own change of time. 2
A note on the suboptimality of pathdependent payoffs in Lévy markets
, 2008
"... Cox & Leland (2000) used techniques from the field of stochastic control theory to show that in the particular case of a Brownian motion for the asset logreturns risk averse decision makers with a fixed investment horizon prefer pathindependent payoffs over pathdependent ones. In this note we pr ..."
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Cited by 2 (0 self)
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Cox & Leland (2000) used techniques from the field of stochastic control theory to show that in the particular case of a Brownian motion for the asset logreturns risk averse decision makers with a fixed investment horizon prefer pathindependent payoffs over pathdependent ones. In this note we provide a novel and simple proof for the Cox & Leland result and we will extend it to general Lévy markets in case pricing is based on the Esscher transform (exponential tilting). It is also shown that in these markets optimal pathindependent payoffs are increasing with the underlying final asset value. We provide examples that allow explicit verification of our theoretical findings and also show that the inefficiency cost of pathdependent payoffs can be significant. Our results indicate that pathdependent investment payoffs, the use of which is widespread in financial markets, do not offer good value from the investor’s point of view. 1
Austrian Academy of Sciences
"... Summary. The onefactor Gaussian model is well known not to fit the prices of the different tranches of a collateralized debt obligation (CDO) simultaneously, leading to the implied correlation smile. Recently, other onefactor models based on different distributions have been proposed. Moosbrucker ..."
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Summary. The onefactor Gaussian model is well known not to fit the prices of the different tranches of a collateralized debt obligation (CDO) simultaneously, leading to the implied correlation smile. Recently, other onefactor models based on different distributions have been proposed. Moosbrucker [12] used a onefactor variancegamma (VG) model, Kalemanova et al. [7] and Guégan and Houdain [6] worked with a normal inverse Gaussian (NIG) factor model, and Baxter [3] introduced the Brownian variancegamma (BVG) model. These models bring more flexibility into the dependence structure and allow tail dependence. We unify these approaches, describe a generic onefactor Lévy model, and work out the large homogeneous portfolio (LHP) approximation. Then we discuss several examples and calibrate a battery of models to market data. Key words: Lévy processes; collateralized debt obligation (CDO); credit risk; credit default; large homogeneous portfolio approximation. 1
IMPLIED LÉVY VOLATILITY
, 2008
"... This paper introduces the concept of implied Lévy volatility, hereby extending the intuitive BlackScholes implied volatility into a more general context. More precisely, Lévy implied time and space volatility are introduced and a study of the shape of implied Lévy volatilities is made. Model perfor ..."
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This paper introduces the concept of implied Lévy volatility, hereby extending the intuitive BlackScholes implied volatility into a more general context. More precisely, Lévy implied time and space volatility are introduced and a study of the shape of implied Lévy volatilities is made. Model performance is studied by analyzing deltahedging strategies for the Normal Inverse Gaussian and the Meixner model, both qualitatively and on historical timeseries of the S&P500. It is shown that under such parameter settings the model performs systematically better.