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74
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.
TimeChanged Lévy Processes and Option Pricing
, 2002
"... As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return ..."
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Cited by 89 (12 self)
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As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. We propose that timechanged Lévy processes be used to simultaneously address these three facets of the underlying asset return process. We show that our framework encompasses almost all of the models proposed in the option pricing literature. Despite the generality of our approach, we show that it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.
The Generalized Hyperbolic Model: Financial Derivatives and Risk Measures
 MATHEMATICAL FINANCE – BACHELIER CONGRESS 2000, GEMAN
, 1998
"... Statistical analysis of data from the nancial markets shows that generalized hyperbolic (GH) distributions allow a more realistic description of asset returns than the classical normal distribution. GH distributions contain as subclasses hyperbolic as well as normal inverse Gaussian (NIG) distributi ..."
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Cited by 40 (5 self)
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Statistical analysis of data from the nancial markets shows that generalized hyperbolic (GH) distributions allow a more realistic description of asset returns than the classical normal distribution. GH distributions contain as subclasses hyperbolic as well as normal inverse Gaussian (NIG) distributions which have recently been proposed as basic ingredients to model price processes. GH distributions generate in a canonical way Levy processes, i.e. processes with stationary and independent increments. We introduce a model for price processes which is driven by generalized hyperbolic Levy motions. This GH model is a generalization of the hyperbolic model developed by Eberlein and Keller (1995). It is incomplete. We derive an option pricing formula for GH driven models using the Esscher transform as martingale measure and compare the prices with classical BlackScholes prices. The objective of this study is to examine the consistency of our model assumptions with the empirically obser...
Term structure models driven by general Lévy processes
, 1999
"... ) in the form f(t, T ) = f(0, T ) + Z t 0 # # (#(s, T ))# 2 (s, T )ds  Z t 0 # 2 (s, T )dL s . Here #(u) = log(E[exp(uL 1 )]) denotes the log of the moment generating function of L(L 1 ) and # 2 the partial derivative of # with respect to maturity. If r(t) = f(t, t) denotes the short rat ..."
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Cited by 39 (4 self)
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) in the form f(t, T ) = f(0, T ) + Z t 0 # # (#(s, T ))# 2 (s, T )ds  Z t 0 # 2 (s, T )dL s . Here #(u) = log(E[exp(uL 1 )]) denotes the log of the moment generating function of L(L 1 ) and # 2 the partial derivative of # with respect to maturity. If r(t) = f(t, t) denotes the short rate process we finally get the following representation for the bond price process P (t, T ) = P (0, T ) exp Z t 0 (r(s)  #(#(s,
Optimal Portfolio Choice Under Loss Aversion
 REVIEW OF ECONOMICS & STATISTICS
, 2000
"... Prospect theory and loss aversion play a dominant role in behavioral finance. In this paper we derive closedform solutions for optimal portfolio choice under loss aversion. When confronted with gains a loss averse investor behaves similar to a portfolio insurer. When confronted with losses, the inv ..."
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Cited by 33 (3 self)
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Prospect theory and loss aversion play a dominant role in behavioral finance. In this paper we derive closedform solutions for optimal portfolio choice under loss aversion. When confronted with gains a loss averse investor behaves similar to a portfolio insurer. When confronted with losses, the investor aims at maximizing the probability that terminal wealth exceeds his aspiration level. Our analysis indicates that a representative agent model with loss aversion cannot resolve the equity premium puzzle. We also extend the martingale methodology to allow for more general utility functions and provide a simple approach to incorporate skewed and fattailed return distributions.
A Simple Option Formula for General JumpDiffusion and Other Exponential Levy Processes
 Other Exponential Lévy Processes,” Environ Financial Systems and OptionCity.net
, 2001
"... Option values are wellknown to be the integral of a discounted transition density times a payoff function; this is just martingale pricing. It's usually done in 'Sspace', where S is the terminal security price. But, for L6vy processes the Sspace transition densities are often very complicated, in ..."
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Cited by 32 (3 self)
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Option values are wellknown to be the integral of a discounted transition density times a payoff function; this is just martingale pricing. It's usually done in 'Sspace', where S is the terminal security price. But, for L6vy processes the Sspace transition densities are often very complicated, involving many special functions and infinite summations. Instead, we show that it's much easier to compute the option value as an integral in Fourier space  and interpret this as a Parseval identity. The formula is especially simple because (i) it's a single integration for any payoff and (ii) the integrand is typically a compact expressions with just elementary functions. Our approach clarifies and generalizes previous work using characteristic functions and Fourier inversions. For example, we show how the residue calculus leads to several variation formulas, such as a wellknown, but less numerically efficient, 'BlackScholes style' formula for call options. The result applies to any Europeanstyle, simple or exotic option (without pathdependence) under any L6vy process with a known characteristic function.
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
Specification Analysis of Option Pricing Models Based on TimeChanged Lévy Processes
, 2003
"... We analyze the specifications of option pricing models based on timechanged Lévy processes. We classify option pricing models based on the structure of the jump component in the underlying return process, the source of stochastic volatility, and the specification of the volatility process itself. O ..."
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Cited by 28 (3 self)
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We analyze the specifications of option pricing models based on timechanged Lévy processes. We classify option pricing models based on the structure of the jump component in the underlying return process, the source of stochastic volatility, and the specification of the volatility process itself. Our estimation of a variety of model specifications indicates that to better capture the behavior of the S&P 500 index options, we must incorporate a high frequency jump component in the return process and generate stochastic volatilities from two different sources, the jump component and the diffusion component.
Minimax and Minimal Distance Martingale Measures and Their Relationship to Portfolio Optimization
, 2000
"... In this paper we give a characterization of minimal distance martingale measures with respect to fdivergence distances in a general semimartingale market model. We provide necessary and sufficient conditions for minimal distance martingale measures and determine them explicitly for exponential ..."
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Cited by 28 (3 self)
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In this paper we give a characterization of minimal distance martingale measures with respect to fdivergence distances in a general semimartingale market model. We provide necessary and sufficient conditions for minimal distance martingale measures and determine them explicitly for exponential L'evy processes with respect to several classical distances. It is shown that the minimal distance martingale measures are equivalent to minimax martingale measures with respect to related utility functions and that optimal portfolios can be characterized by them. Related results in the context of continuoustime diffusion models were first obtained by He and Pearson (1991b) and Karatzas et al. (1991) and in a general semimartingale setting by Kramkov and Schachermayer (1999). Finally parts of the results are extended to utilitybased hedging.