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622
Regularity of the obstacle problem for a fractional power of the Laplace operator
 Comm. Pure Appl. Math
"... Given a function ϕ and s ∈ (0, 1), we will study the solutions of the following obstacle problem: • u ≥ ϕ in Rn, • (−)su ≥ 0 in Rn, • (−)su(x) = 0 for those x such that u(x)> ϕ(x), • limx →+ ∞ u(x) = 0. We show that when ϕ is C1,s or smoother, the solution u is in the space C1,α for every α & ..."
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Cited by 137 (4 self)
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Given a function ϕ and s ∈ (0, 1), we will study the solutions of the following obstacle problem: • u ≥ ϕ in Rn, • (−)su ≥ 0 in Rn, • (−)su(x) = 0 for those x such that u(x)> ϕ(x), • limx →+ ∞ u(x) = 0. We show that when ϕ is C1,s or smoother, the solution u is in the space C1,α for every α < s. In the case where the contact set {u = ϕ} is convex, we prove the optimal regularity result u ∈ C1,s. When ϕ is only C1,β for a β < s, we prove
Smoothness of scale functions for spectrally negative Lévy processes
, 2006
"... Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that stan ..."
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Cited by 84 (17 self)
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Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.
Some remarks on first passage of Lévy processes, the American put and pasting principles
 Annals of Appl. Probability
"... The purpose of this article is to provide, with the help of a fluctuation identity, a generic link between a number of known identities for the first passage time and overshoot above/below a fixed level of a Lévy process and the solution of Gerber and Shiu [Astin ..."
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Cited by 74 (5 self)
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The purpose of this article is to provide, with the help of a fluctuation identity, a generic link between a number of known identities for the first passage time and overshoot above/below a fixed level of a Lévy process and the solution of Gerber and Shiu [Astin
Stochastic volatility with leverage: fast likelihood inference
 Journal of Econometrics
, 2007
"... Kim, Shephard, and Chib (1998) provided a Bayesian analysis of stochastic volatility models based on a fast and reliable Markov chain Monte Carlo (MCMC) algorithm. Their method ruled out the leverage effect, which is known to be important in applications. Despite this, their basic method has been ex ..."
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Cited by 67 (19 self)
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Kim, Shephard, and Chib (1998) provided a Bayesian analysis of stochastic volatility models based on a fast and reliable Markov chain Monte Carlo (MCMC) algorithm. Their method ruled out the leverage effect, which is known to be important in applications. Despite this, their basic method has been extensively used in the financial economics literature and more recently in macroeconometrics. In this paper we show how the basic approach can be extended in a novel way to stochastic volatility models with leverage without altering the essence of the original approach. Several illustrative examples are provided.
A finite difference scheme for option pricing in jump diffusion and exponential Lévy models
, 2003
"... We present a finite difference method for solving parabolic partial integrodierential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a timeinhomogeneous jumpdiffusio ..."
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Cited by 64 (2 self)
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We present a finite difference method for solving parabolic partial integrodierential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a timeinhomogeneous jumpdiffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicitimplicit finite dierence scheme to solve the equation and study stability and convergence of the schemes proposed, using the notion of viscosity solution. Our convergence analysis requires neither the smoothness of the solution nor the nondegeneracy of coefficients and applies to European and barrier options in jumpdiffusion and pure jump models used in the literature. Numerical tests are performed with smooth and nonsmooth initial conditions.
A novel pricing method for European options based on Fouriercosine series expansions
 SIAM J. SCI. COMPUT
, 2008
"... Here we develop an option pricing method for European options based on the Fouriercosine series, and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fouriercosine expansion of the density function. In most cases, ..."
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Cited by 55 (14 self)
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Here we develop an option pricing method for European options based on the Fouriercosine series, and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fouriercosine expansion of the density function. In most cases, the convergence rate of the COS method is exponential and the computational complexity is linear. Its range of application covers different underlying dynamics, including Lévy processes and the Heston stochastic volatility model, and various types of option contracts. We will present the method and its applications in two separate parts. The first one is this paper, where we deal with European options in particular. In a followup paper we will present its application to options with earlyexercise features.
OLD AND NEW EXAMPLES OF SCALE FUNCTIONS FOR SPECTRALLY Negative Lévy Processes
, 2009
"... We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussi ..."
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Cited by 53 (13 self)
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We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as crossreferencing their analytical behaviour against known general considerations.
What’s vol got to do with it
 Review of Financial Studies
, 2011
"... Uncertainty plays a key role in economics, finance, and decision sciences. Financial markets, in particular derivative markets, provide fertile ground for understanding how perceptions of economic uncertainty and cashflow risk manifest themselves in asset prices. We demonstrate that the variance pre ..."
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Cited by 52 (3 self)
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Uncertainty plays a key role in economics, finance, and decision sciences. Financial markets, in particular derivative markets, provide fertile ground for understanding how perceptions of economic uncertainty and cashflow risk manifest themselves in asset prices. We demonstrate that the variance premium, defined as the difference between the squared VIX index and expected realized variance, captures attitudes toward uncertainty. We show conditions under which the variance premium displays significant time variation and return predictability. A calibrated, generalized LongRun Risks model generates a variance premium with time variation and return predictability that is consistent with the data, while simultaneously matching the levels and volatilities of the market return and risk free rate. Our evidence indicates an important role for transient nonGaussian shocks to fundamentals that affect agents ’ views of economic uncertainty and prices. We thank seminar participants at Wharton, the CREATES workshop ‘New Hope for the CCAPM?’,
Nonparametric estimation for Lévy processes from lowfrequency observations
 Bernoulli 15, 223–248. J.M. BARDET AND D. SURGAILIS
, 2009
"... We suppose that a Lévy process is observed at discrete time points. A rather general construction of minimumdistance estimators is shown to give consistent estimators of the LévyKhinchine characteristics as the number of observations tends to infinity, keeping the observation distance fixed. For a ..."
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Cited by 44 (8 self)
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We suppose that a Lévy process is observed at discrete time points. A rather general construction of minimumdistance estimators is shown to give consistent estimators of the LévyKhinchine characteristics as the number of observations tends to infinity, keeping the observation distance fixed. For a specific C 2criterion this estimator is rateoptimal. The connection with deconvolution and inverse problems is explained. A key step in the proof is a uniform control on the deviations of the empirical characteristic function on the whole real line. 2000 Mathematics Subject Classification. Primary 62G15; secondary 62M15. Keywords and Phrases. LévyKhinchine characteristics, density estimation, minimum distance estimator, deconvolution. Short title. Nonparametric estimation for Lévy processes. 1 1.