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24
Meixner Processes in Finance
, 2001
"... In the BlackScholes option price model Brownian motion and the underlying Normal distribution play a fundamental role. Empirical evidence however shows that the normal distribution is a very poor model to fit reallife data. In order to achieve a better fit we replace the Brownian motion by a speci ..."
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Cited by 23 (5 self)
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In the BlackScholes option price model Brownian motion and the underlying Normal distribution play a fundamental role. Empirical evidence however shows that the normal distribution is a very poor model to fit reallife data. In order to achieve a better fit we replace the Brownian motion by a special Levy process: the Meixner process. We show that the underlying Meixner distribution allows an almost perfect fit to the data by performing a number of statistical tests. We discuss properties of the driving Meixner process. Next, we give a valuation formula for derivative securities, state the analogue of the BlackScholes di#erential equation, and compare the obtained prices with the classical BlackScholes prices. Throughout the text the method is illustrated by the modeling of the Nikkei225 Index. Similar analysis for other indices are given in the appendix. 1
Infinitely Divisible Laws Associated With Hyperbolic Functions
, 2000
"... The infinitely divisible distributions on R + of random variables C t , S t and T t with Laplace transforms ` 1 cosh p 2 ' t ; / p 2 sinh p 2 ! t ; and / tanh p 2 p 2 ! t respectively are characterized for various t ? 0 in a number of different ways: by simple relations bet ..."
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Cited by 18 (5 self)
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The infinitely divisible distributions on R + of random variables C t , S t and T t with Laplace transforms ` 1 cosh p 2 ' t ; / p 2 sinh p 2 ! t ; and / tanh p 2 p 2 ! t respectively are characterized for various t ? 0 in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their L'evy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for t = 1 or 2 in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a onedimensional Brownian motion. The distributions of C¹ and S³ are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and ...
Shrink Globally, Act Locally: Sparse Bayesian Regularization and Prediction
, 2010
"... We use Lévy processes to generate joint prior distributions for a location parameter β = (β1,..., βp) as p grows large. This approach, which generalizes normal scalemixture priors to an infinitedimensional setting, has a number of connections with mathematical finance and Bayesian nonparametrics. ..."
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Cited by 16 (5 self)
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We use Lévy processes to generate joint prior distributions for a location parameter β = (β1,..., βp) as p grows large. This approach, which generalizes normal scalemixture priors to an infinitedimensional setting, has a number of connections with mathematical finance and Bayesian nonparametrics. We argue that it provides an intuitive framework for generating new regularization penalties and shrinkage rules; for performing asymptotic analysis on existing models; and for simplifying proofs of some classic results on normal scale mixtures.
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.
Polynomials of Meixner’s type in infinite dimensions—Jacobi fields and orthogonality measures
"... Jacobi fields and orthogonality measures ..."
Selfdecomposability and option pricing
 Mathematical Finance 17, 31–57 (2007). ADDITIVE TIMECHANGES OF FELLER PROCESSES 7
"... The riskneutral process is modeled by a four parameter selfsimilar process of independent increments with a selfdecomposable law for its unit time distribution. Six different processes in this general class are theoretically formulated and empirically investigated. We show that all six models are ..."
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Cited by 8 (2 self)
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The riskneutral process is modeled by a four parameter selfsimilar process of independent increments with a selfdecomposable law for its unit time distribution. Six different processes in this general class are theoretically formulated and empirically investigated. We show that all six models are capable of adequately synthesizing European option prices across the spectrum of strikes and maturities at a point of time. Considerations of parameter stability over time suggest a preference for two of these models. Currently, there are several option pricing models with 6–10 free parameters that deliver a comparable level of performance in synthesizing option prices. The dimension reduction attained here should prove useful in studying the variation over time of option prices.
CGMY and Meixner subordinators are absolutely continuous with respect to one sided stable subordinators
, 2006
"... We describe the CGMY and Meixner processes as time changed Brownian motions. The CGMY uses a time change absolutely continuous with respect to the onesided stable (Y/2) subordinator while the Meixner time change is absolutely continuous with respect to the one sided stable (1/2) subordinator. The r ..."
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Cited by 7 (0 self)
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We describe the CGMY and Meixner processes as time changed Brownian motions. The CGMY uses a time change absolutely continuous with respect to the onesided stable (Y/2) subordinator while the Meixner time change is absolutely continuous with respect to the one sided stable (1/2) subordinator. The required time changes may be generated by simulating the requisite onesided stable subordinator and throwing away some of the jumps as described in Rosinski (2001). 1
Orthogonal decompositions for Lévy processes with an application to the . . .
, 2002
"... It is well known that between all processes with independent increments, essentially only the Brownian motion and the Poisson process possess the chaotic representation property (CRP). Thus, a natural question appears: What is an appropriate analog of the CRP in the case of a general Lévy process. A ..."
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Cited by 6 (4 self)
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It is well known that between all processes with independent increments, essentially only the Brownian motion and the Poisson process possess the chaotic representation property (CRP). Thus, a natural question appears: What is an appropriate analog of the CRP in the case of a general Lévy process. At least three approaches are possible here. The first one, due to Itô, uses the CRP of the Brownian motion and the Poisson process, as well as the representation of a Lévy process through those processes. The second approach, due to Nualart and Schoutens, consists in representing any squareintegrable random variable as a sum of multiple stochastic integrals constructed with respect to a family of orthogonalized centered power jumps processes. The third approach, never applied before to the Lévy processes, uses the idea of orthogonalization of polynomials with respect to a probability measure defined on the dual of a nuclear space. The main aims of the present paper are to develop the three approaches in the case of a general (Rvalued) Lévy process on a Riemannian manifold and (what is more important) to understand a relationship between these approaches. We apply the obtained results to the gamma, Pascal, and Meixner processes, in which case the analysis related to the orthogonalized polynomials becomes essentially simpler and richer than in the general 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 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.
Generic Lévy OneFactor Models for the Joint Modelling of Prepayment and Default: Modelling LCDX
, 2008
"... In this paper, we introduce a new robust model for modelling and pricing LCDX tranches. We extend the generic onefactor model of [1], which was developed for modelling and pricing of a synthetic CDO of CDSs, to a model for tranched portfolio of loanonly CDSs (LCDSs). The essential difference is th ..."
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Cited by 5 (2 self)
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In this paper, we introduce a new robust model for modelling and pricing LCDX tranches. We extend the generic onefactor model of [1], which was developed for modelling and pricing of a synthetic CDO of CDSs, to a model for tranched portfolio of loanonly CDSs (LCDSs). The essential difference is that now also the possibility of prepayments is built in. As a main advantage, the proposed model allows to trade LCDX tranches expressed in base correlations.