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149
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.
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.
Consumptionbased asset pricing with higher cumulants,” manuscript
, 2008
"... I extend the EpsteinZinlognormal consumptionbased assetpricing model to allow for general i.i.d. consumption growth processes. Information about the higher moments—equivalently, cumulants—of consumption growth is encoded in the cumulantgenerating function (CGF). I express four observable quantit ..."
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Cited by 38 (9 self)
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I extend the EpsteinZinlognormal consumptionbased assetpricing model to allow for general i.i.d. consumption growth processes. Information about the higher moments—equivalently, cumulants—of consumption growth is encoded in the cumulantgenerating function (CGF). I express four observable quantities (the equity premium, riskless rate, consumptionwealth ratio and mean consumption growth) and the HansenJagannathan bound in terms of the CGF, and present applications. Models in which consumption is subject to occasional disasters can be handled easily and flexibly within the framework. The importance of higher cumulants is a doubleedged sword: those model parameters which are most important for asset prices, such as disaster parameters, are also the hardest to calibrate. It is therefore desirable to make statements which do not depend on a particular calibrated consumption process. First, I use properties of the CGF to derive restrictions on the timepreference rate and elasticity of intertemporal substitution that must hold in any EpsteinZini.i.d. model which is consistent with the observable quantities. Second, I show that “good deal ” bounds on the maximal Sharpe
Credit spreads, optimal capital structure, and implied volatility with endogenous default and jump risk
, 2005
"... We propose a twosided jump model for credit risk by extending the LelandToft endogenous default model based on the geometric Brownian motion. The model shows that jump risk and endogenous default can have significant impacts on credit spreads, optimal capital structure, and implied volatility of e ..."
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Cited by 34 (6 self)
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We propose a twosided jump model for credit risk by extending the LelandToft endogenous default model based on the geometric Brownian motion. The model shows that jump risk and endogenous default can have significant impacts on credit spreads, optimal capital structure, and implied volatility of equity options: (1) The jump and endogenous default can produce a variety of nonzero credit spreads, including upward, humped, and downward shapes; interesting enough, the model can even produce, consistent with empirical findings, upward credit spreads for speculative grade bonds. (2) The jump risk leads to much lower optimal debt/equity ratio; in fact, with jump risk, highly risky firms tend to have very little debt. (3) The twosided jumps lead to a variety of shapes for the implied volatility of equity options, even for long maturity options; and although in generel credit spreads and implied volatility tend to move in the same direction under exogenous default models, but this may not be true in presence of endogenous default and jumps. In terms of mathematical contribution, we give a proof of a version of the “smooth fitting ” principle for the jump model, justifying a conjecture first suggested by Leland and Toft under the Brownian model. 1
Applied Stochastic Processes and Control for JumpDiffusions: Modeling, Analysis and Computation
 Analysis and Computation, SIAM Books
, 2007
"... Abstract. An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions usi ..."
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Cited by 30 (7 self)
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Abstract. An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions using Wiener processes. Then jump perturbations are added using simple Poisson processes constructing the theory of simple jumpdiffusions. Next, markedjumpdiffusions are treated using compound Poisson processes to include random marked jumpamplitudes in parallel with the equivalent Poisson random measure formulation. Otherwise, the approach is quite applied, using basic principles with no abstractions beyond Poisson random measure. This treatment is suitable for those in classical applied mathematics, physical sciences, quantitative finance and engineering, but have trouble getting started with the abstract measuretheoretic literature. The approach here builds upon the treatment of continuous functions in the regular calculus and associated ordinary differential equations by adding nonsmooth and jump discontinuities to the model. Finally, the stochastic optimal control of markedjumpdiffusions is developed, emphasizing the underlying assumptions. The survey concludes with applications in biology and finance, some of which are canonical, dimension reducible problems and others are genuine nonlinear problems. Key words. Jumpdiffusions, Wiener processes, Poisson processes, random jump amplitudes, stochastic differential equations, stochastic chain rules, stochastic optimal control AMS subject classifications. 60G20, 93E20, 93E03 1. Introduction. There
Retrieving Lévy processes from option prices: Regularization of an illposed inverse problem
, 2000
"... We propose a stable nonparametric method for constructing an option pricing model of exponential Lévy type, consistent with a given data set of option prices. After demonstrating the illposedness of the usual and least squares version of this inverse problem, we suggest to regularize the calibratio ..."
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Cited by 22 (4 self)
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We propose a stable nonparametric method for constructing an option pricing model of exponential Lévy type, consistent with a given data set of option prices. After demonstrating the illposedness of the usual and least squares version of this inverse problem, we suggest to regularize the calibration problem by reformulating it as the problem of finding an exponential Lévy model that minimizes the sum of the pricing error and the relative entropy with respect to a prior exponential Lévy model. We prove the existence of solutions for the regularized problem and show that it yields solutions which are continuous with respect to the data, stable with respect to the choice of prior and converge to the minimumentropy least square solution of the initial problem.
Forecasting spot electricity prices: a comparison of parametric and semiparametric time series models
 International Journal of Forecasting
"... comparison of parametric and semiparametric time series models ..."
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Cited by 21 (0 self)
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comparison of parametric and semiparametric time series models