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237
Meromorphic Lévy processes and their fluctuation identities
 Annals of Applied Probability
, 2011
"... The last couple of years has seen a remarkable number of new, explicit examples of the Wiener–Hopf factorization for Lévy processes where previously there had been very few. We mention, in particular, the many cases of spectrally negative Lévy processes in [Sixth Seminar on Stochastic Analysis, ..."
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Cited by 37 (10 self)
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The last couple of years has seen a remarkable number of new, explicit examples of the Wiener–Hopf factorization for Lévy processes where previously there had been very few. We mention, in particular, the many cases of spectrally negative Lévy processes in [Sixth Seminar on Stochastic Analysis,
A Fast and Accurate FFTBased Method for Pricing EarlyExercise Options under Lévy Processes
 SIAM JOURNAL OF SCIENTIFIC COMPUTING
, 2008
"... A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the wellknown riskneutral valuation formula by reco ..."
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Cited by 36 (9 self)
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A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the wellknown riskneutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the Fast Fourier Transform (FFT). This novel pricing method, which we dub the Convolution method, CONV for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within many regular affine models, among which the class of exponential Lévy models. For an Mtimes exercisable Bermudan option, the overall complexity is O(MN log 2 (N)) with N grid points used to discretise the price of the underlying asset. American options are priced efficiently by applying Richardson extrapolation to the prices of Bermudan options.
Fast deterministic pricing of options on Lévy driven assets
 M2AN Math. Model. Numer. Anal
, 2002
"... A partial integrodifferential equation (PIDE) ∂tu + A[u] = 0 for European contracts on assets with general jumpdiffusion price process of Lévy type is derived. The PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ ..."
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Cited by 36 (3 self)
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A partial integrodifferential equation (PIDE) ∂tu + A[u] = 0 for European contracts on assets with general jumpdiffusion price process of Lévy type is derived. The PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θscheme in time and a wavelet Galerkin method with N degrees of freedom in space. The full Galerkin matrix for A can be replaced with a sparse matrix in the wavelet basis, and the linear systems for each time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for M time steps is bounded by O(MN(ln N) 2) operations and O(N ln(N)) memory. The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solution in the same complexity as finite difference approximations of the standard BlackScholes equation. Computational examples for various Lévy price processes (VG, CGMY) are presented. 1
Characterization of dependence of multidimensional Lévy processes using Lévy copulas
 J. Multivariate Anal
, 2006
"... This paper suggests to use Lévy copulas to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a kind of Sklar’s theorem states that the law of a general multivariate Lévy ..."
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Cited by 35 (2 self)
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This paper suggests to use Lévy copulas to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a kind of Sklar’s theorem states that the law of a general multivariate Lévy process is obtained by combining arbitrary univariate Lévy processes with an arbitrary Lévy copula. We construct parametric families of Lévy copulas and prove a limit theorem, which indicates how to obtain the Lévy copula of a multidimensional Lévy process X from the ordinary copulas of the random vectors Xt for fixed t.
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 33 (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
Spectral calibration of exponential Lévy models
 Finance and Stochastics
"... This research was supported by the Deutsche ..."
Pricing earlyexercise and discrete barrier options by Fouriercosine series expansions
 Numerische Mathematik
"... We present a pricing method based on Fouriercosine expansions for earlyexercise and discretelymonitored barrier options. The method works well for exponential Lévy asset price models. The error convergence is exponential for processes characterized by very smooth (C ∞ [a, b] ∈ R) transitional pr ..."
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Cited by 28 (8 self)
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We present a pricing method based on Fouriercosine expansions for earlyexercise and discretelymonitored barrier options. The method works well for exponential Lévy asset price models. The error convergence is exponential for processes characterized by very smooth (C ∞ [a, b] ∈ R) transitional probability density functions. The computational complexity is O((M − 1)N log N) with N a (small) number of terms from the series expansion, and M, the number of earlyexercise/monitoring dates. This paper is the followup of [22] in which we presented the impressive performance of the Fouriercosine series method for European options. 1
ImplicitExplicit Numerical Schemes for JumpDiffusion Processes
 Calcolo
, 2004
"... We study the numerical approximation of viscosity solutions for Parabolic IntegroDifferential Equations (PIDE). Similar models arise in option pricing, to generalize the BlackScholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jum ..."
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Cited by 19 (5 self)
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We study the numerical approximation of viscosity solutions for Parabolic IntegroDifferential Equations (PIDE). Similar models arise in option pricing, to generalize the BlackScholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Due to the nonlocal nature of the integral term, unconditionally stable implicit difference scheme are not practically feasible. Here we propose to use ImplicitExplicit (IMEX) RungeKutta methods for the time integration to solve the integral term explicitly, giving higher order accuracy schemes under weak stability timestep restrictions. Numerical tests are presented to show the computational efficiency of the approximation.
Time Changed Markov Processes in Unified CreditEquity Modeling ∗
, 2008
"... This paper develops a novel class of hybrid creditequity models with statedependent jumps, localstochastic volatility and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time changed Markov diffusion process with statede ..."
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Cited by 17 (4 self)
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This paper develops a novel class of hybrid creditequity models with statedependent jumps, localstochastic volatility and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time changed Markov diffusion process with statedependent local volatility and killing rate (default intensity). When the time change is a Lévy subordinator, the stock price process exhibits jumps with statedependent Lévy measure. When the time change is a time integral of an activity rate process, the stock price process has localstochastic volatility and default intensity. When the time change process is a Lévy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has statedependent jumps, localstochastic volatility and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the