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249
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 33 (1 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.
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 31 (3 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
C.W.: A novel pricing method for European options based on Fouriercosine series expansions
 SIAM J. Sci. Comput
, 2008
"... Abstract. 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 mos ..."
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Cited by 30 (11 self)
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Abstract. 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.
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 29 (9 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.
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 27 (11 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.
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 21 (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
Numerical methods for controlled HamiltonJacobiBellman PDEs in finance
 Journal of Computational Finance
"... Many nonlinear option pricing problems can be formulated as optimal control problems, leading to HamiltonJacobiBellman (HJB) or HamiltonJacobiBellmanIsaacs (HJBI) equations. We show that such formulations are very convenient for developing monotone discretization methods which ensure convergenc ..."
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Cited by 21 (11 self)
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Many nonlinear option pricing problems can be formulated as optimal control problems, leading to HamiltonJacobiBellman (HJB) or HamiltonJacobiBellmanIsaacs (HJBI) equations. We show that such formulations are very convenient for developing monotone discretization methods which ensure convergence to the financially relevant solution, which in this case is the viscosity solution. In addition, for the HJB type equations, we can guarantee convergence of a Newtontype (Policy) iteration scheme for the nonlinear discretized algebraic equations. However, in some cases, the Newtontype iteration cannot be guaranteed to converge (for example, the HJBI case), or can be very costly (for example for jump processes). In this case, we can use a piecewise constant control approximation. While we use a very general approach, we also include numerical examples for the specific interesting case of option pricing with unequal borrowing/lending costs and stock borrowing fees.
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 20 (1 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?’,
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 16 (7 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