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26
On American options under the Variance Gamma process
 Applied Mathematical Finance
, 2004
"... We consider American options in a market where the underlying asset follows a Variance Gamma process. We give a sufficient condition for the failure of the smooth fit principle for finite horizon call options. We also propose a second order accurate finitedifference method to find the American opti ..."
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Cited by 14 (4 self)
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We consider American options in a market where the underlying asset follows a Variance Gamma process. We give a sufficient condition for the failure of the smooth fit principle for finite horizon call options. We also propose a second order accurate finitedifference method to find the American option price and the exercise boundary. The problem is formulated as a Linear Complementarity Problem and numerically solved by a convenient splitting. Computations have been accelerated with the help of the Fast Fourier Transform. A stability analysis shows that the scheme is conditionally stable, with a mild stability condition of the form k = O(log(h)  −1). The theoretical results are verified numerically throughout a series of numerical experiments. Keywords: Integrodifferential equations, Variance Gamma, finite differences, FFT.
A SemiLagrangian approach for American Asian options under jump diffusion
 SIAM Journal on Scientific Computing
, 2003
"... version 1.7 A semiLagrangian method is presented to price continuously observed fixed strike Asian options. At each timestep a set of one dimensional partial integral differential equations (PIDEs) is solved and the solution of each PIDE is updated using semiLagrangian timestepping. CrankNicolson ..."
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Cited by 14 (7 self)
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version 1.7 A semiLagrangian method is presented to price continuously observed fixed strike Asian options. At each timestep a set of one dimensional partial integral differential equations (PIDEs) is solved and the solution of each PIDE is updated using semiLagrangian timestepping. CrankNicolson and second order backward differencing timestepping schemes are studied. Monotonicity and stability results are derived. With low volatility values, it is observed that the nonsmoothness at the strike in the payoff affects the convergence rate; subquadratic convergence rate is observed.
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 10 (2 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.
Numerical Valuation of European and American Options under Kou’s JumpDiffusion Model
"... Numerical methods are developed for pricing European and American options under Kou’s jumpdiffusion model which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is logdoubleexponentially distributed. The price of a European o ..."
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Cited by 9 (4 self)
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Numerical methods are developed for pricing European and American options under Kou’s jumpdiffusion model which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is logdoubleexponentially distributed. The price of a European option is given by a partial integrodifferential equation (PIDE) while American options lead to a linear complementarity problem (LCP) with the same operator. Spatial differential operators are discretized using finite differences on nonuniform grids and time stepping is performed using the implicit Rannacher scheme. For the evaluation of the integral term easy to implement recursion formulas are derived which have optimal computational cost. When pricing European options the resulting dense linear systems are solved using a stationary iteration. For American options two ways to solve the LCPs are described: an operator slitting method and a penalty method. Numerical experiments confirm that the developed methods are very efficient as fairly accurate option prices can be computed in a few milliseconds on a PC.
Fast and accurate pricing of barrier options under lévy processes
, 2008
"... apport de recherche ..."
The discontinuous Galerkin method for fractal conservation laws, submitted
, 2009
"... Abstract. We propose, analyze, and demonstrate a discontinuous Galerkin method for fractional conservation laws. Various stability estimates are established along with error estimates for regular solutions of linear equations. Moreover, in the nonlinear case and when piecewise constant elements are ..."
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Cited by 4 (2 self)
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Abstract. We propose, analyze, and demonstrate a discontinuous Galerkin method for fractional conservation laws. Various stability estimates are established along with error estimates for regular solutions of linear equations. Moreover, in the nonlinear case and when piecewise constant elements are utilized, we prove a rate of convergence toward the unique entropy solution. We present numerical results for different types of solutions of linear and nonlinear fractional conservation laws.
Numerical Solution of Two Asset Jump Diffusion Models for Option Valuation
, 2005
"... Under the assumption that two financial assets evolve by correlated finite activity jumps superimposed on correlated Brownian motion, the value of a contingent claim written on these two assets is given by a two dimensional parabolic partial integrodifferential equation (PIDE). An implicit, finite ..."
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Cited by 3 (0 self)
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Under the assumption that two financial assets evolve by correlated finite activity jumps superimposed on correlated Brownian motion, the value of a contingent claim written on these two assets is given by a two dimensional parabolic partial integrodifferential equation (PIDE). An implicit, finite difference method is derived in this paper. This approach avoids a dense linear system solution by combining a fixed point iteration scheme with an FFT. The method prices both American and European style contracts independent (under some simple restrictions) of the option payoff and distribution of jumps. Convergence under the localization from the infinite to a finite domain is investigated, as are the theoretical conditions for the stability of the discrete approximation under maximum and von Neumann analysis. The analysis shows that the fixed point iteration is rapidly convergent under typical market parameters. The rapid convergence of the fixed point iteration is verified in some numerical tests. These tests also indicate that the method used to localize the PIDE is inexpensive and easily implemented. Keywords: Twoasset, option pricing, partial integrodifferential equation, finite difference, American option, jump diffusion.
Componentwise splitting methods for pricing American options under stochastic volatility
 Int. J. Theor. Appl. Finance
, 2007
"... Abstract A linear complementarity problem (LCP) is formulated for the price of American options under the Bates model which combines the Heston stochastic volatility model and the Merton jumpdiffusion model. A finite difference discretization is described for the partial derivatives and a simple qu ..."
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Cited by 2 (0 self)
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Abstract A linear complementarity problem (LCP) is formulated for the price of American options under the Bates model which combines the Heston stochastic volatility model and the Merton jumpdiffusion model. A finite difference discretization is described for the partial derivatives and a simple quadrature is used for the integral term due to jumps. A componentwise splitting method is generalized for the Bates model. It is leads to solution of sequence of onedimensional LCPs which can be solved very efficiently using the Brennan and Schwartz algorithm. The numerical experiments demonstrate the componentwise splitting method to be essentially as accurate as the PSOR method, but order of magnitude faster. Furthermore, pricing under the Bates model is less than twice more expensive computationally than under the Heston model in the experiments. 1
Differencequadrature schemes for nonlinear degenerate parabolic integroPDE
 SIAM J. Numer. Anal
, 2010
"... Abstract. We derive and analyze monotone differencequadrature schemes for Bellman equations of controlled Lévy (jumpdiffusion) processes. These equations are fully nonlinear, degenerate parabolic integroPDEs interpreted in the sense of viscosity solutions. We propose new “direct ” discretization ..."
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Cited by 2 (0 self)
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Abstract. We derive and analyze monotone differencequadrature schemes for Bellman equations of controlled Lévy (jumpdiffusion) processes. These equations are fully nonlinear, degenerate parabolic integroPDEs interpreted in the sense of viscosity solutions. We propose new “direct ” discretizations of the nonlocal part of the equation that give rise to monotone schemes capable of handling singular Lévy measures. Furthermore, we develop a new general theory for deriving error estimates for approximate solutions of integroPDEs, which thereafter is applied to the proposed differencequadrature schemes. Contents
NUMERICAL APPROXIMATION OF CAUCHY PROBLEMS FOR MULTIDIMENSIONAL PDES WITH UNBOUNDED COEFFICIENTS ARISING IN FINANCIAL MATHEMATICS
"... Abstract. In this article, we study the numerical approximation of the solution of the Cauchy problem for a multidimensional linear parabolic PDE of second order, with unbounded time and spacedependent coefficients. The PDE free term and the initial data are also allowed to grow. Under the assumpti ..."
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Cited by 1 (1 self)
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Abstract. In this article, we study the numerical approximation of the solution of the Cauchy problem for a multidimensional linear parabolic PDE of second order, with unbounded time and spacedependent coefficients. The PDE free term and the initial data are also allowed to grow. Under the assumption that the PDE does not degenerate, using the L 2 theory of solvability in weighted Sobolev spaces, the PDE problem’s weak solution is approximated in space, with the use of finitedifference methods. Making also use of finite differences (with both the explicit and implicit schemes), the approximation in time is considered in abstract spaces for evolution equations, and then specified to the secondorder parabolic PDE problem. The rate of convergence is estimated for the approximation in space and time. 1.