Results 1  10
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65
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.
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 22 (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.
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 17 (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.
Robust Numerical Valuation of European and American Options under the CGMY Process
, 2007
"... We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Lévy process. For processes of finite variation, quadratic convergence is obtained as the mesh and time step are refined. For infinite variation processe ..."
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Cited by 15 (1 self)
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We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Lévy process. For processes of finite variation, quadratic convergence is obtained as the mesh and time step are refined. For infinite variation processes, better than first order accuracy is achieved. The jump component in the neighborhood of log jump size zero is specially treated by using a Taylor expansion approximation and the drift term is dealt with using a semiLagrangian scheme. The resulting Partial IntegroDifferential Equation (PIDE) is then solved using a preconditioned BiCGSTAB method coupled with a fast Fourier transform. Proofs of fully implicit timestepping stability and monotonicity are provided. The convergence properties of the BiCGSTAB scheme are discussed and compared with a fixed point iteration. Numerical tests showing the convergence and performance of this method for European and American options under processes of finite and infinite variation are presented.
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 12 (5 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.
The discontinuous Galerkin method for fractal conservation laws
, 2009
"... 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, ..."
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Cited by 11 (8 self)
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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.
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 8 (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
Numerical valuation of American options under the CGMY process. In Exotic option pricing and advanced Lévy models
, 2005
"... American put options written on an underlying stock following a CarrMadanGemanYor (CGMY) process are considered. It is known that American option prices satisfy a Partial IntegroDifferential Equation (PIDE) on a moving domain. These equations are reformulated as a Linear Complementarity Problem, ..."
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Cited by 7 (2 self)
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American put options written on an underlying stock following a CarrMadanGemanYor (CGMY) process are considered. It is known that American option prices satisfy a Partial IntegroDifferential Equation (PIDE) on a moving domain. These equations are reformulated as a Linear Complementarity Problem, and solved iteratively by an implicitexplicit type of iteration based on a convenient splitting of the IntegroDifferential operator. The solution to the discrete complementarity problems is found by the BrennanSchwartz algorithm and computations are accelerated by the Fast Fourier Transform. The method is illustrated throughout a series of numerical 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 6 (1 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