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100
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 66 (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.
Quadratic Convergence For Valuing American Options Using A Penalty Method
 SIAM J. Sci. Comput
, 2002
"... . The convergence of a penalty method for solving the discrete regularized American option valuation problem is studied. Su#cient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution. These cond ..."
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Cited by 42 (4 self)
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. The convergence of a penalty method for solving the discrete regularized American option valuation problem is studied. Su#cient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution. These conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem. The e#ciency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the American constraint explicitly. Convergence rates are studied as the timestep and mesh size tend to zero. It is observed that an implicit treatment of the American constraint does not converge quadratically (as the timestep is reduced) if constant timesteps are used. A timestep selector is suggested which restores quadratic convergence. Key words. American option, penalty iteration, linear complementarity AMS subject classifications. 65M12, 65M60, 91B28 Revised: May 18, 2001 1.
A semiLagrangian approach for natural gas storage valuation and optimal operation
, 2006
"... The valuation of a gas storage facility is characterized as a stochastic control problem, resulting in a HamiltonJacobiBellman (HJB) equation. In this paper, we present a semiLagrangian method for solving the HJB equation for a typical gas storage valuation problem. The method is able to handle a ..."
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Cited by 30 (5 self)
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The valuation of a gas storage facility is characterized as a stochastic control problem, resulting in a HamiltonJacobiBellman (HJB) equation. In this paper, we present a semiLagrangian method for solving the HJB equation for a typical gas storage valuation problem. The method is able to handle a wide class of spot price models that exhibit meanreverting, seasonality dynamics and price jumps. We develop fully implicit and CrankNicolson timestepping schemes based on a semiLagrangian approach and prove the convergence of fully implicit timestepping to the viscosity solution of the HJB equation. We show that fully implicit timestepping is equivalent to a discrete control strategy, which allows for a convenient interpretation of the optimal controls. The semiLagrangian approach avoids the nonlinear iterations required by an implicit finite difference method without requiring additional cost. Numerical experiments are presented for several variants of the basic scheme.
Penalty and frontfixing methods for the numerical solution of American option problems.
, 2001
"... In this paper we introduce two methods for the efficient and accurate numerical solution of BlackScholes models of American options; A penalty method and a frontfixing scheme. In the penalty approach the free and moving boundary is removed by adding a small, and continuous penalty term to the Blac ..."
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Cited by 27 (0 self)
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In this paper we introduce two methods for the efficient and accurate numerical solution of BlackScholes models of American options; A penalty method and a frontfixing scheme. In the penalty approach the free and moving boundary is removed by adding a small, and continuous penalty term to the BlackScholes equation. Then the problem can be solved on a fixed domain and thus removing the difficulties associated with a moving boundary. To gain insight in the accuracy of the method, we apply it to similar situations where the approximate solutions can be compared with analytical solutions. For explicit, semiimplicit and fullyimplicit numerical schemes, we prove that the numerical option values generated by the penalty method mimics the basic properties of the analytical solution of the American option problem. In the frontfixing method we apply a change of variables to transform the American put problem into a nonlinear parabolic differential equation posed on a fixed domain. We propose both an implicit and an explicit scheme for solving this latter equation. Finally, the performance of the schemes are illustrated through a series of numerical experiments.
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 multigrid for linear complementarity problems with application to American–style options
 ELECTRON. TRANS. NUMER. ANAL
, 2003
"... We discuss a nonlinear multigrid method for a linear complementarity problem. The convergence is improved by a recombination of iterants. The problem under consideration deals with option pricing from mathematical finance. Linear complementarity problems arise from socalled Americanstyle options. ..."
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Cited by 21 (1 self)
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We discuss a nonlinear multigrid method for a linear complementarity problem. The convergence is improved by a recombination of iterants. The problem under consideration deals with option pricing from mathematical finance. Linear complementarity problems arise from socalled Americanstyle options. A 2D convectiondiffusion type operator is discretized with the help of second order upwind discretizations. The properties of smoothers are analyzed with Fourier twogrid analysis. Numerical solutions obtained for the option pricing problem are compared with reference results.
Error estimates for approximate solutions to Bellman equations associated with controlled jumpdiffusions
 NUMER. MATH
, 2006
"... We derive error estimates for approximate (viscosity) solutions of Bellman equations associated to controlled jumpdiffusion processes, which are fully nonlinear integropartial differential equations. Two main results are obtained: (i) error bounds for a class of monotone approximation schemes, w ..."
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Cited by 21 (5 self)
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We derive error estimates for approximate (viscosity) solutions of Bellman equations associated to controlled jumpdiffusion processes, which are fully nonlinear integropartial differential equations. Two main results are obtained: (i) error bounds for a class of monotone approximation schemes, which under some assumptions includes finite difference schemes, and (ii) bounds on the error induced when the original Lévy measure is replaced by a finite measure with compact support, an approximation process that is commonly used when designing numerial schemes for integropartial differential equations. Our proofs use and extend techniques introduced by Krylov and BarlesJakobsen.
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.
Numerical valuation of options with jumps in the underlying
, 2005
"... A jumpdiffusion model for a singleasset market is considered. Under this assumption the value of a European contingency claim satisfies a general partial integrodifferential equation (PIDE). The equation is localized and discretized in space using finite differences and finite elements and in tim ..."
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Cited by 19 (3 self)
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A jumpdiffusion model for a singleasset market is considered. Under this assumption the value of a European contingency claim satisfies a general partial integrodifferential equation (PIDE). The equation is localized and discretized in space using finite differences and finite elements and in time by the second order backward differentiation formula (BDF2). The resulting system is solved by an iterative method based on a simple splitting of the matrix. Using the fast Fourier transform, the amount of work per iteration may be reduced to O(n log 2 n) and only O(n) entries need to be stored for each time level. Numerical results showing the quadratic convergence of the methods are given for Merton’s model and Kou’s model.