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54
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 31 (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.
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 24 (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 17 (4 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.
Discrete Asian Barrier Options
, 1998
"... . A partial differential equation method based on using auxiliary variables is described for pricing discretely monitored Asian options with or without barrier features. The barrier provisions can be applied to either the underlying asset or to the average. They may also be of either instantaneous o ..."
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Cited by 14 (4 self)
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. A partial differential equation method based on using auxiliary variables is described for pricing discretely monitored Asian options with or without barrier features. The barrier provisions can be applied to either the underlying asset or to the average. They may also be of either instantaneous or delayed effect (i.e. Parisian style). Numerical examples demonstrate that this method can be used for pricing floating strike, fixed strike, American, or European options. In addition, examples are provided which indicate that an upstream biased quadratic interpolation is superior to linear interpolation for handling the jump conditions at observation dates. Moreover, it is shown that defining the auxiliary variable as the average rather than the running sum is more rapidly convergent for AmericanAsian options. Keywords: Asian options, Barrier options, Parisian options, PDE option pricing Running Title: Discrete Asian Barrier Options Acknowledgment: This work was supported by the Nation...
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.
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 14 (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 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.
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 12 (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.
Shout Options: A Framework For Pricing Contracts Which Can Be Modified By The Investor
 J. Comp. Appl. Math
, 1999
"... A shout option may be broadly defined as a financial contract which can be modified by its holder according to specified rules. In a simple example, the holder could have the right to set the strike of an option equal to the current value of the underlying asset. In such a case, the holder effective ..."
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Cited by 12 (9 self)
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A shout option may be broadly defined as a financial contract which can be modified by its holder according to specified rules. In a simple example, the holder could have the right to set the strike of an option equal to the current value of the underlying asset. In such a case, the holder effectively has the right to select when to take ownership of an atthemoney option. More generally, the holder could have multiple rights along these lines, in some cases with a limit placed on the number of rights which may be exercised within a given time period (e.g. four times per year). The value of these types of contracts can be estimated by solving a system of interdependent linear complementarity problems. This paper describes a general framework for the valuation of complex types of shout options. Numerical issues related to interpolation and choice of timestepping method are considered in detail. Some illustrative examples are provided.