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Convergence remedies for nonsmooth payoffs in option pricing
 J. Computational Finance
"... Discontinuities in the payoff function (or its derivatives) can cause inaccuracies for numerical schemes when pricing financial contracts. In particular, large errors may occur in the estimation of the hedging parameters. Three methods of dealing with discontinuities are discussed in this paper: ave ..."
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Cited by 17 (1 self)
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Discontinuities in the payoff function (or its derivatives) can cause inaccuracies for numerical schemes when pricing financial contracts. In particular, large errors may occur in the estimation of the hedging parameters. Three methods of dealing with discontinuities are discussed in this paper: averaging the initial data, shifting the grid, and a projection method. By themselves, these techniques are not sufficient to restore expected behaviour. However, when combined with a special timestepping method, high accuracy is achieved. Examples are provided for one and two factor option pricing problems. 1
On the rate of convergence of discretetime contingent claims
 Mathematical Finance
"... This paper characterizes the rate of convergence of discretetime multinomial option prices. We show that the rate of convergence depends on the smoothness of option payoff functions, and is much lower than commonly believed because option payoff functions are often of allornothing type and are no ..."
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Cited by 16 (0 self)
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This paper characterizes the rate of convergence of discretetime multinomial option prices. We show that the rate of convergence depends on the smoothness of option payoff functions, and is much lower than commonly believed because option payoff functions are often of allornothing type and are not continuously differentiable. To improve the accuracy, we propose two simple methods, an adjustment of the discretetime solution prior to maturity and smoothing of the payoff function, which yield solutions that converge to their continuoustime limit at the maximum possible rate enjoyed by smooth payoff functions. We also propose an intuitive approach that systematically derives multinomial models by matching the moments of a normal distribution. A highly accurate trinomial model also is provided for interest rate derivatives. Numerical examples are carried out to show that the proposed methods yield fast and accurate results.
Convergence of spectral and finite difference methods for initialboundary value problems
 SIAM J. Scientific Computing
"... Abstract. The general theory of compatibility conditions for the differentiability of solutions to initialboundary value problems is well known. This paper introduces the application of that theory to numerical solutions of partial differential equations and its ramifications on the performance of ..."
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Cited by 6 (3 self)
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Abstract. The general theory of compatibility conditions for the differentiability of solutions to initialboundary value problems is well known. This paper introduces the application of that theory to numerical solutions of partial differential equations and its ramifications on the performance of highorder methods. Explicit application of boundary conditions (BCs) that are independent of the initial condition (IC) results in the compatibility conditions not being satisfied. Since this is the case in most science and engineering applications, it is shown that not only does the error in a spectral method, as measured in the maximum norm, converge algebraically, but the accuracy of finite differences is also reduced. For the heat equation with a parabolic IC and Dirichlet BCs, we prove that the Fourier method converges quadratically in the neighborhood of t = 0 and the boundaries and quartically for large t when the firstorder compatibility conditions are violated. For the same problem, the Chebyshev method initially yields quartic convergence and exponential convergence for t>0. In contrast, the wave equation subject to the same conditions results in inferior convergence rates with all spectral methods yielding quadratic convergence for all t. These results naturally direct attention to finite difference methods that are also algebraically convergent. In the case of the wave equation, we prove that a secondorder finite difference method is reduced to 4/3order convergence and numerically show that a fourthorder finite difference scheme is apparently reduced to 3/2order. Finally, for the wave equation subject to general ICs and zero BCs, we give a conjecture on the error for a secondorder finite difference scheme, showing that an O(N −2 log N) convergence is possible.
From Finite Differences to Finite Elements  A short history of numerical analysis of partial differential equations
, 1999
"... This is an account of the history of numerical analysis of partial differential equations, starting with the 1928 paper of Courant, Friedrichs, and Lewy, and proceeding with the development of first finite difference and then finite element methods. The emphasis is on mathematical aspects such as ..."
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Cited by 5 (0 self)
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This is an account of the history of numerical analysis of partial differential equations, starting with the 1928 paper of Courant, Friedrichs, and Lewy, and proceeding with the development of first finite difference and then finite element methods. The emphasis is on mathematical aspects such as stability and convergence analysis.
Numerical Methods for Nonlinear Equations in Option Pricing
, 2003
"... This thesis explores numerical methods for solving nonlinear partial di#erential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and e#cient techniques that converge to the financially relevant solution for both one and two factor problems. To illust ..."
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Cited by 5 (0 self)
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This thesis explores numerical methods for solving nonlinear partial di#erential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and e#cient techniques that converge to the financially relevant solution for both one and two factor problems. To illustrate the underlying concepts, two nonlinear models are examined in detail: uncertain volatility and passport options.
Pricing Discrete European Barrier Options Using Lattice Random Walks
, 2002
"... This paper designs a numerical procedure to price discrete European barrier options in BlackScholes model. The pricing problem is divided in a series of initial value problems, one for each monitoring time. Each initial value problem is solved by replacing the driving Brownian motion by a latti ..."
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Cited by 1 (1 self)
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This paper designs a numerical procedure to price discrete European barrier options in BlackScholes model. The pricing problem is divided in a series of initial value problems, one for each monitoring time. Each initial value problem is solved by replacing the driving Brownian motion by a lattice random walk. Some results