On smoothing of the Crank–Nicolson scheme and higher order schemes for pricing barrier options

Wade, Bruce A.; Khaliq, Abdul; Yousuf, M.; Vigo‐Aguiar, Jesús; Deininger, Rolf A.

doi:10.1016/j.cam.2006.04.034

Cited by 52 publications

(32 citation statements)

References 12 publications

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“…For such a big number of monitoring dates and close barriers obtaining quickly high order accurate results for the option value is not a trivial task for values of the underlying asset close or at the barriers, [21]. The computational time is more than satisfactory, see Table 2.…”

Section: Numerical Resultsmentioning

confidence: 97%

“…We have compared the results with those obtained by other numerical methods in Finance such as the Monte Carlo simulations, 10 trinomial trees of Cheuk and Vorst, [9], the Crank-Nicolson method applied in [11,21].…”

Section: Numerical Resultsmentioning

confidence: 97%

“…Formally, there are three examples respectively when the two barriers are L = 95 and U = 110, as it is explored in [21], when L = 95 and U = 125 used in [20,22], and the third case is when L = 95 and U = 140, [13].…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The second relation (20) of Theorem 3.1 is proved in an analogous way using formulas (11) and (12), and presenting K = U + (K − U) in Eq. (21).…”

Section: Algorithm For Numerical Valuationmentioning

confidence: 99%

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Numerical valuation of discrete double barrier options

Milev

Tagliani

2010

Journal of Computational and Applied Mathematics

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a b s t r a c tIn the present paper we explore the problem for pricing discrete barrier options utilizing the Black-Scholes model for the random movement of the asset price. We postulate the problem as a path integral calculation by choosing approach that is similar to the quadrature method. Thus, the problem is reduced to the estimation of a multi-dimensional integral whose dimension corresponds to the number of the monitoring dates.We propose a fast and accurate numerical algorithm for its valuation. Our results for pricing discretely monitored one and double barrier options are in agreement with those obtained by other numerical and analytical methods in Finance and literature. A desired level of accuracy is very fast achieved for values of the underlying asset close to the strike price or the barriers.The method has a simple computer implementation and it permits observing the entire life of the option.

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Section: Numerical Resultsmentioning

confidence: 97%

Section: Numerical Resultsmentioning

confidence: 97%

Section: Numerical Resultsmentioning

confidence: 99%

“…The second relation (20) of Theorem 3.1 is proved in an analogous way using formulas (11) and (12), and presenting K = U + (K − U) in Eq. (21).…”

Section: Algorithm For Numerical Valuationmentioning

confidence: 99%

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Numerical valuation of discrete double barrier options

Milev

Tagliani

2010

Journal of Computational and Applied Mathematics

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“…[15,22,23,32,34]. Motivated by the aforementioned articles, we approximate the matrix exponential function using the (0, 3)-Padé scheme at a few initial steps, usually between two and four.…”

Section: The Modified Time-stepping Schemesmentioning

confidence: 99%

Smoothing schemes for reaction-diffusion systems with nonsmooth data

Khaliq

Martín-Vaquero

Wade

et al. 2009

Journal of Computational and Applied Mathematics

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Cox and Matthews [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002) 430-455] developed a class of Exponential Time Differencing Runge-Kutta schemes (ETDRK) for nonlinear parabolic equations; Kassam and Trefethen [A.K. Kassam, Ll. N. Trefethen, Fourth-order time stepping for stiff pdes, SIAM J. Sci. Comput. 26 (2005) 1214-1233] have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth-order (ETDRK4) scheme. They use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials, but this approach creates new difficulties in choosing and evaluating the contour for larger problems. Neither treatment addresses problems with nonsmooth data, where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully. Such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications, including computational chemistry and financial engineering. We introduce a new version of the fourth-order Cox-Matthews, Kassam-Trefethen ETDRK4 scheme designed to eliminate the remaining computational difficulties. This new scheme utilizes an exponential time differencing Runge-Kutta ETDRK scheme using a diagonal Padé approximation of matrix exponential functions, while to deal with the problem of nonsmooth data we use several steps of an ETDRK scheme using a sub-diagonal Padé formula. The new algorithm improves computational efficiency with respect to evaluation of the high degree polynomial functions of matrices, having an advantage of splitting the matrix polynomial inversion problem into a sum of linear problems that can be solved in parallel. In this approach it is only required that several backward Euler linear problems be solved, in serial or parallel. Numerical experiments are described to support the new scheme.

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Approximation of single‐barrier options partial differential equations using feed‐forward neural network

Umeorah

Clément

2022

Appl Stoch Models Bus & Ind

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Artificial neural networks are generally employed in the numerical solution of differential equation problems. In this article, we propose an approach that deals with the combination of the feed-forward neural network method and the optimization technique in solving the partial differential equation arising from the valuation of barrier options. The methodology entails transforming the extended Black-Scholes partial differential equations (PDE), which defines a barrier option, into a constrained optimization problem, and then proposing a trial solution that reduces the differential equation problem to an unconstrained one. This trial function consists of the adjustable and non-adjustable neural network parameters. We design it to be differentiable, analytic, and satisfy the initial and boundary conditions of the corresponding option pricing PDE. We compare the corresponding option values to the Monte-Carlo simulated values, Crank-Nicolson finite-difference values and the exact Black-Scholes prices.Numerical results presented in this research show that neural networks can sufficiently solve PDE-related problems with sufficient precision and accuracy. Furthermore, they can be applied in the fast and accurate valuation of financial derivatives without closed analytic forms.

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On smoothing of the Crank–Nicolson scheme and higher order schemes for pricing barrier options

Cited by 52 publications

References 12 publications

Numerical valuation of discrete double barrier options

Numerical valuation of discrete double barrier options

Smoothing schemes for reaction-diffusion systems with nonsmooth data

Approximation of single‐barrier options partial differential equations using feed‐forward neural network

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