The numerical approximation of nonlinear Black–Scholes model for exotic path-dependent American options with transaction cost

Yousuf, M.; Kleefeld, B.

doi:10.1080/00207160.2012.688115

Cited by 38 publications

(25 citation statements)

References 31 publications

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“…Using Duhamel's principle, an approach similar to Kleefeld et al and Yousuf et al , we can show that

u_{normalα} (t_{n}), 1 \leq α \leq m_{o}, 0 < n \leq N

satisfy the following recurrent formula:

u_{normalα} (t_{n + 1}) = e^{- k A_{normalα}} u_{normalα} (t_{n}) + \int_{0}^{k} e^{- A_{α} false(k - normalτ false)} F_{normalα} (u_{1} (t_{n} + τ), u_{2} (t_{n} + τ), \dots, u_{m_{o}} (t_{n} + τ), t_{n} + τ) d τ

The integral in (2.6) can be approximated by a class of exponential time differencing (ETD) numerical schemes, see for example and references therein. Denoting the approximation to

u_{normalα} (t_{n})

u_{α, n}

and the approximation to

a_{normalα} (t_{n})

a_{α, n}

, the second order exponential time differencing Runge–Kutta semidiscrete scheme is given by ,

u_{α,}

…”

Section: Partial Integral Differential Equationsmentioning

confidence: 99%

“…To resolve this difficulty, an efficient version of the methods is developed using partial fraction decomposition technique. This technique not only avoid these difficulties but also increase the computational efficiency of the method, see Yousuf et al for more details.…”

Section: Partial Integral Differential Equationsmentioning

confidence: 99%

“…A numerical scheme which is not L –stable, may produce unwanted spurious oscillations in case of nonsmooth initial data unless the restriction

Δ t \leq {(Δ x)}^{2}

is applied. Yousuf et al addressed this problem by proposing the use of a Padé (0, 2) rational approximation and developed an L –stable method which does not require this restriction. The resulting linear systems were solved using block–tridiagonal solver.…”

Section: Introductionmentioning

confidence: 99%

“…The pricing model of American options is a linear complementary problem (LCP) , on an unbounded domain, for which closed–form solution is not known. The linear complementarity problem can be transformed into a nonlinear parabolic partial differential equation by adding a penalty term

\frac{1}{ε} \max {(g (S) - V (S, t)), 0}

proposed by Zvan et al , where g ( S ) is the initial condition or the payoff function.…”

Section: Introductionmentioning

confidence: 99%

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Numerical solution of systems of partial integral differential equations with application to pricing options

Yousuf

2018

Numerical Methods Partial

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We introduce and analyze a strongly stable numerical method designed to yield good performance under challenging conditions of irregular or mismatched initial data for solving systems of coupled partial integral differential equations (PIDEs). Spatial derivatives are approximated using second order central difference approximations by treating the mixed derivative terms in a special way. The integral operators are approximated using one and two–dimensional trapezoidal rule on an equidistant grid. Computational complexity of the method for solving large systems of PIDEs is discussed. A detailed treatment for the consistency, stability, and convergence of the proposed method is provided. Two asset American option under regime–switching with jump–diffusion model when solved using a penalty term, leads to a system of two dimensional PIDEs with mixed derivatives. This model involves double probability density function which brings more challenges to the numerical solution in already a complicated partial integral differential equation. The complexity of the dense jump probability generator, the nonlinear penalty term and the regime–switching terms are treated efficiently, while maintaining the stability and convergence of the method. The impact of the jump intensity and other parameters is shown in the graphs. Numerical experiments are performed to demonstrated efficiency, accuracy, and reliability of the proposed approach.

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“…Using Duhamel's principle, an approach similar to Kleefeld et al and Yousuf et al , we can show that

u_{normalα} (t_{n}), 1 \leq α \leq m_{o}, 0 < n \leq N

satisfy the following recurrent formula:

u_{normalα} (t_{n + 1}) = e^{- k A_{normalα}} u_{normalα} (t_{n}) + \int_{0}^{k} e^{- A_{α} false(k - normalτ false)} F_{normalα} (u_{1} (t_{n} + τ), u_{2} (t_{n} + τ), \dots, u_{m_{o}} (t_{n} + τ), t_{n} + τ) d τ

The integral in (2.6) can be approximated by a class of exponential time differencing (ETD) numerical schemes, see for example and references therein. Denoting the approximation to

u_{normalα} (t_{n})

u_{α, n}

and the approximation to

a_{normalα} (t_{n})

a_{α, n}

, the second order exponential time differencing Runge–Kutta semidiscrete scheme is given by ,

u_{α,}

…”

Section: Partial Integral Differential Equationsmentioning

confidence: 99%

Section: Partial Integral Differential Equationsmentioning

confidence: 99%

“…A numerical scheme which is not L –stable, may produce unwanted spurious oscillations in case of nonsmooth initial data unless the restriction

Δ t \leq {(Δ x)}^{2}

Section: Introductionmentioning

confidence: 99%

\frac{1}{ε} \max {(g (S) - V (S, t)), 0}

proposed by Zvan et al , where g ( S ) is the initial condition or the payoff function.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical solution of systems of partial integral differential equations with application to pricing options

Yousuf

2018

Numerical Methods Partial

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show abstract

“…Analytical and numerical methods have been used for European and American option pricing such as finite difference (and the others mesh methods) [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the binomial/triniomial trees [20][21][22][23], nevertheless some authors have also proposed the use of meshfree algorithms based on radial basis functions [24][25][26][27][28] and on quasi radial basis functions [29]. To solve the dynamical systems, analytical solutions are the first choice, but are applicable to sample cases, not for American options.…”

Section: Introductionmentioning

confidence: 99%

Pricing European and American Options Using a Very Fast and Accurate Scheme: The Meshless Local Petrov–Galerkin Method

Rad

Parand

Abbasbandy

2015

Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci.

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In this paper, a method for the numerical pricing of American and European options under the Black-Scholes model is introduced. This approach is meshless local Petrov-Galerkin (MLPG) based on local weak form and the moving least squares approximations. The MLPG offers advantages over conventional and strong meshless methods of radial basis function approximations. In this paper, the American option which is a free boundary problem, is reduced to a fixed boundary problem using a Richardson extrapolation technique. Then a time stepping method is employed for the time derivative. Finally numerical results are presented in two test cases. These experiments show that MLPG approach is accurate and fast, and performs significantly better compared to the conventional radial basis functions collocation methods.

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Bayesian Support Vector Machines for Economic Modeling: Application to Option Pricing

Marwala

2013

Advanced Information and Knowledge Processing

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The numerical approximation of nonlinear Black–Scholes model for exotic path-dependent American options with transaction cost

Cited by 38 publications

References 31 publications

Numerical solution of systems of partial integral differential equations with application to pricing options

Numerical solution of systems of partial integral differential equations with application to pricing options

Pricing European and American Options Using a Very Fast and Accurate Scheme: The Meshless Local Petrov–Galerkin Method

Bayesian Support Vector Machines for Economic Modeling: Application to Option Pricing

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