General properties of solutions to inhomogeneous Black–Scholes equations with discontinuous maturity payoffs

Hyong-Chol, O; Jo, Jong-Jun; Kim, Ji-Sok

doi:10.1016/j.jde.2015.08.036

Cited by 7 publications

(2 citation statements)

References 24 publications

(12 reference statements)

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“…Over the last decades, not only financial engineers but also mathematicians have paid special attention to the valuation of derivative financial instruments. Indeed, since being introduced by Fischer Black and Myron Scholes in 1973, the Black-Scholes model based on partial differential equation has been widely employed in modern mathematical finance and become a common-sense approach for pricing options as well as many other financial securities [8]. This mathematical model was derived from the principle that yielding profits from making portfolios of both short and long positions in options as well as their underlying stocks should not be possible, if option prices are rightly priced in the market [2].…”

Section: Introductionmentioning

confidence: 99%

A finite-difference scheme for initial boundary value problem of the Gamma equation in the pricing of financial derivatives

Hieu¹,

Hanh²,

Thanh³

2020

J. Math. Computer Sci.

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In the article, we consider the initial boundary value problem for the Gamma equation, which can be derived by transforming the nonlinear Black-Scholes equation for option price into a quasilinear parabolic equation for the second derivative of the option price. We develop unconditionally monotone finite-difference schemes of second-order of local approximation on uniform grids for the initial boundary value problem for the Gamma equation. Two-side estimates of the solution of the scheme are established. By means of regularization principle, the previous results are generalized for construction of unconditionally monotone finite-difference scheme (the maximum principle is satisfied without constraints on relations between the coefficients and grid parameters) of the second-order of approximation on uniform grids for this equation. With the help of difference maximum principle, the two-side estimates for difference solution are obtained at the arbitrary non-sign-constant input data of the problem. A priori estimate in the maximum norm C is proved. It is interesting to note that the proven two-side estimates for difference solution are fully consistent with the differential problem, and the maximal and minimal values of the difference solution do not depend on the diffusion and convection coefficients. Computational experiments, confirming the theoretical conclusions, are given.

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Section: Introductionmentioning

confidence: 99%

A finite-difference scheme for initial boundary value problem of the Gamma equation in the pricing of financial derivatives

Hieu¹,

Hanh²,

Thanh³

2020

J. Math. Computer Sci.

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show abstract

“…Email address: hieulm@due.edu.vn https//doi.org/ 10.25073/2588-1124/vnumap. 4364 Black and Myron Scholes in 1973, the Black-Scholes model based on partial differential equation has been widely employed in modern mathematical finance and become a common-sense approach for pricing options as well as many other financial securities [1]. This mathematical model was derived from the principle that yielding profits from making portfolios of both short and long positions in options as well as their underlying stocks should not be possible, if option prices are rightly priced in the market [2].…”

Section: Introductionmentioning

confidence: 99%

Finite-Difference Scheme for Initial Boundary Value Problems in Financial Mathematics

Hieu¹,

Hanh²,

Thanh³

2019

MaP

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We develop unconditionally monotone nite-difference schemes of second-order of local approxi-mation on uniform grids for the initial boundary problem value for the Gamma equation. Two-sideestimates of the solution of the scheme are established. We consider the initial boundary valueproblem for the so called Gamma equation, which can be derived by transforming the nonlinearBlack-Scholes equation for option price into a quasilinear parabolic equation for the second derivativeof the option price, and for its exact solution the two-side estimates are obtained. By means of regu-larization principle, the previous results are generalized for construction of unconditionally monotonenite-difference scheme (the maximum principle is satised without constraints on relations betweenthe coeffcients and grid parameters) of second order of approximation on uniform grids for this equa-tion. With the help of difference maximum principle, the two-side estimates for difference solutionare obtained at the arbitrary non-sign-constant input data of the problem. A priori estimate in themaximum norm C is proved. It is interesting to note that the proven two-side estimates for differ-ence solution are fully consistent with differential problem, and the maximal and minimal values ofthe difference solution do not depend on the diffusion and convection coeffcients. Computationalexperiments, conrming the theoretical conclusions, are given.

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Design of interval observers and controls for PDEs using finite-element approximations

et al. 2018

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Synthesis of interval state estimators is investigated for the systems described by a class of parabolic Partial Differential Equations (PDEs). First, a finite-element approximation of a PDE is constructed and the design of an interval observer for the derived ordinary differential equation is given. Second, the interval inclusion of the state function of the PDE is calculated using the error estimates of the finite-element approximation. Finally, the obtained interval estimates are used to design a dynamic output stabilizing control. The results are illustrated by numerical experiments with an academic example and the Black-Scholes model of financial market.

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General properties of solutions to inhomogeneous Black–Scholes equations with discontinuous maturity payoffs

Cited by 7 publications

References 24 publications

A finite-difference scheme for initial boundary value problem of the Gamma equation in the pricing of financial derivatives

A finite-difference scheme for initial boundary value problem of the Gamma equation in the pricing of financial derivatives

Finite-Difference Scheme for Initial Boundary Value Problems in Financial Mathematics

Design of interval observers and controls for PDEs using finite-element approximations

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