Computers and Mathematics with Applications, ISSN 0898-1221, 04/2018, Volume 75, Issue 8, pp. 2874 - 2887

In this paper a time-fractional Black–Scholes equation is examined. We transform the initial value problem into an equivalent integral–differential equation...

Fractional differential equation | Singularity | Adapted mesh | Black–Scholes equation | Option valuation | AMERICAN | MATHEMATICS, APPLIED | FINITE-VOLUME METHOD | SPLINE COLLOCATION | ERROR ANALYSIS | Black-Scholes equation | DOUBLE-BARRIER OPTIONS | Analysis | Differential equations

Fractional differential equation | Singularity | Adapted mesh | Black–Scholes equation | Option valuation | AMERICAN | MATHEMATICS, APPLIED | FINITE-VOLUME METHOD | SPLINE COLLOCATION | ERROR ANALYSIS | Black-Scholes equation | DOUBLE-BARRIER OPTIONS | Analysis | Differential equations

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 04/2017, Volume 73, Issue 7, pp. 1566 - 1575

The purpose of this work is to apply the results developed by Chemin and David (2013, 2015), to the Black–Scholes equation. This latter equation being directly...

Control | Black–Scholes equation | Shape parameters | Black-Scholes equation | MATHEMATICS, APPLIED | Mathematics - Analysis of PDEs | Analysis of PDEs | Mathematics | Optimization and Control

Control | Black–Scholes equation | Shape parameters | Black-Scholes equation | MATHEMATICS, APPLIED | Mathematics - Analysis of PDEs | Analysis of PDEs | Mathematics | Optimization and Control

Journal Article

Annals of Operations Research, ISSN 0254-5330, 10/2019, Volume 281, Issue 1, pp. 229 - 251

In this article, two general results are provided about the multidimensional Black–Scholes partial differential equation: its fundamental solution is derived,...

Black–Scholes multidimensional equation | Mixed derivative | Multivariate normal distribution | Parabolic | Heat equation | Theory of Computation | Dimension | Multiasset option | Business and Management | Option on the maximum | Double barrier | Operations Research/Decision Theory | Combinatorics | Gaussian distribution | Research | Differential equations, Partial | Mathematical research | Operators (mathematics) | Studies | Operations research | Partial differential equations | Multivariate analysis | Normal distribution

Black–Scholes multidimensional equation | Mixed derivative | Multivariate normal distribution | Parabolic | Heat equation | Theory of Computation | Dimension | Multiasset option | Business and Management | Option on the maximum | Double barrier | Operations Research/Decision Theory | Combinatorics | Gaussian distribution | Research | Differential equations, Partial | Mathematical research | Operators (mathematics) | Studies | Operations research | Partial differential equations | Multivariate analysis | Normal distribution

Journal Article

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 09/2018, Volume 62, pp. 164 - 173

In this paper, we investigate the non-linear Black–Scholes equation: and show that the one can be reduced to the equation by an appropriate point...

Symmetry reduction | Black–Scholes equation | Exact solutions | LIE-ALGEBRAS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | PHYSICS, FLUIDS & PLASMAS | Black-Scholes equation | OPTIONS | PHYSICS, MATHEMATICAL | Algebra | Quantitative Finance - Mathematical Finance

Symmetry reduction | Black–Scholes equation | Exact solutions | LIE-ALGEBRAS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | PHYSICS, FLUIDS & PLASMAS | Black-Scholes equation | OPTIONS | PHYSICS, MATHEMATICAL | Algebra | Quantitative Finance - Mathematical Finance

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 01/2020, Volume 363, pp. 464 - 484

This paper presents a high order numerical method based on a uniform mesh to obtain a highly accurate result for generalized Black–Scholes equation arising in...

European call option | Stability analysis | Black–Scholes equation | Compact finite difference method | Convergence analysis | Crank–Nicolson method | SCHEME | MATHEMATICS, APPLIED | SPLINE COLLOCATION METHOD | Crank-Nicolson method | STABILITY | Black-Scholes equation | CONVERGENCE | Financial markets | Analysis | Methods | Differential equations

European call option | Stability analysis | Black–Scholes equation | Compact finite difference method | Convergence analysis | Crank–Nicolson method | SCHEME | MATHEMATICS, APPLIED | SPLINE COLLOCATION METHOD | Crank-Nicolson method | STABILITY | Black-Scholes equation | CONVERGENCE | Financial markets | Analysis | Methods | Differential equations

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 06/2016, Volume 40, Issue 11-12, pp. 5819 - 5834

In recent years, the Finite Moment Log Stable(FMLS), KoBoL and CGMY models, which follow a jump process or a Lévy process, have become the most popular...

Numerical simulation | European double barrier option | Fractional Black–Scholes model | Stability and convergence | Tempered fractional derivative | Fast bi-conjugate gradient stabilized method | Fractional Black-Scholes model | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | MODELS | CONVERGENCE | DIFFUSION EQUATION | Numerical analysis | Analysis | Models | Accuracy | Computer simulation | Movements | Barriers | Mathematical models | Raw materials | Computational efficiency | Convergence

Numerical simulation | European double barrier option | Fractional Black–Scholes model | Stability and convergence | Tempered fractional derivative | Fast bi-conjugate gradient stabilized method | Fractional Black-Scholes model | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | MODELS | CONVERGENCE | DIFFUSION EQUATION | Numerical analysis | Analysis | Models | Accuracy | Computer simulation | Movements | Barriers | Mathematical models | Raw materials | Computational efficiency | Convergence

Journal Article

Moscow University Physics Bulletin, ISSN 0027-1349, 5/2016, Volume 71, Issue 3, pp. 237 - 244

Operational solutions to fractional-order ordinary differential equations and to partial differential equations of the Black–Scholes and of Fourier heat...

Black–Scholes equation | inverse operator | Theoretical, Mathematical and Computational Physics | heat conduction | Laguerre and Hermite polynomials | Physics | Schrödinger equation | MATRIX | PHYSICS, MULTIDISCIPLINARY | PLANAR UNDULATOR | Black-Scholes equation | ALGEBRAIC EQUATIONS | EXPONENTIAL FORM | POLYNOMIALS | CONSTANT MAGNETIC-FIELD | Schrodinger equation | REPRESENTATION-THEORY | DIFFUSION-EQUATIONS | HIGH HARMONIC-GENERATION | Quantum theory | Differential equations

Black–Scholes equation | inverse operator | Theoretical, Mathematical and Computational Physics | heat conduction | Laguerre and Hermite polynomials | Physics | Schrödinger equation | MATRIX | PHYSICS, MULTIDISCIPLINARY | PLANAR UNDULATOR | Black-Scholes equation | ALGEBRAIC EQUATIONS | EXPONENTIAL FORM | POLYNOMIALS | CONSTANT MAGNETIC-FIELD | Schrodinger equation | REPRESENTATION-THEORY | DIFFUSION-EQUATIONS | HIGH HARMONIC-GENERATION | Quantum theory | Differential equations

Journal Article

European Journal of Operational Research, ISSN 0377-2217, 07/2016, Volume 252, Issue 1, pp. 183 - 190

In this paper, we develop a fast and accurate numerical method for pricing of the three-asset equity-linked securities options. The option pricing model is...

Non–uniform grid | Equity–linked securities | Option pricing | Operator splitting method | Black–Scholes partial differential equation | Black-Scholes partial differential equation | Non-uniform grid | Equity-linked securities | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | STOCHASTIC VOLATILITY | OPTIONS | GREEKS | Analysis | Pricing | Monte Carlo method | Monte Carlo methods | Numerical analysis | Computer simulation | Mathematical analysis | Mathematical models | Computational efficiency | Finite difference method

Non–uniform grid | Equity–linked securities | Option pricing | Operator splitting method | Black–Scholes partial differential equation | Black-Scholes partial differential equation | Non-uniform grid | Equity-linked securities | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | STOCHASTIC VOLATILITY | OPTIONS | GREEKS | Analysis | Pricing | Monte Carlo method | Monte Carlo methods | Numerical analysis | Computer simulation | Mathematical analysis | Mathematical models | Computational efficiency | Finite difference method

Journal Article

Computers & Mathematics with Applications, ISSN 0898-1221, 04/2017, Volume 73, Issue 7, p. 1566

The purpose of this work is to apply the results developed by Chemin and David (2013, 2015), to the Black-Scholes equation. This latter equation being directly...

Studies | Black-Scholes equation | Stochastic models | Heat conductivity

Studies | Black-Scholes equation | Stochastic models | Heat conductivity

Journal Article

Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, 10/2019, Volume 532

Merton has proposed a model for contingent claims on a firm as an option on the firms value, and is based on a generalization of the Black–Scholes stochastic...

Option pricing models | Oscillator Hamiltonian | Merton's equation | Put–call parity | Beyond Black–Scholes | PHYSICS, MULTIDISCIPLINARY | Put-call parity | Beyond Black-Scholes

Option pricing models | Oscillator Hamiltonian | Merton's equation | Put–call parity | Beyond Black–Scholes | PHYSICS, MULTIDISCIPLINARY | Put-call parity | Beyond Black-Scholes

Journal Article

2011, ISBN 9780465028153, xiv, 298

Book

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 02/2017, Volume 311, pp. 11 - 37

Our aim in this paper is to approximate the price of an American call option written on a dividend-paying stock close to expiry using an asymptotic analytic...

Transparent boundary conditions | Repeated integral of the complementary error functions | Black–Scholes equation | American call option | Poincaré asymptotic expansion | Free boundary value problem | ARTIFICIAL BOUNDARY METHOD | MATHEMATICS, APPLIED | APPROXIMATIONS | Black-Scholes equation | VALUATION | MODEL | Poincare asymptotic expansion | DIVIDENDS | Numerical analysis | Pricing | Differential equations | Asymptotic expansions | Equivalence | Asymptotic properties | Mathematical analysis | Exact solutions | Mathematical models

Transparent boundary conditions | Repeated integral of the complementary error functions | Black–Scholes equation | American call option | Poincaré asymptotic expansion | Free boundary value problem | ARTIFICIAL BOUNDARY METHOD | MATHEMATICS, APPLIED | APPROXIMATIONS | Black-Scholes equation | VALUATION | MODEL | Poincare asymptotic expansion | DIVIDENDS | Numerical analysis | Pricing | Differential equations | Asymptotic expansions | Equivalence | Asymptotic properties | Mathematical analysis | Exact solutions | Mathematical models

Journal Article

13.
Full Text
The asymptotic behavior of the solutions of the Black–Scholes equation as volatility σ→0

Computers and Mathematics with Applications, ISSN 0898-1221, 2019, Volume 78, Issue 3, pp. 1037 - 1050

The aim of this paper is to explore the asymptotic properties of the solutions to the Black–Scholes equation. This paper focuses on the basic properties of...

Volatility | Black–Scholes equation | Asymptotic behavior | Asymptotic properties | Formulas (mathematics)

Volatility | Black–Scholes equation | Asymptotic behavior | Asymptotic properties | Formulas (mathematics)

Journal Article

Journal of Difference Equations and Applications, ISSN 1023-6198, 07/2015, Volume 21, Issue 7, pp. 547 - 552

We construct the exact finite difference representation for a second-order, linear, Cauchy-Euler ordinary differential equation. This result is then used to...

34A05 | sub-equations | 91G60 | 39A12 | 65N06 | Black-Scholes equation | exact finite difference schemes | 65L12 | Cauchy-Euler equation | Cauchy–Euler equation | Black–Scholes equation | MATHEMATICS, APPLIED

34A05 | sub-equations | 91G60 | 39A12 | 65N06 | Black-Scholes equation | exact finite difference schemes | 65L12 | Cauchy-Euler equation | Cauchy–Euler equation | Black–Scholes equation | MATHEMATICS, APPLIED

Journal Article

Journal of Nonlinear Science, ISSN 0938-8974, 8/2019, Volume 29, Issue 4, pp. 1563 - 1619

High-dimensional partial differential equations (PDEs) appear in a number of models from the financial industry, such as in derivative pricing models, credit...

Knightian uncertainty | Theoretical, Mathematical and Computational Physics | Classical Mechanics | Economic Theory/Quantitative Economics/Mathematical Methods | Mathematics | Numerical method | Black–Scholes–Barenblatt equation | Nonlinear expectation | Deep learning | Hamiltonian–Jacobi–Bellman equation | Analysis | HJB equation | Mathematical and Computational Engineering | Second-order backward stochastic differential equation | 2BSDE | G -Brownian motion | G-Brownian motion | BROWNIAN-MOTION | MATHEMATICS, APPLIED | Black-Scholes-Barenblatt equation | ORDER NUMERICAL SCHEMES | Hamiltonian-Jacobi-Bellman equation | BSDES | SIMULATION | PHYSICS, MATHEMATICAL | DISCRETIZATION | MECHANICS | DEEP NEURAL-NETWORKS | DISCRETE-TIME APPROXIMATION | Investment analysis | Big data | Algorithms | Differential equations | Machine learning

Knightian uncertainty | Theoretical, Mathematical and Computational Physics | Classical Mechanics | Economic Theory/Quantitative Economics/Mathematical Methods | Mathematics | Numerical method | Black–Scholes–Barenblatt equation | Nonlinear expectation | Deep learning | Hamiltonian–Jacobi–Bellman equation | Analysis | HJB equation | Mathematical and Computational Engineering | Second-order backward stochastic differential equation | 2BSDE | G -Brownian motion | G-Brownian motion | BROWNIAN-MOTION | MATHEMATICS, APPLIED | Black-Scholes-Barenblatt equation | ORDER NUMERICAL SCHEMES | Hamiltonian-Jacobi-Bellman equation | BSDES | SIMULATION | PHYSICS, MATHEMATICAL | DISCRETIZATION | MECHANICS | DEEP NEURAL-NETWORKS | DISCRETE-TIME APPROXIMATION | Investment analysis | Big data | Algorithms | Differential equations | Machine learning

Journal Article

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 07/2014, Volume 19, Issue 7, pp. 2200 - 2211

The complete group classification of a generalization of the Black–Scholes–Merton model is carried out by making use of the underlying equivalence and...

Lie point symmetry | Group classification | Computer assisted research | Black–Scholes–Merton equation | Black-Scholes-Merton equation | MATHEMATICS, APPLIED | PHYSICS, FLUIDS & PLASMAS | SYMMETRY ANALYSIS | DIFFERENTIAL-EQUATIONS | LIE | PHYSICS, MATHEMATICAL | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | TRANSFORMATIONS | Computer simulation | Mathematical analysis | Classification | Nonlinearity | Mathematical models | Transformations | Terminals | Invariants | Mathematics - Analysis of PDEs

Lie point symmetry | Group classification | Computer assisted research | Black–Scholes–Merton equation | Black-Scholes-Merton equation | MATHEMATICS, APPLIED | PHYSICS, FLUIDS & PLASMAS | SYMMETRY ANALYSIS | DIFFERENTIAL-EQUATIONS | LIE | PHYSICS, MATHEMATICAL | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | TRANSFORMATIONS | Computer simulation | Mathematical analysis | Classification | Nonlinearity | Mathematical models | Transformations | Terminals | Invariants | Mathematics - Analysis of PDEs

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 06/2015, Volume 69, Issue 12, pp. 1407 - 1419

This paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system. In this...

Closed-form analytical solution | Fractional partial differential equation | Double barrier options | AMERICAN | MATHEMATICS, APPLIED | DIFFUSION | VALUATION | DRIVEN | Analysis | Pricing | Mathematical analysis | Barriers | Exact solutions | Black-Scholes equation | Mathematical models | Derivatives | Convergence

Closed-form analytical solution | Fractional partial differential equation | Double barrier options | AMERICAN | MATHEMATICS, APPLIED | DIFFUSION | VALUATION | DRIVEN | Analysis | Pricing | Mathematical analysis | Barriers | Exact solutions | Black-Scholes equation | Mathematical models | Derivatives | Convergence

Journal Article

Computational Economics, ISSN 0927-7099, 4/2018, Volume 51, Issue 4, pp. 961 - 972

We present an accurate and efficient finite difference method for solving the Black–Scholes (BS) equation without boundary conditions. The BS equation is a...

Economics | Behavioral/Experimental Economics | Far field boundary conditions | Black–Scholes equation | Operations Research/Decision Theory | Economic Theory/Quantitative Economics/Mathematical Methods | Computer Appl. in Social and Behavioral Sciences | Math Applications in Computer Science | Finite difference method | AMERICAN | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MANAGEMENT | Black-Scholes equation | ECONOMICS | OPTIONS | SCHEMES | Hedging (Finance) | Algorithms | Analysis | Methods | Differential equations | Partial differential equations | Vanilla | Linearity | Pricing | Boundary conditions | Dirichlet problem | Computational grids

Economics | Behavioral/Experimental Economics | Far field boundary conditions | Black–Scholes equation | Operations Research/Decision Theory | Economic Theory/Quantitative Economics/Mathematical Methods | Computer Appl. in Social and Behavioral Sciences | Math Applications in Computer Science | Finite difference method | AMERICAN | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MANAGEMENT | Black-Scholes equation | ECONOMICS | OPTIONS | SCHEMES | Hedging (Finance) | Algorithms | Analysis | Methods | Differential equations | Partial differential equations | Vanilla | Linearity | Pricing | Boundary conditions | Dirichlet problem | Computational grids

Journal Article

Social Studies of Science, ISSN 0306-3127, 12/2003, Volume 33, Issue 6, pp. 831 - 868

This paper describes and analyses the history of the fundamental equation of modern financial economics: the Black-Scholes (or Black-Scholes-Merton) option...

Stock prices | Economic models | Investors | Economic theory | Stock options | Market prices | Capital asset pricing models | Finance | Financial economics | Financial portfolios | Social studies of finance | Black-Scholes | Option pricing | Performativity | Bricolage | RISK | HISTORY & PHILOSOPHY OF SCIENCE | BACHELIER,LOUIS | OPTION PRICES | EQUILIBRIUM | EFFICIENT CAPITAL-MARKETS | SCIENCE | option pricing | bricolage | UTILITY | performativity | social studies of finance

Stock prices | Economic models | Investors | Economic theory | Stock options | Market prices | Capital asset pricing models | Finance | Financial economics | Financial portfolios | Social studies of finance | Black-Scholes | Option pricing | Performativity | Bricolage | RISK | HISTORY & PHILOSOPHY OF SCIENCE | BACHELIER,LOUIS | OPTION PRICES | EQUILIBRIUM | EFFICIENT CAPITAL-MARKETS | SCIENCE | option pricing | bricolage | UTILITY | performativity | social studies of finance

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 01/2018, Volume 41, Issue 2, pp. 697 - 704

This work presents a new model of the fractional Black‐Scholes equation by using the right fractional derivatives to model the terminal value problem. Through...

fractional derivative | Black‐Scholes equation | asset pricing models | initial value problem | mathematical finance | terminal value problem | MATHEMATICS, APPLIED | PARTIAL-DIFFERENTIAL-EQUATIONS | Black-Scholes equation | EUROPEAN OPTIONS

fractional derivative | Black‐Scholes equation | asset pricing models | initial value problem | mathematical finance | terminal value problem | MATHEMATICS, APPLIED | PARTIAL-DIFFERENTIAL-EQUATIONS | Black-Scholes equation | EUROPEAN OPTIONS

Journal Article

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