Conservation laws for the Black–Scholes equation

Edelstein, R. M.; Govinder, K. S.

doi:10.1016/j.nonrwa.2008.10.064

Cited by 15 publications

(11 citation statements)

References 24 publications

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“…We now restrict

f false(x, u_{x} false) = p_{u_{x}} false(x, u_{x} false)

and

g false(x, u_{x} false) = p_{x} false(x, u_{x} false)

, which allows us to write Equation in conserved form . Although other conserved forms can be obtained without this restriction, our choice ensures a clean, conserved form, viz,

u_{t} = {([], P false(x, u_{x} false))}_{x} .

Introducing a potential dependent variable, v yields the auxiliary system

v_{x} = u,

v_{t} = P false(x, u_{x} false),

which admits a three‐dimensional principal Lie algebra

H_{1} = \frac{\partial}{\partial t},

H_{2} = \frac{\partial}{\partial v},

H_{3} = \frac{\partial}{\partial u} + x \frac{\partial}{\partial v} .

…”

Section: Preliminary Potential Group Classificationmentioning

confidence: 93%

On the method of preliminary group classification applied to the nonlinear heat equation ut=f(x,ux)uxx+g(x,ux)

Edelstein

Govinder

2020

Math Methods in App Sciences

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We apply the method of preliminary group classification to a specific class of the nonlinear heat equation, ie, ut=ffalse(x,uxfalse)uxx+gfalse(x,uxfalse). This results in an optimal system of one‐dimensional subalgebras, which will be used to find specific forms of the equation, which admit more symmetries and so allow us to find additional group invariant solutions. We extend this work by determining the potential symmetries of a closely related system, which gives rise to many new, additional solutions of the original equation.

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“…We now restrict

f false(x, u_{x} false) = p_{u_{x}} false(x, u_{x} false)

and

g false(x, u_{x} false) = p_{x} false(x, u_{x} false)

, which allows us to write Equation in conserved form . Although other conserved forms can be obtained without this restriction, our choice ensures a clean, conserved form, viz,

u_{t} = {([], P false(x, u_{x} false))}_{x} .

Introducing a potential dependent variable, v yields the auxiliary system

v_{x} = u,

v_{t} = P false(x, u_{x} false),

which admits a three‐dimensional principal Lie algebra

H_{1} = \frac{\partial}{\partial t},

H_{2} = \frac{\partial}{\partial v},

H_{3} = \frac{\partial}{\partial u} + x \frac{\partial}{\partial v} .

…”

Section: Preliminary Potential Group Classificationmentioning

confidence: 93%

On the method of preliminary group classification applied to the nonlinear heat equation ut=f(x,ux)uxx+g(x,ux)

Edelstein

Govinder

2020

Math Methods in App Sciences

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“…[40][41][42][43] In a great number of papers, the authors have used the classical symmetry method and the conservation law multiplier method to study PDEs. [44][45][46][47][48][49][50][51]…”

Section: Introductionmentioning

confidence: 99%

Classical symmetries and conservation laws for the dissipative Dullin‐Gottwald‐Holm equation with arbitrary coefficients

Camacho

Rosa

Gandarias

et al. 2018

Math Methods in App Sciences

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“…Then optimal system of subalgebras were constructed and invariant solutions of the Black-Scholes equation were computed. In [6] the classical Lie point symmetries and potential symmetries were found and were used to obtain new solutions to the Black-Scholes equation. Also conservation laws were calculated by utilizing the Lie point symmetries of the equation using the direct approach [7].…”

Section: Introductionmentioning

confidence: 99%

Lie Symmetry Reductions and Exact Solutions of an Option-Pricing Equation for Large Agents

2015

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In this paper, we study a nonlinear partial differential equation that models the one-factor term structure option-pricing for large agents from the Lie symmetry stand point. This equation was modelled by Jonsson and Keppo (Appl Math Financ 9:261-272, 2002) and is a nonlinear modified Black-Scholes partial differential equation. We first determine an optimal system of one-dimensional subalgebras. We then use it to obtain symmetry reductions and families of group-invariant solutions of the underlying equation.Mathematics Subject Classification. 35A09, 35C05, 35C06, 76M60.

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Conservation laws for the Black–Scholes equation

Cited by 15 publications

References 24 publications

On the method of preliminary group classification applied to the nonlinear heat equation ut=f(x,ux)uxx+g(x,ux)

On the method of preliminary group classification applied to the nonlinear heat equation ut=f(x,ux)uxx+g(x,ux)

Classical symmetries and conservation laws for the dissipative Dullin‐Gottwald‐Holm equation with arbitrary coefficients

Lie Symmetry Reductions and Exact Solutions of an Option-Pricing Equation for Large Agents

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