Lie symmetries, group-invariant solutions and conservation laws of the Vasicek pricing equation of mathematical finance

Khalique, Chaudry Masood; Motsepa, Tanki

doi:10.1016/j.physa.2018.03.053

Cited by 14 publications

(7 citation statements)

References 14 publications

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“…Whereas, the Lie symmetry approach allows us to find complete local one-parameter transformation groups that can be used to find exact solutions or to reduce nonlinear DEs to linear DEs. Concerning the relationship between Lie symmetry methods and financial mathematics and economic models, we would like to note the presence of several recent studies [12][13][14][15][16][17][18][19][20][21][22], which represent the applications and utility of well-known concepts of Lie theory to several problems from mathematical finance.…”

Section: Theory and Backgroundmentioning

confidence: 99%

On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique

Aziz

2021

Mathematical Problems in Engineering

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The classical Cox–Ingersoll–Ross (CIR) bond-pricing model is based on the evolution space-time dependent partial differential equation (PDE) which represents the standard European interest rate derivatives. In general, such class of evolution partial differential equations (PDEs) has generally been resolved by classical methods of PDEs and by ansatz-based techniques which have been previously applied in a similar context. The author here shows the application of an invariant approach, a systematic method based on deductive group-theoretical analysis. The invariant technique reduces the scalar linear space-time dependent parabolic PDE to one of the four classical Lie canonical forms. This method leads us to exactly solve the scalar linear space-time dependent parabolic PDE representing the CIR model. It was found that CIR PDE is transformed into the first canonical form, which is the heat equation. Under the proper choice of emerging parameters of the model, the CIR equation is also reduced to the second Lie canonical form. The equivalence transformations which map the CIR PDE into the different canonical forms are deduced. With the use of these equivalence transformations, the invariant solutions of the underlying model are found by using some well-known results of the heat equation and the second Lie canonical form. Furthermore, the Cauchy initial-value model of the CIR problem along with the terminal condition is discussed and closed-form solutions are deduced. Finally, the conservation laws associated with the CIR equation are derived by using the general conservation theorem.

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Section: Theory and Backgroundmentioning

confidence: 99%

On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique

Aziz

2021

Mathematical Problems in Engineering

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“…The volatility (σ ), rate of mean reversion (κ) and long-term mean variance (θ ) are real and positive constants. Khalique and Motsepa [4] proposed new group invariant solutions of the generalized Vasicek model by changing α = σ 2 2 and γ = −1 in the Vasicek equation through the Lie symmetry method. This paper will study the solutions of the generalized CIR equation.…”

Section: Introductionmentioning

confidence: 99%

“…This paper will study the solutions of the generalized CIR equation. According to the change of variables in [4], the generalized CIR equation is given as…”

Section: Introductionmentioning

confidence: 99%

Lie Symmetries, Optimal System, and Invariant Solutions of the Generalized Cox-Ingersoll-Ross Equation

Tang¹,

Chong²,

Kee³

2022

Proceedings of the International Conference on Mathematical Sciences and Statistics 2022 (ICMSS 2022)

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The Cox-Ingersoll-Ross (CIR) model is a short-rate model and is widely used in the finance field to predict the movement of interest rates in bond pricing models. This paper exploited Lie symmetry analysis to solve the generalized CIR model by determining the infinitesimal generators. Lie symmetry is one of the powerful tools to solve the partial differential equation (PDE) analytically by reducing the PDE into a lower form. Besides, an optimal system of one-dimensional subalgebras is constructed and then used to reduce the generalized CIR equation by introducing the similarity variables. Lastly, the invariant solutions are obtained by solving the reduced equation.

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“…Sinkala and colleagues in [9] computed new prices for the bond PDE model with special consideration given to Vasicek and CIR models. In recent years, Khalique et al [10] have proposed new invariant solutions and conservation laws for the Vasicek pricing equation model. Lie symmetry analysis, as originated in studies by mathematician Sophus Lie, has proved in studies to be one of the prominent tools for obtaining analytical solutions for differential equations.…”

Section: Introductionmentioning

confidence: 99%

Symmetry Analysis of an Interest Rate Derivatives PDE Model in Financial Mathematics

Kaibe

O’Hara

2019

Symmetry

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We perform Lie symmetry analysis to a zero-coupon bond pricing equation whose price evolution is described in terms of a partial differential equation (PDE). As a result, using the computer software package SYM, run in conjunction with Mathematica, a new family of Lie symmetry group and generators of the aforementioned pricing equation are derived. We furthermore compute the exact invariant solutions which constitute the pricing models for the bond by making use of the derived infinitesimal generators and the associated similarity reduction equations. Using known solutions, we again compute more solutions via group point transformations.

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Lie symmetries, group-invariant solutions and conservation laws of the Vasicek pricing equation of mathematical finance

Cited by 14 publications

References 14 publications

On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique

On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique

Lie Symmetries, Optimal System, and Invariant Solutions of the Generalized Cox-Ingersoll-Ross Equation

Symmetry Analysis of an Interest Rate Derivatives PDE Model in Financial Mathematics

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