Lie Theory: Applications to problems in Mathematical Finance and Economics

Hernández, Isabel Carrasco; Mateos, Consuelo; Núñez, Juan; Villalón, Ángel Francisco Tenorio

doi:10.1016/j.amc.2008.12.025

Cited by 13 publications

(8 citation statements)

References 18 publications

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“…This is to be juxtaposed with with the numerous applications of Lie theory of continuous symmetries to problems arising in the field of Economics (see e.g. [40] and references therein).…”

Section: Kovacic's Algorithmmentioning

confidence: 99%

Classes of elementary function solutions to the CEV model I

Melas

2019

Journal of Computational and Applied Mathematics

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In the equity markets the stock price volatility increases as the stock price declines. The classical Black−Scholes−Merton (BSM) option pricing model does not reconcile with this association. Cox introduced the constant elasticity of variance (CEV) model in 1975, in order to capture this inverse relationship between the stock price and its volatility. An important parameter in the model is the parameter β, the elasticity of volatility. The CEV model subsumes some of the previous option pricing models. For β = 0, β = −1/2, and β = −1 the CEV model reduces respectively to the BSM model, the square−root model of Cox and Ross, and the Bachelier model. Both in the case of the BSM model and in the case of the CEV model it has become traditional to begin a discussion of option pricing by starting with the vanilla European calls and puts. The pricing formulas for these financial instruments give concrete information about the pricing of options only after the employment of some intermediate approximation scheme. However, there are simpler solutions to both models than those pertaining to the standard calls and puts. Mathematically, it makes sense to investigate the simpler cases first. Furthermore we do not allow ourselves to be drawn into any rash generalizations or inferences from the vanilla European case by prematurely focusing on those cases and we obtain concrete information for the pricing of options without needing to introduce any intermediate approximation schemes. In the case of BSM model simpler solutions are the log and power solutions. These contracts, despite the simplicity of their mathematical description, are attracting increasing attention as a trading instrument. Similar simple solutions have not been studied so far in a systematic fashion for the CEV model. We use Kovacic's algorithm to derive, for all half−integer values of β, all solutions "in quadratures" of the CEV ordinary differential equation. These solutions give rise, by separation of variables, 1 emelas@econ.uoa.gr to simple solutions to the CEV partial differential equation. In particular, when β = ..., − 5 2 , −2, − 3 2 , −1, 1, 3 2 , 2, 5 2 , ..., we obtain four classes of of denumerably infinite elementary function solutions, when β = − 1 2 and β = 1 2 we obtain two classes of of denumerably infinite elementary function solutions, whereas, when β = 0 we find two elementary function solutions. In the derived solutions we have also dispensed with the unnecessary assumption made in the the BSM model asserting that the underlying asset pays no dividends during the life of the option.

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“…This is to be juxtaposed with with the numerous applications of Lie theory of continuous symmetries to problems arising in the field of Economics (see e.g. [40] and references therein).…”

Section: Kovacic's Algorithmmentioning

confidence: 99%

Classes of elementary function solutions to the CEV model I

Melas

2019

Journal of Computational and Applied Mathematics

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“…The fundamental solutions to zero-coupon bound-pricing equations have been investigated by Pooe et al [5]. In recent time, the group approach has been widely applied to other partial differential equations (PDEs) from finance, for example, Goard [6], Lekalakala et al [7], and a very interesting review has been written by Hernández et al [8].…”

Section: Introductionmentioning

confidence: 99%

Invariant approach to optimal investment–consumption problem: the constant elasticity of variance (CEV) model

Bakkaloglu

Aziz

Fatima

et al. 2016

Math Methods in App Sciences

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Communicated by G. FranssensThe optimal investment-consumption problem under the constant elasticity of variance (CEV) model is solved using the invariant approach. Firstly, the invariance criteria for scalar linear second-order parabolic partial differential equations in two independent variables are reviewed. The criteria is then employed to reduce the CEV model to one of the four Lie canonical forms. It is found that the invariance criteria help in transforming the original equation to the second Lie canonical form and with a proper parameter selection; the required transformation converts the original equation to the first Lie canonical form that is the heat equation. As a consequence, we find some new classes of closed-form solutions of the CEV model for the case of reduction into heat equation and also into second Lie canonical form. The closed-form analytical solution of the Cauchy initial value problems for the CEV model under investigation is also obtained. Copyright

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“…Further applications of the Lie group analysis in mathematical finance were subsequently explored by a number of papers, for example, Goard [13], Pooe et al [14], Goard et al [15], Leach et al [16], and Sinkala et al [17]. A recent review of the applications of the Lie theory to problems in mathematical finance and economics can be found in [18].…”

Section: Introductionmentioning

confidence: 99%

Lie-Algebraic Approach for Pricing Zero-Coupon Bonds in Single-Factor Interest Rate Models

2013

Journal of Applied Mathematics

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The Lie-algebraic approach has been applied to solve the bond pricing problem in single-factor interest rate models. Four of the popular single-factor models, namely, the Vasicek model, Cox-Ingersoll-Ross model, double square-root model, and Ahn-Gao model, are investigated. By exploiting the dynamical symmetry of their bond pricing equations, analytical closed-form pricing formulae can be derived in a straightfoward manner. Time-varying model parameters could also be incorporated into the derivation of the bond price formulae, and this has the added advantage of allowing yield curves to be fitted. Furthermore, the Lie-algebraic approach can be easily extended to formulate new analytically tractable single-factor interest rate models.

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Lie Theory: Applications to problems in Mathematical Finance and Economics

Cited by 13 publications

References 18 publications

Classes of elementary function solutions to the CEV model I

Classes of elementary function solutions to the CEV model I

Invariant approach to optimal investment–consumption problem: the constant elasticity of variance (CEV) model

Lie-Algebraic Approach for Pricing Zero-Coupon Bonds in Single-Factor Interest Rate Models

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