Invariance properties of a general bond-pricing equation

Abstract

We perform the group classification of a bond-pricing partial differential equation of mathematical finance to discover the combinations of arbitrary parameters that allow the partial differential equation to admit a nontrivial symmetry Lie algebra. As a result of the group classification we propose "natural" values for the arbitrary parameters in the partial differential equation, some of which validate the choices of parameters in such classical models as that of Vasicek and Cox-Ingersoll-Ross. For each set of these natural parameter values we compute the admitted Lie point symmetries, identify the corresponding symmetry Lie algebra and solve the partial differential equation.

Algebraic properties of evolution partial differential equations modelling prices of commodities

“…The algebra [11] is {sl(2, R) ⊕ s W } ⊕ s ∞A 1 , where W is the Weyl-Heisenberg algebra denoted as A 3,3 in the Mubarakzyanov classification scheme [21][22][23][24]. Equation (10): (14).…”

“…In a sense we look at the 'worst-case' scenario. To discuss the algebras we have two exemplars, videlicet (14) and (15). It is of course informative to look at the parent equations, as it were, to see how the numbers of Lie point symmetries do vary with dimension and the introduction of the terms that are important for the problem being modelled.…”

“…As we see in the next section, this reduction is insufficient to hinder integrability for the class of problem considered in this paper. If we exclude the solution symmetries, the nonzero Lie Bracket relations of the symmetries listed in (14) and (15) are …”

The algebraic properties of the space-and time-dependent one-factor model of commodities

“…The detailed analysis for the Lie symmetries of the three models, which were proposed by Schwartz, and the generalisation to the n-factor model can be found in [3]. Other Financial models which have been studied with the use of group invariants can be found in [4,5,6,7,8,9,10,11,12] and references therein.…”

Painlevé analysis, Lie symmetries, and exact solutions for the time-dependent coefficients Gardner equations

“…It is known that to find exact solutions of the NLEEs is always one of the central themes in mathematics and physics. In the past few decades, there is noticeable progress in this field, and various methods have been developed, such as the inverse scattering transformation (IST) [1], Darboux and Bäcklund transformations [2], Hirota's bilinear method [2][3][4], Lie symmetry analysis [5][6][7][8][9][10][11][12], CK method [13,14], and so on.…”