Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility

We perform a classification of the Lie point symmetries for the Black--Scholes--Merton Model for European options with stochastic volatility, $\sigma$, in which the last is defined by a stochastic differential equation with an Orstein--Uhlenbeck term. In this model, the value of the option is given by a linear (1 + 2) evolution partial differential equation in which the price of the option depends upon two independent variables, the value of the underlying asset, $S$, and a new variable, $y$. We find that for arbitrary functional form of the volatility, $\sigma(y)$, the (1 + 2) evolution equation always admits two Lie point symmetries in addition to the automatic linear symmetry and the infinite number of solution symmetries. However, when $\sigma(y)=\sigma_{0}$ and as the price of the option depends upon the second Brownian motion in which the volatility is defined, the (1 + 2) evolution is not reduced to the Black--Scholes--Merton Equation, the model admits five Lie point symmetries in addition to the linear symmetry and the infinite number of solution symmetries. We apply the zeroth-order invariants of the Lie symmetries and we reduce the (1 + 2) evolution equation to a linear second-order ordinary differential equation. Finally, we study two models of special interest, the Heston model and the Stein--Stein model.Comment: Published version, 14pages, 4 figure

Ermakov's Superintegrable Toy and Nonlocal Symmetries

We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2,R). The number of point symmetries is insufficient and the algebra unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the complete symmetry group from this system. Four of the required symmetries are nonlocal and the algebra is the direct sum of a one-dimensional Abelian algebra with the semidirect sum of a two-dimensional solvable algebra with a two-dimensional Abelian algebra. The problem illustrates the difficulties which can arise in very elementary systems. Our treatment demonstrates the existence of possible routes to overcome these problems in a systematic fashion.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Analytic Behaviour of Competition among Three Species

We analyse the classical model of competition between three species studied by May and Leonard ({\it SIAM J Appl Math} \textbf{29} (1975) 243-256) with the approaches of singularity analysis and symmetry analysis to identify values of the parameters for which the system is integrable. We observe some striking relations between critical values arising from the approach of dynamical systems and the singularity and symmetry analyses.Comment: 14 pages, to appear in Journal of Nonlinear Mathematical Physic

Noether's Theorem and Symmetry

In Noether's original presentation of her celebrated theorm of 1918 allowance was made for the dependence of the coefficient functions of the differential operator which generated the infinitesimal transformation of the Action Integral upon the derivatives of the depenent variable(s), the so-called generalised, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to point transformations only. In recent decades this dimunition of the power of Noether's Theorem has been partly countered, in particular in the review of Sarlet and Cantrijn. In this special issue we emphasise the generality of Noether's Theorem in its original form and explore the applicability of even more general coefficient functions by alowing for nonlocal terms. We also look for the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence upon the independent variablesComment: 23 pages, to appear in Symmetry in the special issue "Noether's Theorem and Symmetry", dedicated for the 100 years from the publication of E. Noether's original work on the invariance of the functional of the Calculus of Variation

Similarity solutions and Conservation laws for the Bogoyavlensky-Konopelchenko Equation by Lie point symmetries

The 1 + 2 dimensional Bogoyavlensky-Konopelchenko Equation is investigated for its solution and conservation laws using the Lie point symmetry analysis. In the recent past, certain work has been done describing the Lie point symmetries for the equation and this work seems to be incomplete (Ray S (2017) Compt. Math. Appl. 74, 1157). We obtained certain new symmetries and corresponding conservation laws. The travelling-wave solution and some other similarity solutions are studied.Comment: 12 pages. Accepted for publication in Quaestiones Mathematica

Parsimonious Kernel Fisher Discrimination

By applying recent results in optimization transfer, a new algorithm for kernel Fisher Discriminant Analysis is provided that makes use of a non-smooth penalty on the coefficients to provide a parsimonious solution. The algorithm is simple, easily programmed and is shown to perform as well as or better than a number of leading machine learning algorithms on a substantial benchmark. It is then applied to a set of extreme small-sample-size problems in virtual screening where it is found to be less accurate than a currently leading approach but is still comparable in a number of cases

A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators

We explore a nonlocal connection between certain linear and nonlinear ordinary differential equations (ODEs), representing physically important oscillator systems, and identify a class of integrable nonlinear ODEs of any order. We also devise a method to derive explicit general solutions of the nonlinear ODEs. Interestingly, many well known integrable models can be accommodated into our scheme and our procedure thereby provides further understanding of these models.Comment: 12 pages. J. Phys. A: Math. Gen. 39 (2006) in pres