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

A direct approach to the construction of standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients

We present a direct approach to the construction of Lagrangians for a large class of one-dimensional dynamical systems with a simple dependence (monomial or polynomial) on the velocity. We rederive and generalize some recent results and find Lagrangian formulations which seem to be new. Some of the considered systems (e.g., motions with the friction proportional to the velocity and to the square of the velocity) admit infinite families of different Lagrangian formulations.Comment: 17 page

Classification of real three-dimensional Lie bialgebras and their Poisson-Lie groups

Classical r-matrices of the three-dimensional real Lie bialgebras are obtained. In this way all three-dimensional real coboundary Lie bialgebras and their types (triangular, quasitriangular or factorizable) are classified. Then, by using the Sklyanin bracket, the Poisson structures on the related Poisson-Lie groups are obtained.Comment: 17 page

Group-invariant solutions of a nonlinear acoustics model

Based on a recent classification of subalgebras of the symmetry algebra of the Zabolotskaya-Khokhlov equation, all similarity reductions of this equation into ordinary differential equations are obtained. Large classes of group-invariant solutions of the equation are also determined, and some properties of the reduced equations and exact solutions are discussed.Comment: 14 page

Examination of a Size-Change Test for Photovoltaic Encapsulation Materials: Preprint

We examine a proposed test standard that can be used to evaluate the maximum representative change in linear dimensions of sheet encapsulation products for photovoltaic modules (resulting from their thermal processing). The proposed protocol is part of a series of material-level tests being developed within Working Group 2 of the Technical Committee 82 of the International Electrotechnical Commission. The characterization tests are being developed to aid module design (by identifying the essential characteristics that should be communicated on a datasheet), quality control (via internal material acceptance and process control), and failure analysis. Discovery and interlaboratory experiments were used to select particular parameters for the size-change test. The choice of a sand substrate and aluminum carrier is explored relative to other options. The temperature uniformity of +/- 5C for the substrate was confirmed using thermography. Considerations related to the heating device (hot-plate or oven) are explored. The time duration of 5 minutes was identified from the time-series photographic characterization of material specimens (EVA, ionomer, PVB, TPO, and TPU). The test procedure was revised to account for observed effects of size and edges. The interlaboratory study identified typical size-change characteristics, and also verified the absolute reproducibility of +/- 5% between laboratories

Deformations of N=2 super-conformal algebra and supersymmetric two-component Camassa-Holm equation

This paper is concerned with a link between central extensions of N=2 superconformal algebra and a supersymmetric two-component generalization of the Camassa--Holm equation. Deformations of superconformal algebra give rise to two compatible bracket structures. One of the bracket structures is derived from the central extension and admits a momentum operator which agrees with the Sobolev norm of a coadjoint orbit element. The momentum operator induces via Lenard relations a chain of conserved hamiltonians of the resulting supersymmetric Camassa-Holm hierarchy.Comment: Latex, 21 pages, version to appear in J. Phys.

Results from the Second International Module Inter-comparison

Most photovoltaic (PV) manufacturers trace their peakwatt rating through calibrations/measurements performed at recognized terrestrial calibration facilities. This paper summarizes the results of one such measurement performed by many different calibration facilities. The participants were selected from around the world based on their designation as a national PV calibration facility, prior participation in inter-comparisons, or as an ISO 17025-accredited PV module qualification or certification facility. Each facility was sent the same devices and was requested to treat them as a regular measurement. The modules were selected from newer thin-film manufacturers - ones that might stretch or exceed the current scope of capabilities of the different participants. A concentrator module was even included as part of the set. Short-circuit current (Isc), open-circuit voltage (Voc), fill factor (FF), and peak power (Pmax) results are reported

Continuous Symmetries of Difference Equations

Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict to point symmetries, but to allow them to also transform the lattice.Comment: Review articl