arXiv subjects

Evangelos Melas

Publications and source records attributed to Evangelos Melas.

9 recordsLinked to original sources

arXiv2026

A Differential-Geometric Framework for Risk-Optimal Asset Reallocation

We develop a differential-geometric framework for risk-optimal portfolio reallocation. A long-only portfolio is represented as a point of the probability simplex, endowed with a positive-definite Riemannian metric combining market covariance risk with position-dependent concentration risk. The cumulative risk of a rebalancing trajectory is identified with its Riemannian length, so the least-risk transition between a current allocation and a Markowitz target is a geodesic. We compare this route with direct linear rebalancing and projected gradient ascent. When only market covariance risk is priced, the metric is constant and flat, and the geodesic is exactly the straight-line path. Once position-dependent risk is introduced, the geometry becomes curved and the geodesic weakly dominates competing paths with the same endpoints. A Fisher-Rao concentration term produces modest but systematic savings, while an endogenous crowding metric creates non-convex risk ridges that geodesics can bypass through temporary diversification. Numerical experiments, Monte Carlo transitions, and regression analysis show that the largest gains occur when the direct path crosses strongly crowded regions. The framework provides a general geometric formulation of transition management and can accommodate richer risk metrics and transaction-cost structures.

math.DG↗

arXiv2022

Elementary functions solutions to the Bachelier model generated by Lie point symmetries

Under the recent negative interest rate situation, the Bachelier model has been attracting attention and adopted for evaluating the price of interest rate options. In this paper we find the Lie point symmetries of the Bachelier partial differential equation (PDE) and use them in order to generate new classes of denumerably infinite elementary function solutions to the Bachelier model from elementary function solutions to it which we derived in a previous publication.

math.GM↗

arXiv2022

Representations of the Bondi-Metzner-Sachs group in three space-time dimensions in the Hilbert topology I. Determination of the representations

The original Bondi$-$Metzner-Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian 4-dim space$-$times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). Here, the analogue $B(2,1)$ of $B$ in 3 space$-$time dimensions is properly defined. We study its representation theory in the Hilbert topology by using an infinite-dimensional extension of Wigner-Mackey theory. We obtain the necessary data in order to construct the strongly continuous irreducible unitary representations (IRS) of B(2,1). The main results of the representation theory are: The IRS are induced from ``little groups'' which are compact. There is one infinite connected ``little group'', the special orthogonal group SO(2). There are infinite non$-$connected finite discrete ``little groups'', the cyclic groups C_{n} of even order. The inducing construction is exhaustive notwithstanding the fact that B(2,1) is not locally compact in the employed Hilbert topology. B(2,1) is also derived, in Klein's sense, as the automorphism group of the ``strong conformal geometry'' of future null infinity. Besides the Hilbert topology other reasonable topologies are given to B(2,1) and their physical relevance is discussed. Connections of the IRS of B(2,1) with geometry are outlined.

math-ph↗

arXiv2018

Nonlinear Invariants of Planar Point Clouds Transformed by Matrices

The goal of this paper is to present invariants of planar point clouds, that is functions which take the same value before and after a linear transformation of a planar point cloud via a $2 \times 2$ invertible matrix. In the approach we adopt here, these invariants are functions of two variables derived from the least squares straight line of the planar point cloud under consideration. A linear transformation of a point cloud induces a nonlinear transformation of these variables. The said invariants are solutions to certain Partial Differential Equations, which are obtained by employing Lie theory. We find cloud invariants in the general case of a four$-$parameter transformation matrix, as well as, cloud invariants of various one$-$parameter sets of transformations which can be practically implemented. Case studies and simulations which verify our findings are also provided.

math.GM↗

arXiv2018

Classes of elementary function solutions to the CEV model. I

The CEV model subsumes some of the previous option pricing models. An important parameter in the model is the parameter b, the elasticity of volatility. For b=0, b=-1/2, and b=-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. 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 b, all solutions "in quadratures" of the CEV ordinary differential equation. These solutions give rise, by separation of variables, to simple solutions to the CEV partial differential equation. In particular, when b=...,-5/2,-2,-3/2,-1, 1, 3/2, 2, 5/2,..., we obtain four classes of denumerably infinite elementary function solutions, when b=-1/2 and b=1/2 we obtain two classes of denumerably infinite elementary function solutions, whereas, when b=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.

q-fin.MF↗

arXiv2017

On the Liouvillian solutions to the perturbation equations of the Schwarzschild black hole

We use Kovacic's algorithm to obtain all Liouvillian solutions, i.e., essentially all solutions in terms of quadratures, of the master equation which governs the evolution of first order perturbations of the Schwarzschild geometry. We show that all solutions in quadratures of this equation contain a polynomial solution to an associated ordinary differential equation (ODE). This ODE, apart from a few trivial cases, falls into the confluent Heun class. In the case of the gravitational perturbations, for the Liouvillian solution $χ\int \frac {{\rm d}r_{\!\ast}}{χ^{2}}$, we find in "closed form" the polynomial solution P to the associated confluent Heun ODE. We prove that the Liouvillian solution $χ\int \frac {{\rm d}r_{\!\ast}}{χ^{2}}$ is a product of elementary functions, one of them being the polynomial P. We extend previous results by Hautot and use the extended results we derive in order to prove that P admits a finite expansion in terms of truncated confluent hypergeometric functions of the first kind. We also prove, by using the extended results we derive, that P admits also a finite expansion in terms of associated Laguerre polynomials. We prove, save for two unresolved cases, that the Liouvillian solutions $χ$ and $χ\int \frac {{\rm d}r_{\!\ast}}{χ^{2}}$, initially found by Chandrasekhar, are the only Liouvillian solutions to the master equation. We improve previous results in the literature on this problem and compare our results with theirs. Comments are made for a more efficient implementation of Kovacic's algorithm to any second order ODE with rational function coefficients. Our results set the stage for deriving similar results in other black hole geometries 4-dim and higher.

math-ph↗

arXiv2017

On the representation theory of the Bondi-Metzner-Sachs group and its variants in three space-time dimensions

The original Bondi-Metzner-Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian radiating 4-dim space-times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). In 1973, with this motivation, P. J. McCarthy classified all relativistic B-invariant-systems in terms of strongly continuous irreducible unitary repesentations (IRS) of B. Here we introduce the analogue B(2,1) of the BMS group B in 3 space-time dimensions. B(2,1) itself admits thirty-four analogues both real in all signatures and in complex space-times. In order to find the IRS of both B(2,1) and its analogues we need to extend Wigner-Mackey's theory of induced representations. The necessary extension is described and is reduced to the solution of three problems. These problems are solved in the case where B(2,1) and its analogues are equipped with the Hilbert topology. The extended theory is necessary in order to construct the IRS of both B and its analogues in any number d of space-time dimensions, d is greater or equal to 3, and also in order to construct the IRS of their supersymmetric counterparts. We use the extended theory to obtain the necessary data in order to construct the IRS of B(2,1): The main results of the representation theory are: The IRS are induced from little groups which are compact. The finite little groups are cyclic groups of even order. The inducing construction is exhaustive notwithstanding the fact that B(2,1) is not locally compact in the employed Hilbert topology.

math-ph↗

arXiv2014

Representations of the ultrahyperbolic BMS group HB. III. Determination of the representations induced from finite little groups

The ordinary Bondi-Metzner-Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian space-times. As such, B is the best candidate for the universal symmetry group of General Relativity. However, in studying quantum gravity, space-times with signatures other than the usual Lorentzian one, and complex space-times, are frequently considered. Generalisations of B appropriate to these other signatures have been defined earlier. In particular, HB, a variant of BMS group appropriate to the ultrahyperbolic signature (+,+,-,-), has been defined in a previous paper where it was shown that all the strongly continuous unitary irreducible representations (IRs) of HB can be obtained with the Wigner-Mackey's inducing method and that all the little groups of HB are compact.Here we describe in detail all the finite little groups of HB and we find all the IRs of HB induced by them.

math.RT↗

arXiv2013

Representations of the ultrahyperbolic BMS group HB.II. Determination of the representations induced from infinite little groups

The ordinary Bondi-Metzner-Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian space-times. As such, B is the best candidate for the universal symmetry group of General Relativity. However, in studying quantum gravity, space-times with signatures other than the usual Lorentzian one, and complex space-times, are frequently considered. Generalisations of B appropriate to these other signatures have been defined earlier. In particular, the generalisation HB, a BMS group appropriate to the ultrahyperbolic signature (+,+,-,-), has been defined in a previous paper where it was shown that all the strongly continuous unitary irreducible representations (IRs) of HB can be obtained with the Wigner-Mackey's inducing method and that all the little groups of HB are compact. Here we describe in detail all the infinite little groups of HB and we find the IRs of HB induced by them.

math.RT↗