scholarly journals Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 455
Author(s):  
Amitesh S. Jayaraman ◽  
Domenico Campolo ◽  
Gregory S. Chirikjian
Keyword(s):  
Lie Group ◽  
Cotangent Bundle ◽  
Diffusion Processes ◽  
Mathematical Finance ◽  
Alternative Interpretation ◽  
Affine Group ◽  
Black Scholes ◽  
Covariance Propagation ◽  
Alternative Means ◽  
Transformation Of Variables

The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker–Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

scholarly journals Group Classification of a General Bond-Option Pricing Equation of Mathematical Finance

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Tanki Motsepa ◽  
Chaudry Masood Khalique ◽  
Motlatsi Molati
Keyword(s):  
Option Pricing ◽  
Three Dimensional ◽  
Mathematical Finance ◽  
Group Classification ◽  
Lie Point Symmetries ◽  
Group Invariant ◽  
Black Scholes ◽  
Group Invariant Solutions ◽  
Bond Option

We carry out group classification of a general bond-option pricing equation. We show that the equation admits a three-dimensional equivalence Lie algebra. We also show that some of the values of the constants which result from group classification give us well-known models in mathematics of finance such as Black-Scholes, Vasicek, and Cox-Ingersoll-Ross. For all such values of these arbitrary constants we obtain Lie point symmetries. Symmetry reductions are then obtained and group invariant solutions are constructed for some cases.

The Duflo non-commutative Fourier transform for the Lorentz group

2020 ◽  
Vol 17 (supp01) ◽  
pp. 2040011
Author(s):  
Giacomo Rosati
Keyword(s):  
Fourier Transform ◽  
Phase Space ◽  
Quantum System ◽  
Lie Group ◽  
Lorentz Group ◽  
Commutative Algebra ◽  
Cotangent Bundle ◽  
Irreducible Representations ◽  
Algebra Representation

For a quantum system whose phase space is the cotangent bundle of a Lie group, like for systems endowed with particular cases of curved geometry, one usually resorts to a description in terms of the irreducible representations of the Lie group, where the role of (non-commutative) phase space variables remains obscure. However, a non-commutative Fourier transform can be defined, intertwining the group and (non-commutative) algebra representation, depending on the specific quantization map. We discuss the construction of the non-commutative Fourier transform and the non-commutative algebra representation, via the Duflo quantization map, for a system whose phase space is the cotangent bundle of the Lorentz group.

scholarly journals On the BKS pairing for Kähler quantizations of the cotangent bundle of a Lie group

2006 ◽  
Vol 234 (1) ◽  
pp. 180-198 ◽  
Author(s):  
Carlos Florentino ◽  
Pedro Matias ◽  
José Mourão ◽  
João P. Nunes
Keyword(s):  
Lie Group ◽  
Cotangent Bundle

scholarly journals Exact simulation of first exit times for one-dimensional diffusion processes

2020 ◽  
Vol 54 (3) ◽  
pp. 811-844
Author(s):  
Samuel Herrmann ◽  
Cristina Zucca
Keyword(s):  
Diffusion Processes ◽  
Exit Time ◽  
Convergent Series ◽  
Mathematical Finance ◽  
Exit Times ◽  
Rejection Sampling ◽  
Girsanov Transformation ◽  
One Dimensional ◽  
Usual Procedure ◽  
First Exit Times

The simulation of exit times for diffusion processes is a challenging task since it concerns many applications in different fields like mathematical finance, neuroscience, reliability… The usual procedure is to use discretization schemes which unfortunately introduce some error in the target distribution. Our aim is to present a new algorithm which simulates exactly the exit time for one-dimensional diffusions. This acceptance-rejection algorithm requires to simulate exactly the exit time of the Brownian motion on one side and the Brownian position at a given time, constrained not to have exit before, on the other side. Crucial tools in this study are the Girsanov transformation, the convergent series method for the simulation of random variables and the classical rejection sampling. The efficiency of the method is described through theoretical results and numerical examples.

scholarly journals Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

2016 ◽  
Author(s):  
Frédéric Barbaresco
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Statistical Physics ◽  
Geometric Theory ◽  
Adjoint Action ◽  
Information Geometry ◽  
Affine Group ◽  
Exponential Families ◽  
Natural Exponential Families ◽  
Fisher Metric

We introduce the Symplectic Structure of Information Geometry based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. Souriau model is based on affine representation of Lie Group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant-Kirillov-Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of Information Geometry for the action of an affine Group for exponentiel families, and provide some illustration of use cases for multivariate Gaussian densities. Information Geometry is presented in the context of seminal work of Fréchet and his Clairait-Legendre equation. Souriau model of Statistical Physics is validated as compatible with Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.

scholarly journals UNIVERSAL RISK BUDGETING

2021 ◽  
Author(s):  
ALEX GARIVALTIS
Keyword(s):  
Side Information ◽  
Covariance Structure ◽  
Mathematical Finance ◽  
Risk Budgeting ◽  
Capital Budget ◽  
New Family ◽  
Black Scholes ◽  
Universal Portfolios ◽  
Universal Portfolio ◽  
Dirichlet Priors

I juxtapose Cover’s vaunted universal portfolio selection algorithm ([Cover, TM (1991). Universal portfolios. Mathematical Finance, 1, 1–29]) with the modern representation of a portfolio as a certain allocation of risk among the available assets, rather than a mere allocation of capital. Thus, I define a Universal Risk Budgeting scheme that weights each risk budget, instead of each capital budget, by its historical performance record, á la Cover. I prove that my scheme is mathematically equivalent to a novel type of [Cover, TM and E Ordentlich (1996). Universal portfolios with side information. IEEE Transactions on Information Theory, 42, 348–363] universal portfolio that uses a new family of prior densities that have hitherto not appeared in the literature on universal portfolio theory. I argue that my universal risk budget, so-defined, is a potentially more perspicuous and flexible type of universal portfolio; it allows the algorithmic trader to incorporate, with advantage, his prior knowledge or beliefs about the particular covariance structure of instantaneous asset returns. Say, if there is some dispersion in the volatilities of the available assets, then the uniform or Dirichlet priors that are standard in the literature will generate a dangerously lopsided prior distribution over the possible risk budgets. In the author’s opinion, the proposed “Garivaltis prior” makes for a nice improvement on Cover’s timeless expert system, that is properly agnostic and open to different risk budgets from the very get-go. Inspired by [Jamshidian, F (1992). Asymptotically optimal portfolios. Mathematical Finance, 2, 131–150], the universal risk budget is formulated as a new kind of exotic option in the continuous time Black–Scholes market, with all the pleasure, elegance, and convenience that entails.

scholarly journals Hyperkähler metrics associated to compact Lie groups

1996 ◽  
Vol 120 (1) ◽  
pp. 61-69 ◽  
Author(s):  
Andrew Dancer ◽  
Andrew Swann
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Left And Right ◽  
Hyperkähler Metrics

It is well known that the cotangent bundle of any manifold has a canonical symplectic structure. If we specialize to the case when the manifold is a compact Lie group G, then this structure is preserved by the actions of G on T*G induced by left and right translation on G. We refer to these as the left and right actions of G on T*G.

scholarly journals Correcting the Bias in the Practitioner Black-Scholes Method

2019 ◽  
Vol 12 (4) ◽  
pp. 157
Author(s):  
Yun Yin ◽  
Peter G. Moffatt
Keyword(s):  
Linear Regression ◽  
Implied Volatility ◽  
Option Price ◽  
Option Prices ◽  
Technical Problems ◽  
Black Scholes ◽  
Option Values ◽  
Non Linear ◽  
Alternative Means ◽  
Log Linear

We address a number of technical problems with the popular Practitioner Black-Scholes (PBS) method for valuing options. The method amounts to a two-stage procedure in which fitted values of implied volatilities (IV) from a linear regression are plugged into the Black-Scholes formula to obtain predicted option prices. Firstly we ensure that the prediction from stage one is positive by using log-linear regression. Secondly, we correct the bias that results from the transformation applied to the fitted values (i.e., the Black-Scholes formula) being a highly non-linear function of implied volatility. We apply the smearing technique in order to correct this bias. An alternative means of implementing the PBS approach is to use the market option price as the dependent variable and estimate the parameters of the IV equation by the method of non-linear least squares (NLLS). A problem we identify with this method is one of model incoherency: the IV equation that is estimated does not correspond to the set of option prices used to estimate it. We use the Monte Carlo method to verify that (1) standard PBS gives biased option values, both in-sample and out-of-sample; (2) using standard (log-linear) PBS with smearing almost completely eliminates the bias; (3) NLLS gives biased option values, but the bias is less severe than with standard PBS. We are led to conclude that, of the range of possible approaches to implementing PBS, log-linear PBS with smearing is preferred on the basis that it is the only approach that results in valuations with negligible bias.

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