scholarly journals Invariant Solutions of Black–Scholes Equation with Ornstein–Uhlenbeck Process

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 847
Author(s):  
Maba Boniface Matadi ◽  
Phumlani Lawrence Zondi
Keyword(s):  
Stochastic Process ◽  
Option Pricing ◽  
Invariant Solution ◽  
Point Of View ◽  
Theoretic Approach ◽  
Invariant Solutions ◽  
Uhlenbeck Process ◽  
Independent Variables ◽  
Black Scholes ◽  
Ornstein Uhlenbeck Process

This paper analyses the model of Black–Scholes option pricing from the point of view of the group theoretic approach. The study identified new independent variables that lead to the transformation of the Black–Scholes equation. Furthermore, corresponding determining equations were constructed and new symmetries were found. As a result, the findings of the study demonstrate of the integrability of the model to present an invariant solution for the Ornstein–Uhlenbeck stochastic process.

Introduction

2017 ◽  
Author(s):  
Roi Wagner
Keyword(s):  
Option Pricing ◽  
Cognitive Theory ◽  
Philosophy Of Mathematics ◽  
Mathematical Cognition ◽  
Point Of View ◽  
Current State ◽  
Mathematical Life ◽  
Black Scholes ◽  
Epistemological Position ◽  
Mathematical Interpretation

This book examines the force of mathematics, what this force builds on, and how it works in practice by discussing mathematics not only from the point of view of applications but also from the point of view of its production. It explores the function of mathematical statements, their epistemological position, consensus in mathematics, and mathematical interpretation and semiosis. It also considers the notion of embodied mathematical cognition as well as the limitations of the cognitive theory of mathematical metaphor in accounting for the formation of actual historical mathematical life worlds. This introduction provides an overview of the current state of the philosophy of mathematics and presents a vignette on option pricing to give a concrete example of how mathematics relates to its wider scientific and practical context, with particular emphasis on the Black-Scholes formula.

scholarly journals On the Entropy of Fractionally Integrated Gauss–Markov Processes

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2031
Author(s):  
Mario Abundo ◽  
Enrica Pirozzi
Keyword(s):  
Dynamical System ◽  
Stochastic Process ◽  
Brownian Motion ◽  
Markov Process ◽  
Markov Processes ◽  
Numerical Approximation ◽  
Uhlenbeck Process ◽  
Sample Paths ◽  
Ornstein Uhlenbeck Process ◽  
Application Fields

This paper is devoted to the estimation of the entropy of the dynamical system {Xα(t),t≥0}, where the stochastic process Xα(t) consists of the fractional Riemann–Liouville integral of order α∈(0,1) of a Gauss–Markov process. The study is based on a specific algorithm suitably devised in order to perform the simulation of sample paths of such processes and to evaluate the numerical approximation of the entropy. We focus on fractionally integrated Brownian motion and Ornstein–Uhlenbeck process due their main rule in the theory and application fields. Their entropy is specifically estimated by computing its approximation (ApEn). We investigate the relation between the value of α and the complexity degree; we show that the entropy of Xα(t) is a decreasing function of α∈(0,1).

On the solution of Kac-type partial differential equations

1994 ◽  
Vol 31 (A) ◽  
pp. 311-324
Author(s):  
Mátyás Arató
Keyword(s):  
Stochastic Process ◽  
Cauchy Problem ◽  
Brownian Motion ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Explicit Form ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
The Cauchy Problem ◽  
Constant Coefficients

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].

Asymptotic distributions for the Ornstein-Uhlenbeck process

1975 ◽  
Vol 12 (01) ◽  
pp. 107-114 ◽  
Author(s):  
John A. Beekman
Keyword(s):  
Stochastic Process ◽  
Numerical Analysis ◽  
Asymptotic Distributions ◽  
Exit Times ◽  
Uhlenbeck Process ◽  
Time Period ◽  
Ornstein Uhlenbeck Process ◽  
Fixed Constant ◽  
First Exit Times ◽  
Asymptotic Forms

This paper gives the asymptotic distributions, as the time period grows infinite, of the first exit times above a fixed constant and from upper and lower constant boundaries for the Ornstein-Uhlenbeck stochastic process. The results of a large amount of numerical analysis illustrate the asymptotic forms.

On the solution of Kac-type partial differential equations

1994 ◽  
Vol 31 (A) ◽  
pp. 311-324
Author(s):  
Mátyás Arató
Keyword(s):  
Stochastic Process ◽  
Cauchy Problem ◽  
Brownian Motion ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Explicit Form ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
The Cauchy Problem ◽  
Constant Coefficients

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].

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