According to previous research, the issue on the applicability of the original Black-Scholes model to the inverse quantity of price can be formulated as the argument of the symmetry between price and its inverse, whether there exists the set of real numbers as the drift and the volatility about the inverse quantity satisfying a certain system of stochastic differential equations. As the result of solving the equations in terms of such real numbers, it is revealed that there exist symmetries between not only them but also the coefficients of two equations. The aim of this article is to reveal in which cases these symmetries exist in the generalized Black-Scholes model, where the coeficients are deterministic or stochastic processes.
Inhaltsverzeichnis (Table of Contents)
- Introduction
- Preparation
- Lemma 1
- Lemma 2
- Argument
- Conclusion and Future Work
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This article aims to investigate the applicability of the generalized Black-Scholes model to the inverse quantity of price and to determine if it exhibits the same symmetries between price and its inverse as the original Black-Scholes model with constant coefficients.
- Symmetries in the generalized Black-Scholes model
- Applicability of the Black-Scholes model to inverse quantities
- Itô lemma and its role in analyzing stochastic processes
- Comparison of coefficients in stochastic differential equations
- Generalization of the Black-Scholes model with deterministic or stochastic coefficients
Zusammenfassung der Kapitel (Chapter Summaries)
- Introduction: This chapter introduces the original Black-Scholes model, describes its application to financial instruments, and highlights the concept of symmetry between price and its inverse. It also outlines the aim of the article to explore these symmetries in the context of the generalized Black-Scholes model.
- Preparation: This chapter introduces two key lemmas that are essential for the arguments presented in the article: Itô lemma and the coefficient comparison method for stochastic differential equations.
- Argument: This chapter provides a detailed mathematical argument demonstrating the applicability of the generalized Black-Scholes model to the inverse quantity of price. It utilizes the Itô lemma to derive the relationship between the coefficients of the two equations and shows that the symmetry exists in all cases.
Schlüsselwörter (Keywords)
The main keywords and focus topics of the article include the Black-Scholes model, foreign currency, applicability, Itô lemma, symmetry, stochastic differential equations, and deterministic or stochastic processes. The article explores the applicability of the Black-Scholes model to inverse quantities, investigates the presence of symmetries in its generalized form, and utilizes mathematical tools such as Itô lemma and coefficient comparison methods to analyze the relationships between various parameters.
- Quote paper
- Anonym (Author), 2020, On the symmetries in the generalized Black-Scholes model with variable coefficients, Munich, GRIN Verlag, https://www.grin.com/document/917156
