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.
Table of Contents
1. Introduction
2. Preparation
3. Argument
4. Conclusion and Future Work
Objectives and Research Themes
This article investigates the applicability of the generalized Black-Scholes model to the inverse quantity of a price by analyzing symmetry relations between a financial instrument and its inverse. The primary research question addresses whether the set of real numbers representing drift and volatility in the original model holds its symmetric structure when generalized to deterministic or stochastic coefficients.
- Mathematical analysis of the symmetry between price and inverse price.
- Application of the Itô lemma in generalized stochastic differential equations.
- Uniqueness proof for the coefficients in the generalized model.
- Investigation of applicability regarding foreign currency valuation.
- Extension of Black-Scholes symmetry to time-dependent and stochastic coefficient models.
Excerpt from the Book
3 Argument
The inverse quantity of price is defined as follows:
Definition 3.
β := S−1
Let S be the price of foreign currency from the view of domestic currency, β express the price of the later seen from the former as mentioned in previous research. This is why the issue on the applicability can be fomulated as the argument on the symmetry between S and β in which cases there exists the set of processes (μ', σ') satisfying the system of equations below:
dS = μSdt + σSdW (1)
dβ = μ'βdt + σ'βdW (2)
with all coefficients depending on some variables generally.
Due to (2) in the system and Definition 3., by substituting x = β, f = μ'β, g = σ'β and h = S = β−1 into Lemma 1.,
dS = {∂S/∂β μ'β + 1/2 ∂2S/∂β2 (σ'β)2 + ∂S/∂t }dt + ∂S/∂β σ'βdW
= {− 1/β2μ'β + 1/2 · 2 · 1/β3 σ'2β2}dt − 1/β2 σ'βdW
= 1/β (σ'2 − μ')dt + 1/β (−σ')dW
= S (σ'2 − μ')dt + S (−σ')dW
By comparing coefficients of (1) and this, validated by Lemma 2.,
μ = σ'2 − μ'
σ = −σ'
Summary of Chapters
1. Introduction: The chapter establishes the stochastic foundation of the original Black-Scholes model and introduces the research problem regarding the symmetry of inverse price quantities.
2. Preparation: This section provides the necessary mathematical tools, specifically the Itô lemma and the method of coefficient comparison, required to analyze the generalized models.
3. Argument: This core chapter formally defines the inverse price quantity and derives the symmetric relationship between the original and inverse processes through rigorous algebraic manipulation.
4. Conclusion and Future Work: The chapter summarizes the proof of applicability for the generalized model and suggests future research into even more general stochastic differential models.
Keywords
Black-Scholes model, foreign currency, applicability, Itô lemma, symmetry, stochastic differential equations, drift, volatility, inverse price, financial instrument, coefficient comparison, Wiener process, stochastic processes, mathematical finance, modeling.
Frequently Asked Questions
What is the core focus of this research?
The research focuses on the applicability of the Black-Scholes model when dealing with the inverse quantity of a price, specifically examining whether the symmetric relations found in the original model persist in a generalized framework.
What are the central thematic areas?
The central themes include stochastic calculus, the symmetry of financial variables, the generalization of drift and volatility coefficients, and the mathematical validation of financial model consistency.
What is the primary goal of the study?
The goal is to determine if the set of processes representing drift and volatility in the generalized Black-Scholes model satisfies the same symmetric expressions as the original constant-coefficient model.
Which scientific methodology is utilized?
The author employs a rigorous mathematical approach, utilizing the Itô lemma to differentiate stochastic processes and the coefficient comparison method to validate the consistency of the derived equations.
What topics are covered in the main section?
The main section covers the formal definition of inverse price quantities, the derivation of differential equations for these quantities, and the subsequent comparison of coefficients to prove the existence and uniqueness of the symmetric relationships.
Which keywords characterize this work?
Key terms include Black-Scholes model, Itô lemma, stochastic differential equations, symmetry, inverse price, and drift/volatility estimation.
How does the generalized model differ from the original?
The original model assumes constant coefficients for drift and volatility, whereas the generalized model allows these coefficients to be deterministic or stochastic processes depending on time, price, or other variables.
What is the significance of the "beautiful symmetric expression" mentioned?
It refers to the elegant mathematical result where the relationship between the drift and volatility of the price and its inverse is uniquely determined by simple additions and sign reversals (e.g., μ + μ' = σ²).
What are the implications for foreign currency markets?
The findings suggest that the Black-Scholes framework is robust enough to be applied to both a currency and its inverse without losing the underlying symmetry, which is essential for consistent option pricing.
What is proposed for future investigation?
The author suggests extending this analysis to even more general models where the drift and volatility are expressed as arbitrary functions of time, price, and other sample space variables.
- 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
