This academic paper focuses on breaking down the magic of the Black-Scholes formula, which is used to value options. The author first introduces basic concepts like options, option strategies and the put-call parity to guide the reader through the underlying, basic concepts. To illustrate the use and the power of the Black-Scholes formula, two examples are calculated to better understand the complex steps involved in finding the call value. Finally, a failure case is presented, to show some pitfalls of this mathematical function.
Table of Contents
1. INTRODUCTION
2. FROM THE BASICS TO THE PARITY
2.1 PUT AND CALL OPTIONS
2.2 PUT-CALL PARITY
3. BLACK-SCHOLES – AN OPTION PRICING MODEL
3.1 ASSUMPTIONS OF THE MODEL AND ITS INFLUENCING FACTORS
3.2 THE BLACK-SCHOLES FORMULA
3.3 THE BLACK-SCHOLES FORMULA IN PRACTICE
3.3.1 Fictional Example
3.3.2 General Electric Example
3.4 EXCURSION TO THE GREEKS
4. CONCLUSION
Objectives and Research Focus
This paper aims to provide a comprehensive walkthrough of the Black-Scholes model, explaining its theoretical foundations, its mathematical application in option pricing, and its practical limitations in real-world financial markets.
- Theoretical overview of put and call options and put-call parity.
- Detailed breakdown of the Black-Scholes model assumptions and formula components.
- Practical demonstration of the formula using both fictional and real-world market data.
- Introduction to "The Greeks" as sensitivity measures for option pricing.
- Critical analysis of model failure through the case study of Long-Term Capital Management (LTCM).
Excerpt from the Book
3.3 THE BLACK-SCHOLES FORMULA IN PRACTICE
The author first wants to illustrate the above explained formula using a fictional examples. Afterwards, the Black-Scholes model will be applied to a real world case.
3.3.1 Fictional Example
What is the price of a call option, given the following values:
S = 80, R = 5%, σ = 0.4068, X = 80 and t = 180/365 days
Now where we have derived the values for d1 and d2, one can now substitude the values into the basic Black-Scholes formula.
C = 80*N(0.2297) – Xe-rtN(-0.058)
Using the Standard Distribution Table, the N(d1) and N(d2) values can be derived as being:
N(d1) = 0.5908 and N(d2) = 0.4769.
However, the present value of the strike price still needs to be calculated beforehand:
PV(X) = 80*e-(0.05)*(180/365)
PV (X) = 78.05
Further, all variable are known, and the formula can be finally solved:
C = 80*0.5908 – 78.05*0.4769
C = $10.05
Hence, the value of the call for the details mentioned above is $10.05 (Allen, Brealey & Myers, 2007).
Summary of Chapters
1. INTRODUCTION: Outlines the historical development of option pricing theory and the significance of the Black-Scholes model in modern finance.
2. FROM THE BASICS TO THE PARITY: Defines fundamental option types and explains the put-call parity relationship essential for understanding option pricing.
3. BLACK-SCHOLES – AN OPTION PRICING MODEL: Details the core assumptions, mathematical structure, and practical application of the Black-Scholes formula, including sensitivity factors.
4. CONCLUSION: Evaluates the utility of the model while cautioning against its blind application, illustrated by the failure of the LTCM fund.
Keywords
Black-Scholes Model, Option Pricing, Financial Derivatives, Put-Call Parity, Stock Price, Strike Price, Volatility, The Greeks, Delta, Gamma, Theta, Vega, Risk-Free Rate, Long-Term Capital Management, LTCM.
Frequently Asked Questions
What is the primary focus of this work?
The work provides a technical and practical walkthrough of the Black-Scholes option pricing model, detailing its history, components, and application.
What are the central themes discussed?
The paper covers option basics, the mathematical derivation of the Black-Scholes formula, real-world pricing examples, and the risks associated with reliance on theoretical financial models.
What is the main research objective?
The objective is to explain how to calculate option values using the Black-Scholes formula and to illustrate the importance of understanding the model's limitations through case studies.
Which scientific methodology is applied?
The paper utilizes a descriptive and analytical approach, combining financial theory with practical calculation examples and a retrospective case study analysis.
What does the main body of the paper cover?
It covers the transition from basic option definitions to complex pricing formulas, provides step-by-step calculations for fictional and real-world scenarios, and defines sensitivity metrics known as "The Greeks".
How would you describe the document's keywords?
The keywords focus on quantitative finance, specific components of option pricing, and the historical context of model-based financial failures.
What role does the put-call parity play in the calculation of an option?
Put-call parity establishes a mathematical relationship between the prices of puts, calls, and the underlying asset, allowing one to derive the value of a put if the call price and other variables are known.
How does the author characterize the failure of LTCM?
The author uses LTCM as a cautionary example of how reliance on purely mathematical models, while powerful, can lead to catastrophic losses when unexpected, random market events occur.
What are "The Greeks" in the context of this paper?
The Greeks are sensitivity measures (Delta, Gamma, Theta, Vega) used to quantify how an option's price changes in response to fluctuations in underlying variables like price, volatility, and time.
- Quote paper
- Cornelius Kirsche (Author), 2012, Black-Scholes Formula: A Walkthrough, Munich, GRIN Verlag, https://www.hausarbeiten.de/document/197990
