Option Theory with Stochastic Analysis: An Introduction to Mathematical Finance

ISBN-13: 9783540405023 / Angielski / Miękka / 2003 / 162 str.

Fred Espen Benth

Opis

Since 1972 and the appearance of the famous Black & Scholes option pric- ing formula, derivatives have become an integrated part of everyday life in the financial industry. Options and derivatives are tools to control risk ex- posure, and used in the strategies of investors speculating in markets like fixed-income, stocks, currencies, commodities and energy. A combination of mathematical and economical reasoning is used to find the price of a derivatives contract. This book gives an introduction to the theory of mathematical finance, which is the modern approach to analyse options and derivatives. Roughly speaking, we can divide mathematical fi- nance into three main directions. In stochastic finance the purpose is to use economic theory with stochastic analysis to derive fair prices for options and derivatives. The results are based on stochastic modelling of financial as- sets, which is the field of empirical finance. Numerical approaches for finding prices of options are studied in computational finance. All three directions are presented in this book. Algorithms and code for Visual Basic functions are included in the numerical chapter to inspire the reader to test out the theory in practice. The objective of the book is not to give a complete account of option theory, but rather relax the mathematical rigour to focus on the ideas and techniques.

Recenzje

From the reviews:

"This is a ... book concerned solely with describing the mathematics of option pricing and I found it a delight to read. It is very well written, quite comprehensive and non-rigorous so that it can be used on courses aimed at a variety of students. ... The book includes a healthy number of exercises and there are fully worked solutions to most of these." (David Applebaum, The Mathematical Gazette, Vol. 90 (517), 2006)

"The book provides an introduction to the basic ideas of the mathematical theory of financial options valuation, or, more concretely, to the Black-Scholes theory of pricing contingent claims on equity. ... The text is a brief, neat, carefully written introduction to the fundamentals of the mathematics and the modelling of the analysis of options pricing." (José Lúis Fernandez Perez, Zentralblatt MATH, Vol. 1042 (17), 2004)

Spis treści

1 Introduction.- 1.1 An Introduction to Options in Finance.- 1.1.1 Empirical Finance.- 1.1.2 Stochastic Finance.- 1.1.3 Computational Finance.- 1.2 Some Useful Material from Probability Theory.- 2 Statistical Analysis of Data from the Stock Market.- 2.1 The Black & Scholes Model.- 2.2 Logarithmic Returns from Stocks.- 2.3 Scaling Towards Normality.- 2.4 Heavy-Tailed and Skewed Logreturns.- 2.5 Logreturns and the Normal Inverse Gaussian Distribution.- 2.6 An Alternative to the Black & Scholes Model.- 2.7 Logreturns and Autocorrelation.- 2.8 Conclusions Regarding the Choice of Stock Price Model.- 3 An Introduction to Stochastic Analysis.- 3.1 The Itô Integral.- 3.2 The Itô Formula.- 3.3 Geometric Brownian Motion as the Solution of a Stochastic Differential Equation.- 3.4 Conditional Expectation and Martingales.- 4 Pricing and Hedging of Contingent Claims.- 4.1 Motivation from One-Period Markets.- 4.2 The Black & Scholes Market and Arbitrage.- 4.3 Pricing and Hedging of Contingent Claims X= f(S(T)).- 4.3.1 Derivation of the Black & Scholes Partial Differential Equation.- 4.3.2 Solution of the Black & Scholes Partial Differential Equation.- 4.3.3 The Black & Scholes Formula for Call Options.- 4.3.4 Hedging of Call Options.- 4.3.5 Hedging of General Options.- 4.3.6 Implied Volatility.- 4.4 The Girsanov Theorem and Equivalent Martingale Measures.- 4.5 Pricing and Hedging of General Contingent Claims.- 4.5.1 An Example: a Chooser Option.- 4.6 The Markov Property and Pricing of General Contingent Claims.- 4.7 Contingent Claims on Many Underlying Stocks.- 4.8 Completeness, Arbitrage and Equivalent Martingale Measures.- 4.9 Extensions to Incomplete Markets.- 4.9.1 Energy Markets and Incompleteness.- 5 Numerical Pricing and Hedging of Contingent Claims.- 5.1 Pricing and Hedging with Monte Carlo Methods.- 5.1.1 Pricing and Hedging of Contingent Claims with Payoff of the Form f(ST).- 5.1.2 The Accuracy’ of Monte Carlo Methods.- 5.1.3 Pricing of Contingent Claims on Many Underlying Stocks.- 5.1.4 Pricing of Path-Dependent Claims.- 5.2 Pricing and Hedging with the Finite Difference Method.- A Solutions to Selected Exercises.- References.

Blurb

The objective of this textbook is to provide a very basic and accessible introduction to option pricing, invoking only a minimum of stochastic analysis. Although short, it covers the theory essential to the statistical modeling of stocks, pricing of derivatives (general contingent claims) with martingale theory, and computational finance including both finite-difference and Monte Carlo methods. The reader is led to an understanding of the assumptions inherent in the Black & Scholes theory, of the main idea behind deriving prices and hedges, and of the use of numerical methods to compute prices for exotic contracts. Finally, incomplete markets are also discussed, with references to different practical/theoretical approaches to pricing problems in such markets.
The author's style is compact and to-the-point, requiring of the reader only basic mathematical skills. In contrast to many books addressed to an audience with greater mathematical experience, it can appeal to many practitioners, e.g. in industry, looking for an introduction to this theory without too much detail.
It dispenses with introductory chapters summarising the theory of stochastic analysis and processes, leading the reader instead through the stochastic calculus needed to perform the basic derivations and understand the basic tools
It focuses on ideas and methods rather than full rigour, while remaining mathematically correct.
The text aims at describing the basic assumptions (empirical finance) behind option theory, something that is very useful for those wanting actually to apply this. Further, it includes a big section on pricing using both the pde-approach and the martingale approach (stochastic finance).
Finally, the reader is presented the two main approaches for numerical computation of option prices (computational finance). In this chapter, Visual Basic code is supplied for all methods, in the form of an add-in for Excel.
The book can be used at an introductory level in Universities. Exercises (with solutions) are added after each chapter.