Option Theory with Stochastic Analysis: An Introduction to Mathematical Finance / Edition 1

by Fred Espen Benth

Overview

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 exposure, and used in the strategies of investors speculating in markets like fixed-income, sks, 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 shastic finance the purpose is to use economic theory with shastic analysis to derive fair prices for options and derivatives. The results are based on shastic 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.

Product Details

ISBN-13:	9783540405023
Publisher:	Springer Berlin Heidelberg
Publication date:	03/19/2004
Series:	Universitext
Edition description:	Softcover reprint of the original 1st ed. 2004
Pages:	162
Product dimensions:	6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

1 Introduction.- 1.1 An Introduction to Options in Finance.- 1.2 Some Useful Material from Probability Theory.- 2 Statistical Analysis of Data from the Sk Market.- 2.1 The Black & Scholes Model.- 2.2 Logarithmic Returns from Sks.- 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 Auorrelation.- 2.8 Conclusions Regarding the Choice of Sk Price Model.- 3 An Introduction to Shastic Analysis.- 3.1 The Itô Integral.- 3.2 The Itô Formula.- 3.3 Geometric Brownian Motion as the Solution of a Shastic 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)).- 4.4 The Girsanov Theorem and Equivalent Martingale Measures.- 4.5 Pricing and Hedging of General Contingent Claims.- 4.6 The Markov Property and Pricing of General Contingent Claims.- 4.7 Contingent Claims on Many Underlying Sks.- 4.8 Completeness, Arbitrage and Equivalent Martingale Measures.- 4.9 Extensions to Incomplete Markets.- 5 Numerical Pricing and Hedging of Contingent Claims.- 5.1 Pricing and Hedging with Monte Carlo Methods.- 5.2 Pricing and Hedging with the Finite Difference Method.- A Solutions to Selected Exercises.- References.

Editorial Reviews

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)

From the Publisher