Mathematics of Financial Markets, Volume 10

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Springer Science & Business Media, 2005 - Business & Economics - 352 pages
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This work is aimed at an audience with a sound mathematical background wishing to learn about the rapidly expanding ?eld of mathematical ?nance. Its content is suitable particularly for graduate students in mathematics who have a background in measure theory and probability. The emphasis throughout is on developing the mathematical concepts required for the theory within the context of their application. No attempt is made to cover the bewildering variety of novel (or ‘exotic’) ?nancial - struments that now appear on the derivatives markets; the focus throu- out remains on a rigorous development of the more basic options that lie at the heart of the remarkable range of current applications of martingale theory to ?nancial markets. The ?rst ?ve chapters present the theory in a discrete-time framework. Stochastic calculus is not required, and this material should be accessible to anyone familiar with elementary probability theory and linear algebra. The basic idea of pricing by arbitrage (or, rather, by non-arbitrage) is presented in Chapter 1. The unique price for a European option in a single-period binomial model is given and then extended to multi-period binomial models. Chapter 2 introduces the idea of a martingale measure for price processes. Following a discussion of the use of self-?nancing tr- ing strategies to hedge against trading risk, it is shown how options can be priced using an equivalent measure for which the discounted price p- cess is a martingale.
 

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Contents

Pricing by Arbitrage
1
Martingale Measures
27
The First Fundamental Theorem
57
Complete Markets
87
Discretetime American Options
105
ContinuousTime Stochastic Calculus
131
ContinuousTime European Options
167
The American Put Option
223
Bonds and Term Structure
247
ConsumptionInvestment Strategies
285
Measures of Risk
303
Bibliography
329
Index
342
308
349
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About the author (2005)

P. Ekkehard Kopp is Professor of Mathematics, and a former Pro-Vice-Chancellor, at the University of Hull.