Mathematics of Financial Markets, Volume 10
Springer Science & Business Media, 2005 - Business & Economics - 352 pages
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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Pricing by Arbitrage
The First Fundamental Theorem
Discretetime American Options
ContinuousTime Stochastic Calculus
ContinuousTime European Options
The American Put Option
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admissible strategy American option American put option arbitrage arbitrage opportunity assume Black-Scholes bond bounded Brownian motion Chapter claim H coherent risk measure condition Consequently consider contingent claim continuous continuous-time convergence convex deﬁned Deﬁnition denote differential equation equivalent martingale measure European call option exercise ﬁltration ﬁnite finite market model ﬁrst function given hedging strategy Hence implies initial interest rate investor Lemma local martingale measure Q non-negative numéraire option price portfolio price process probability measure probability space Proof Proposition put option random variable result follows right-continuous risk-neutral measure riskless risky asset satisﬁes self-financing strategy sequence solution standard Brownian motion stock price supermartingale Suppose Theorem trading strategy unique value process vector viable wealth process Write Zero Zero coupon bond