Mathematics of Financial MarketsThis work is aimed at an audience with asound mathematical background wishing to leam about the rapidly expanding field of mathematical finance. Its content is suitable particularly for graduate students in mathematics who have a background in measure theory and prob ability. The emphasis throughout is on developing the mathematical concepts re quired for the theory within the context of their application. No attempt is made to cover the bewildering variety of novel (or 'exotic') financial instru ments that now appear on the derivatives markets; the focus throughout 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 financial markets. The first five 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 nonarbitrage) 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 intro duces the idea of a martingale measure for price pro cesses. Following a discussion of the use of self-financing trading strategies to hedge against trading risk, it is shown how options can be priced using an equivalent measure for which the discounted price process is a mar tingale. |
Contents
Bonds and Term Structure | 9 |
Martingale Measures | 23 |
The Fundamental Theorem of Asset Pricing | 45 |
Complete Markets and Martingale Representation | 62 |
Stopping Times and American Options | 75 |
A Review of ContinuousTime Stochastic Calculus | 99 |
xi | 123 |
21 | 132 |
23 | 154 |
32 | 161 |
45 | 172 |
ConsumptionInvestment Strategies | 251 |
59 | 275 |
| 284 | |
| 288 | |
European Options in Continuous Time | 135 |
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Mathematics of Financial Markets, Volume 10 Robert J. Elliott,P. Ekkehard Kopp No preview available - 2005 |
Common terms and phrases
American option arbitrage arbitrage opportunity Black-Scholes bond Brownian motion Chapter Consequently consider consumption contingent claim continuous continuous-time convergence decomposition defined Definition denotes differential discounted equation equivalent martingale measure European call option exercise F-measurable filtration F Finance finite market model function Girsanov's theorem given hedging strategy hence initial interest rate Itô process Lemma local martingale martingale representation measure Q non-negative numéraire o-field optimal stopping option price predictable process price process probability measure probability space Proof Proposition R.J. Elliott random variable result follows risk-neutral measure riskless risky asset S¹(t satisfies self-financing strategy Snell envelope solution standard Brownian motion stochastic integral stock price supermartingale Suppose Theorem tingale trading strategy u)du uniformly integrable unique value process vector viable wealth process Write zero zero coupon bond