Arbitrage Theory in Continuous TimeProfessor Bjork provides an accessible introduction to the classical underpinnings of the central mathematical theory behind modern finance. Combining sound mathematical principles with the necessary economic focus, Arbitrage Theory in Continuous Time is specifically designed for graduate students, and includes solved examples for every new technique presented, numerous exercises, and Further Reading lists for each chapter. -;The second edition of this popular introduction to the classical underpinnings of the mathematics behind finance continues to combine sound mathematical principles with economic applications.Concentrating on the probabilistic theory of continuous arbitrage pricing of financial derivatives, including stochastic optimal control theory and Merton's fund separation theory, the book is designed for graduate students and combines necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter.In this substantially extended new edition Tomas Bj--ouml--;rk has added completely new chapters on measure theory and probability theory, including the Radon-Nikodym Theorem, Girsanov transformations, and stochastic integral martingale representations. There is also an extensive new chapter on the abstract martingale approach to arbitrage theory, including a guided tour through the Delbaen-Schachermayer proof of the first fundamental theorem, as well as a new chapter on the LIBOR and swap marketmodels. Providing two full treatments of arbitrage theory - the classical delta hedging approach and the modern martingale approach - the book is written in such a way that these approaches can be studied independently of each other, thus providing the less mathematically oriented reader with a selfcontained introduction to arbitrage theory, while at the same time allowing the specialist to see the full theory in action.This is the textbook of choice for graduate students and advanced undergraduates studying finance and an invaluable introduction to mathematical finance for mathematicians and professionals in financial markets. |
Contents
1 Introduction | 1 |
2 The Binomial Model | 6 |
3 Stochastic Integrals | 27 |
4 Differential Equations | 52 |
5 Portfolio Dynamics | 69 |
6 Arbitrage Pricing | 76 |
7 Completeness and Hedging | 99 |
8 Parity Relations and Delta Hedging | 108 |
13 Barrier Options | 182 |
14 Stochastic Optimal Control | 198 |
15 Bonds and Interest Rates | 228 |
16 Short Rate Models | 242 |
17 Martingale Models for the Short Rate | 252 |
18 Forward Rate Models | 266 |
19 Change of Numeraire | 274 |
20 Forwards and Futures | 297 |
9 Several Underlying Assets | 119 |
10 Incomplete Markets | 135 |
11 Dividends | 154 |
12 Currency Derivatives | 167 |
303 | |
308 | |
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Common terms and phrases
argument asset assume Assumption Black–Scholes bond bond prices claim complete compute condition consider consisting constant contingent claim contract corresponding coupon course currency defined definition denote derivative determined deterministic differential distribution dividend domestic dynamics equal equation European call example Exercise exists expected expression fact fixed foreign formal formula forward rate function futures given gives hold integral interpretation interval linear martingale measure matrix maturity means measure Q natural notation Note object observed obtain optimal option particular portfolio possible price process pricing function probability problem Proof Proposition prove rate of interest reason relation replicating result risk neutral satisfies short rate Show simple solution solve standard stochastic stochastic differential equations stock price structure T-claim term Theorem theory traded underlying asset variable vector volatility Wiener process write zero