## Option Pricing and Portfolio Optimization: Modern Methods of Financial MathematicsUnderstanding and working with the current models of financial markets requires a sound knowledge of the mathematical tools and ideas from which they are built. Banks and financial houses all over the world recognize this and are avidly recruiting mathematicians, physicists, and other scientists with these skills. The mathematics involved in modern finance springs from the heart of probability and analysis: the Ito calculus, stochastic control, differential equations, martingales, and so on. The authors give rigorous treatments of these topics, while always keeping the applications in mind. Thus, the way in which the mathematics is developed is governed by the way it will be used, rather than by the goal of optimal generality.Indeed, most of purely mathematical topics are treated in extended 'excursions' from the applications into the theory. Thus, with the main topic of financial modelling and optimization in view, the reader also obtains a self-contained and complete introduction to the underlying mathematics. This book is specifically designed as a graduate textbook. It could be used for the second part of a course in probability theory, as it includes an applied introduction to the basics of stochastic processes (martingales and Brownian motion) and stochastic calculus. It would also be suitable for a course in continuous-time finance that assumes familiarity with stochastic processes. The prerequisites are basic probability theory and calculus. Some background in stochastic processes would be useful, but not essential. |

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### Contents

1 | |

The ContinuousTime Market Model | 11 |

Option Pricing | 79 |

Pricing of Exotic Options and Numerical Algorithms | 153 |

Optimal Portfolios | 203 |

247 | |

### Common terms and phrases

algorithm approximation arbitrage arbitrage opportunity assume assumptions binomial model binomial tree Black-Scholes formula Black-Scholes model bounded Brownian motion Cauchy problem complete market compute constant consumption contingent claim continuous-time Corollary corresponding wealth process defined Definition discounted distribution Donsker's theorem equivalent martingale measure European call Exercise existence exotic options expectation final payment final wealth given Hence HJB-equation implies inequality initial wealth investor Ito formula Ito process KA/SH Lemma local martingale market model martingale representation theorem method notation numbers numeraire obtain option price P-almost surely partial differential equation payoff Pi(t portfolio problem portfolio process price process probability measure progressively measurable Proposition random variable Remark replication strategy requirement riskless satisfies Section sequence simple processes solve stochastic control stochastic differential equation stochastic integral stochastic process stock price strike price suitable terminal wealth trading strategy TT,c variance wealth process corresponding yields

### Popular passages

Page ix - Subsequently, the financial markets introduced (and still are introducing!) more and more types of derivatives often having very complex structures. For their quantitative valuation it is essential to have a sound knowledge of mathematical models for financial markets and to be able to handle the corresponding mathematical toolbox. Here. the most important tool has become the Ito calculus.

Page x - We shall present numerous examples of such options in this book, some of them with explicit pricing formulas. For obtaining prices for exotic options where an explicit pricing formula cannot be found. numerical methods are needed. Therefore. in Chapter 3 we also describe the basics of Monte Carlo methods.

Page x - We shall start by introducing the method of option pricing via replication and no arbitrage. This approach is based on the principle that the price of an option should exactly equal the amount of money needed to create the option's payments synthetically.

Page ix - Thus, it was natural that R. Merton and M. Scholes were awarded the Nobel Prize in economics for their work contributing to the Black-Scholes formula.

Page xiii - C0' (the collection of functions that have compact support and that are continuously differentiable with respect to the first variable, and twice continuously differentiable with respect to the second variable).

Page x - One aim of this book is a fast and at the same time rigorous introduction to Ito calculus.

Page x - For understanding most parts of this book a basic course in probability theory is sufficient.