A Benchmark Approach to Quantitative FinanceIn recent years products based on ?nancial derivatives have become an ind- pensabletoolforriskmanagersandinvestors. Insuranceproductshavebecome part of almost every personal and business portfolio. The management of - tual and pension funds has gained in importance for most individuals. Banks, insurance companies and other corporations are increasingly using ?nancial and insurance instruments for the active management of risk. An increasing range of securities allows risks to be hedged in a way that can be closely t- lored to the speci?c needs of particular investors and companies. The ability to handle e?ciently and exploit successfully the opportunities arising from modern quantitative methods is now a key factor that di?erentiates market participants in both the ?nance and insurance ?elds. For these reasons it is important that ?nancial institutions, insurance companies and corporations develop expertise in the area of quantitative ?nance, where many of the as- ciated quantitative methods and technologies emerge. This book aims to provide an introduction to quantitative ?nance. More precisely, it presents an introduction to the mathematical framework typically usedin?nancialmodeling,derivativepricing,portfolioselectionandriskm- agement. It o?ers a uni?ed approach to risk and performance management by using the benchmark approach, which is di?erent to the prevailing paradigm and will be described in a systematic and rigorous manner. This approach uses the growth optimal portfolio as numeraire and the real world probability measure as pricing measure. |
Contents
| 1 | |
Statistical Methods | 55 |
Modeling via Stochastic Processes | 99 |
Diffusion Processes | 133 |
Martingales and Stochastic Integrals | 163 |
The Itô Formula | 204 |
Stochastic Differential Equations | 237 |
Introduction to Option Pricing | 276 |
Portfolio Optimization | 403 |
Modeling Stochastic Volatility | 438 |
Minimal Market Model | 483 |
Markets with Event Risk | 512 |
Numerical Methods | 551 |
Solutions for Exercises | 614 |
References | 667 |
Author Index | 682 |
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Common terms and phrases
approximate asset price benchmark approach Bessel process converges corresponding covariation denote diffusion coefficient diffusion process discounted GOP distributed random variable drift equation equivalent risk neutral estimate European call option finite geometric Brownian motion given growth rate hedge implied volatility independent initial value Itô differential Itô formula Itô integral jump Lévy process locally optimal portfolio log-returns market model Markov martingale matrix neutral probability measure nonnegative Note numeraire obtain option price P)-martingale parameter payoff Platen Poisson process primary security account probability measure quadratic variation quantitative finance real world pricing risk neutral risk neutral probability savings account Sect sequence short rate solution squared Bessel process standard Wiener process stationary density stochastic process strictly positive portfolio Theorem transition density underlying security variance vector Wiener process zero coupon bond