nach oben

2006 | Buch

Kapitel lesen Erstes Kapitel lesen

A Benchmark Approach to Quantitative Finance

verfasst von: Eckhard Platen, David Heath

Verlag: Springer Berlin Heidelberg

Buchreihe : Springer Finance

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

In 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.

Inhaltsverzeichnis

Frontmatter

1. Preliminaries from Probability Theory

Abstract

This chapter reviews some important results from probability theory and fixes notation. First we introduce discrete and continuous random variables and their distributions. Then we discuss functionals of random variables such as moments. Furthermore, we introduce certain classes of distributions and also multivariate distributions together with copulas.

Eckhard Platen, David Heath

2. Statistical Methods

Abstract

We introduce in this chapter further fundamental results from probability theory and statistics which are important in quantitative finance. They are highly relevant for the empirical analysis of financial data. In particular, limit theorems are presented and confidence intervals constructed. Furthermore, the log-returns of a world stock index will be estimated pointing at a stylized empirical fact.

Eckhard Platen, David Heath

3. Modeling via Stochastic Processes

Abstract

In this chapter the fundamental concept of a stochastic process is introduced. We show how stochastic processes can be applied in the context of asset price modeling. The notions of processes with independent increments, stationary processes and Markov processes are explained. Essentially, stochastic processes provide the mathematical framework that allows us to model financial quantities as families of random variables that evolve over time.

Eckhard Platen, David Heath

4. Diffusion Processes

Abstract

In this chapter diffusion processes are introduced. These are potential candidates for the modeling of asset prices, interest rates and other financial quantities. We cover examples on geometric Brownian motion, Ornstein-Uhlenbeck and square root processes.

Eckhard Platen, David Heath

5. Martingales and Stochastic Integrals

Abstract

In this chapter we consider a class of continuous stochastic processes, called martingales, which play a central role in finance. We also define the gains realized from trading as a stochastic integral. Stochastic integration and martingales provide key tools for the analysis of the continuous time evolution of financial markets.

Eckhard Platen, David Heath

6. The Itô Formula

Abstract

The price of a security, for instance, a zero coupon bond which generates some future payoff at a maturity date, is often dependent on the value of an underlying process. In many applications, the effect of changes in the underlying process on this price needs to be quantified. In deterministic calculus this type of problem is handled by the chain rule. In stochastic calculus the corresponding generalization of the chain rule is given by the Itô formula. This stochastic chain rule contains terms reflecting the effect due to the stochastic processes involved having non-zero quadratic variation. In this chapter we introduce, apply and derive the Itô formula. It is widely regarded as the main tool in stochastic calculus and is therefore highly important in quantitative finance.

Eckhard Platen, David Heath

7. Stochastic Differential Equations

Abstract

Stochastic differential equations provide a powerful mathematical framework for the continuous time modeling of asset prices and general financial markets. We consider both scalar and vector stochastic differential equations which allow us to model feedback effects in the market. Explicit solutions will be given in certain cases. Furthermore, questions related to the existence and uniqueness of solutions will be discussed. We also mention stochastic differential equations with jumps which allow us to model event driven uncertainty.

Eckhard Platen, David Heath

8. Introduction to Option Pricing

Abstract

In the previous chapters we have prepared mathematical tools that allow us to model in continuous time the dynamics of financial securities, for instance, stocks. Now, we shall study prices of derived financial securities. A derivative security, for instance an option, is a financial instrument whose value is dependent upon the values of an underlying more fundamental security. In this chapter we give an introduction into derivatives, in particular, European options. For simplicity, we focus our discussion on options under the BS model. Furthermore, we introduce at the end of the chapter important results on squared Bessel processes because these will be crucial for the understanding of the following chapters.

Eckhard Platen, David Heath

9. Various Approaches to Asset Pricing

Abstract

A fundamental result of this chapter is that prices can be generally obtained under the benchmark approach in situations where other approaches are not available. This chapter also clarifies relationships between real world pricing under the benchmark approach and the pricing by other means in the areas of finance and insurance. Furthermore, it presents the Girsanov transformation, the change of numeraire technique and the Feynman-Kac formula, which are all highly relevant to derivative pricing.

Eckhard Platen, David Heath

10. Continuous Financial Markets

Abstract

This and the following three chapters present a range of new concepts and ideas that do not fit under presently prevailing approaches. They derive a general, unified framework for modeling continuous financial markets.

Eckhard Platen, David Heath

11. Portfolio Optimization

Abstract

This chapter derives and extends a range of classical results from portfolio optimization and derivative pricing in incomplete markets in the context of a CFM. First, we consider the question of how wealth should be optimally transferred into the future given the preferences of an investor. This is a central question in economics and finance and leads into the area of portfolio optimization. We shall advocate the GOP as the best long term investment. This is consistent with views formulated in Latané (1959), Breiman (1961), Hakansson (1971) and Thorp (1972).

Eckhard Platen, David Heath

12. Modeling Stochastic Volatility

Abstract

This chapter introduces into the pricing and hedging of derivatives under stochastic volatility. The emphasis is on standard derivatives for various index models. We choose as underlying security a diversified index, which we interpret as GOP.

Eckhard Platen, David Heath

13. Minimal Market Model

Abstract

This chapter derives an alternative model for the long term dynamics of the GOP from basic economic arguments. The discounted GOP drift, which models the long term trend of the economy, is chosen as the key parameter process. This leads to the minimal market model with the discounted GOP forming a time transformed squared Bessel process of dimension four. Its dynamics allows us to explain various empirical stylized facts and other properties relating to the long term behavior of a world stock index.

Eckhard Platen, David Heath

14. Markets with Event Risk

Abstract

After having studied continuous financial markets, this chapter applies the benchmark approach to markets that exhibit jumps due to event risk. It generalizes several results previously obtained to the case of jump diffusion markets (JDMs).

Eckhard Platen, David Heath

15. Numerical Methods

Abstract

This final chapter describes a range of numerical methods that have been used for the pricing of derivative contracts and other tasks in quantitative finance. First we describe random number generation and simulation methods for scenario and Monte Carlo simulation. Finally, we introduce tree methods and numerical schemes for the solution of partial differential equations.

Eckhard Platen, David Heath

16. Solutions for Exercises

Eckhard Platen, David Heath

Backmatter

Titel: A Benchmark Approach to Quantitative Finance
verfasst von: Eckhard Platen
David Heath
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-47856-0
Print ISBN: 978-3-540-26212-1
DOI: https://doi.org/10.1007/978-3-540-47856-0