The Art of Quantitative Finance Vol.2

Volatilities, Stochastic Analysis and Valuation Tools

verfasst von: Gerhard Larcher

Verlag: Springer International Publishing

Buchreihe : Springer Texts in Business and Economics

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

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Über dieses Buch

This textbook provides the necessary techniques from financial mathematics and stochastic analysis for the valuation of more complex financial products and strategies. The author discusses how to make use of mathematical methods to analyse volatilities in capital markets. Furthermore, he illustrates how to apply and extend the Black-Scholes theory to several fields in finance. In the final section of the book, the author introduces the readers to the fundamentals of stochastic analysis and presents examples of applications. This book builds on the previous volume of the author’s trilogy on quantitative finance. The aim of the second volume is to present and discuss more complex and advanced techniques of modern financial mathematics in a way that is intuitive and easy to follow. As in the previous volume, the author provides financial mathematicians with insights into practical requirements when applying financial mathematical techniques in the real world.

Inhaltsverzeichnis

Frontmatter

1. Volatilities

Abstract

Volatility of the underlying is the essential parameter when we have to price derivatives on this underlying. This chapter is dedicated to a detailed study of various concepts of volatility (historical volatility, implied volatility?) and its modelling. Especially we will study the volatilities of the S&P500 index and the volatility index VIX of the S&P500. We also study the dependence between the S&P500 and the VIX and try to model this dependence.

Finally, we will deal with VIX options and VIX futures, and we analyse trading strategies based on combinations of SPX and VIX derivatives.

Gerhard Larcher

2. Extensions of the Black-Scholes Theory to Other Types of Options (Futures Options, Currency Options, American Options, Path-Dependent Options, Multi-asset Options)

Abstract

We extend the basic Black-Scholes formula, which was derived in Volume I for European plain vanilla options to more complex types of derivatives like currency options, futures options, American options, path-dependent options, or multi-asset options. We show how to use these extended formulas to price complex types of options with the help of Monte Carlo methods. For some types of path-dependent options (geometric Asian options, barrier options), we also give explicit valuation formulas. We also discuss refinements of Monte Carlo methods, for example, variance reduction methods for Monte Carlo, or quasi-Monte Carlo methods and their application in option pricing.

Gerhard Larcher

3. Fundamentals: Stochastic Analysis and Applications, Interest Rate Dynamics, and Basic Principles of Pricing Interest Rate Derivatives

Abstract

We start this chapter with a crash course in stochastic analysis. We give a heuristic introduction into the basic principles of these powerful techniques, i.e. we explain in an intuitive way the concepts of stochastic processes, stochastic integration, Ito formula, and stochastic differential equations.

We then apply these tools for modelling interest rates and for pricing interest rate derivatives like caps, floors, or interest rate swaps.

Additionally, we give alternative proofs of the Black-Scholes formula with the help of stochastic analysis, and thereby we gain essential new insights into the dynamics of financial markets. We define complete markets, and we show that the (multidimensional) Black-Scholes market (under certain conditions) is complete. Finally, we consider some examples of incomplete markets, and we analyse possible approaches to the valuation of derivatives in incomplete markets.

Gerhard Larcher

Titel: The Art of Quantitative Finance Vol.2
verfasst von: Gerhard Larcher
Verlag: Springer International Publishing
Electronic ISBN: 978-3-031-23870-3
Print ISBN: 978-3-031-23869-7
DOI: https://doi.org/10.1007/978-3-031-23870-3