Mathematical Finance: Theory, Modeling, Implementation [Hardcover]

 

 

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Disclaimer: As you can see from Amazon RealName (TM), I am the author of the book. The editorial review provided on the back of the book and reproduced on amazon was written by the publisher. However, that editorial review does not provide as much information about the book as I think is necessary. This review hopefully provides you with a more detailed description of the contents and objectives of the book, to help you finding the right book for your needs. In addition you should check the sample chapters available via the books homepage at www.christian-fries.de/finmath/book

The book's main objective is to provide an intuition for the theoretical concepts relevant for derivative pricing and to bridge from the more academic concepts (filtration, random variable, stochastic process) to their application in industry, most notably modeling, calibration and object oriented implementation. It comes with extensive additional material to further explore the key concepts. See the book's home page at christian-fries.de/finmath/book

The book starts like a textbook giving an introduction to probability theory and stochastic processes. However, instead of repeating "Definition-Theorem-Proof" the book often leaves out the proof and adds two special sections: "Motivation" and "Interpretation" (before and after a definition or theorem). The first part makes you acquainted with the mathematical theory and provides the intuition for the fundamental building blocks like random variable, brownian motion, drift and volatility, Ito process, measures, change of measure and numéraire, etc.

In the second part, first applications are, of course, the Black-Scholes model for a single asset. As an excursion important concepts like implied volatility, hedging and the greeks are presented. The results and graphs of these applications may be explored interactively in Java applets on associated web pages.

The third part introduces interest rates, interest rate products and further analytical pricing models. At first, this might come as an arbitrary choice of a specific asset class, namely to focus on interest rates in contrast then equity, foreign exchange (fx), or credit derivatives. However, there is a motivation on why interest rates are a natural choice if one wants to move to more complex derivatives like they have become popular recently: Derivatives feature payments or cash-flows (settlements) at different times, and interest rates are one way to describe the value of future payouts. Mathematically speaking, interest rate products (like bonds or money market accounts) are a natural choice for a numéraire. So interest rates are part of any model (e.g. the black-scholes model for equity and foreign exchange) and considering stochastic interest rates will make these models into hybrid interest rate models.

Before discussing interest rates models (part V) or hybrid models (part VI), the part IV of the book gives a treatment of the numerical implementation of such models. It focuses on Monte-Carlo simulations and their object oriented implementation. Monte-Carlo simulation is one of the most powerful tools in (numerical) derivative pricing. It is also a straight forward approach to implement models, making as few assumption as possible (for example: finite differences, like PDEs and trees are limited to low(er) dimensions). Despite its ubiquitous application, Monte-Carlo simulation brings several disadvantages: a) It is sometimes slower. Given the performance of todays computers, this disadvantage is becoming less important. b) Bermudan options are hard to price. This is solved in Chapter 15. Path-dependent bermudan options are even harder. This is solved in Chapter 16. c) Sensitivities are unstable. This is solved in Chapter 17 and 18.

Part V introduces bigger models, like the LIBOR Market Model, the classical Short Rate Models, Heath-Jarrow-Morton Framework, Cheyette Model and Markov Functional Models. This part focuses a bit on the LIBOR Market Model as it is our workhorse. The calibration of the LIBOR Market Model is discussed (e.g. the calibration to swaption volatility and swap rate covariance) and hints for fast, object oriented implementations are given. Object oriented designs are given in UML diagrams. In "Excursions" concepts like mean-reversion, instantaneous and terminal correlation, multi-factor model, etc. are discussed and illustrated. This part will both endow you with a solid intuition of important model aspects as well as the ability to actually implement such model.

Part VI builds upon the models presented in part V to introduce model extensions like credit spread (credit default) or hybrid models. Examples for hybrid-models are equity-interest rate hybrid model, fx-interest rate hybrid model, multi-currency model. The equity-interest rate hybrid model is essentially a Black-Scholes model (as it was discussed in the second part of the book) with stochastic interest rate modeled by a LIBOR market model (as it was discussed in the fifth part of the book). Since the numéraire is an interest rate product, a Black-Scholes model with stochastic interest rates becomes an interest rate model with an extension.

Part VII gives a short introduction to object oriented implementation.

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