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Options, Futures, and Other Derivatives (5th Edition) John C. Hull 
Rating: 4.5/5 Stars Rank: 3460 (Los Angeles, California USA) May 7, 2005 - 4.0/5 stars The author has written a nice, lively elementary text on mathematical finance. This book can serve as a excellent launching point into the topic. For the next step in the reader's development, I recommend the very good intermediate level treatment by Bjork in "Arbitrage Theory in Continuous Time" 2nd edition. As a capstone for advanced study, I recommend the advanced treatment of Musiela and Rutkowski's "Martingale Methods in Financial Modeling. Hull starts out his 5th edition with several chapters on the basics of the derivative contracts in his study. The contracts introduced are forward and futures contracts, interest rate swaps, and equity options. The basic definitions of each contingency contract is given, as well as characteristics of the markets where these contracts trade. Some basic trading strategies are also studied. The study of the option pricing model problem begins in earnest in Chapter 10. The section on one-step binomial tree model leads to a very intuitive description of risk-neutral valuation. Chapter 11 introduces continuous time stochastic processes in a very intuitive setting. To avoid the hard-core Ito calculus, the author motivates the stochastic differential by considering difference equations. This is a nice technique and makes the material accessible to the beginner. The next highlight is a statement of Ito's lemma. This is not given in full generality, but only stated precisely as needed for Black-Scholes calculations. The appendix gives an intuitive motivation for Ito's lemma based on the multi-dimensional Taylor's formula. This is a nice illustration as Taylor's formula is indeed a component of the formal semi-martingale based proof of Ito's rule. See for example Oksendal, "Stochastic Differential Equations" Chapter 4, or Karatzas & Shreve "Brownian Motion and Stochastic Calculus, Chapter 3. Chapter 12 is devoted to the Black-Scholes-Merton theory of option pricing. The famous Black-Scholes PDE is derived via Ito's rule and application of a delta hedge. The author doesn't directly solve this PDE (via the standard application of the Feyman-Kac formula). Instead a nice proof of the option pricing formula is established in the appendix based on a simple log-normal distribution argument. Chapter 13 discusses option pricing in for other contingency contracts. In Chapter 14, we return to equity options by studying the Greek letters. The reader discovers the Greek letters can be thought of as coefficients of the Black-Scholes PDE and learns some elementary hedging techniques. Chapter 15 discusses implied volatility and volatility smiles. It is here that the astute reader gets his first indication that the Black-Scholes theory for option pricing may not be as robust or "true to market" as the reader may have been lead to believe. (The folks at Long-Term Capital Management learned this hard lesson rather publicly.) A survey of topics of interest follows in the next handful of chapters. The material on value at risk, the GARCH volatility model and exotic options is somewhat superficial. The careless reader will come away feeling he knows quite a bit more than he really does. Martingale theory is touched on in 21 and the Girsanov Theorem is alluded to, but these topics are really too complex and require too many prerequisites for proper treatment in the context. A general multi-variate version of Ito's Rule is stated in the appendix of this chapter. The next section of the book deals with term-structure models and their applications. One-factor models are discussed along with the various limitations of each of these models. This gives a nice historical treatment. The Heath-Jarrow-Morton and Libor Market Model k-factor term-structure frameworks are introduced. Without the supporting martingale theory, the analysis of these models presented here is very limited. The last several chapters of the text are very survey-like and breezily touch on topics such as credit risk, credit derivatives and energy derivatives. There isn't a lot of theory in these chapters at all, but at least the reader is made aware of the existence of these kinds of contingencies. The book wraps up with a cautionary chapter in the form of lessons learned. The unwary reader might see all of the derivative-related train wrecks and say to himself "well, that won't be me". The problem is that it really might be you if you truly (and foolishly) still believe the equity prices always follow geometric Brownian motion. See Lo & MacKinlay "A Non-Random Walk Down Wall Street" for an excellent exposition into the limitations of the basic assumptions underpinning the Black-Scholes-Merton theory. If nothing else, Hull's last chapter should convince you that maybe this isn't the only book you'll ever want to read in your study of mathematical finance.
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