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Financial Calculus: An Introduction to Derivative Pricing

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Financial Calculus: An Introduction to Derivative Pricing

Here is the first rigorous and accessible account of the mathematics behind the pricing, construction, and hedging of derivative securities. With mathematical precision and in a style tailored for market practioners, the authors describe key concepts such as martingales, change of measure, and the Heath-Jarrow-Morton model. Starting from discrete-time hedging on binary trees, the authors develop continuous-time stock models (including the Black-Scholes method). They stress practicalities includi

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2 thoughts on “Financial Calculus: An Introduction to Derivative Pricing”

  1. 40 of 43 people found the following review helpful
    5.0 out of 5 stars
    Absolutely top-notch, June 1, 1999
    By A Customer
    This review is from: Financial Calculus: An Introduction to Derivative Pricing (Hardcover)

    This is an elegant book for students of financial mathematics. You won’t see the tedious Theorem/Proof format so common in other similar textbooks. But what it lacks in rigor it more than makes up for in other more important areas: superb writing, clear explanations and brilliant insight into the most popular valuation models. For instance, the concise but very clear derivation of the Black-Scholes formula should impress anyone who has studied the PDE-based derivation covered by Hull and others.

    There is little discussion of empirical issues. This, in my opinion, was a wise choice by the authors. Any such discussion would severely dilute the strength of the book — namely, the fundamentals of stochastic calculus applied to arbitrage pricing. For those interested in empirical features of the markets, I’d suggest “Econometrics of Financial Markets” (Andy Lo, et al).

    I find it ironic that the punchline for the whole book — a chapter on exotic option valuation where probabilistic techniques such as the reflection principle naturally come into play — did not make it to production. But this excellent chapter is available on the errata Web page under [...]

    This book is a great place to begin study for quantitative MBA students or math students with an interest in option valuation. Supplement this book with Oksendal or Karatzas / Shreve, perhaps, for more in-depth material on stochastic calculus.

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  2. 65 of 66 people found the following review helpful
    4.0 out of 5 stars
    Nice, compact book on financial engineering, March 10, 2001
    By 
    Dr. Lee D. Carlson (Baltimore, Maryland USA) –
    (VINE VOICE)
      
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    This review is from: Financial Calculus: An Introduction to Derivative Pricing (Hardcover)

    This book is an introduction to financial engineering from the standpoint of martingales, and assumes the reader knows only elementary calculus and probability theory. After giving a motivating example entitled “the parable of the bookmaker” the authors clarify in the introduction the difference between pricing derivatives by expected value versus using the concept of arbitrage. Vowing then never to use the strong law of large numbers to price derivatives, discrete processes are take up in the next chapter. The authors do an excellent job of discussing the binomial tree model using only elementary mathematics. Interestingly, they introduce the concept of a filtration in constructing the binomial tree model for pricing. Filtrations are usually introduced formally in other books as a measure theory concept. They then define a martingale using a filtration and a choice of measure. The use of martingales pretty much dominates the rest of the book. They emphasize that a martingale can be a martingale with respect to one measure but not to another. Continuous models are the subject of the next chapter, where the ubiquitous Brownian motion is introduced. The discussion is very lucid and easy to understand, and they explain why the conditions in the definition of Brownian motion make its use nontrivial; namely, one must pay attention to all the marginals conditioned on all the filtrations (or histories). The Ito calculus is then appropriately introduced along with stochastic differential equations. The authors do a good job of discussing the difference between stochastic calculus and Newtonian calculus. Recognizing that the Brownian motion they have defined is with respect to a given measure, they then ask the reader to consider the effect of changing the measure, thus motivating the idea of a Radon-Nikodym derivative. Their discussion is very intuitive and promotes a clear understanding rather than giving a mere formal measure-theoretic definition. Many interesting examples of changes are given. Portfolio construction and the Black-Scholes model follows. Basing their treatment of the Black-Scholes model of martingales gives an interesting and enlightening alternative to the usual ones based on partial differential equations (they do however later show how to obtain the usual equations). The next chapter discusses how to use the Black-Scholes equations to price market securities and how to assess the market price of risk. The discussion is very understandable but not enough exercises are given. Modeling interest rates is the subject of the next chapter. The mathematical treatiment is somewhat more involved than the rest of the the book. Several models of interest rate dynamics are discussed here very clearly, including the Ho/Lee, Vasicek, Cox-Ingersoll-Ross, Black-Karasinski, and Brace-Gatarek-Musiela models. A few of these models were unfamliar to me so I appreciated the author’s detailed discussion. The book ends with a discussion of extensions to the Black-Scholes model. The emphasis is on multiple stock and foreign currency interest-rate models. A brief discussion of the Harrison/Pliska theorem is given with references indicated for the proof. An excellent book and recommended for beginning students or mathematicians interested in entering the field. My sole objection is the paucity of exercises in the last few chapters.

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