Financial Calculus: An Introduction to Derivative Pricing

I view this text as a complete outline or guide to the mathematics and ideas of financial calculus and derivative pricing.This is not meant disparagingly. The progression of concepts is clearly explained which is what the authors purport to do. Though discrete processes are discussed involving for instance binomial coefficients (combinations) in the beginning as examples, the real meat of the subject lies in probability applied to continuous processes. Hence knowledge of measure theoretic probability and martingales is required to rigorously complete the arguments. Brownian motion is used to model market fluctuation which stems from ideas of Bachelier. This motion has a Gaussian distribution as discovered by the eclectic genius of Einstein who had the insight to apply the heat equation in his solution. It models noise for instance in electrical engineering. Any differential equation containing this distribution term is referred to as a stochastic differential equation. A solution of it is called a diffusion. A systematic theory of these was developed by Ito with his so-called Ito calculus. The Black-Scholes equation which takes this Brownian motion fluctuation into account which ultimately lets you balance out risk is developed in the text. This equation surprisingly (or not!) is equivalent to the heat equation (there are numerous derivations of this on the web). The solution of the heat equation expressed as an integral has the Gaussian distribution as kernel or weight (Well how about that! Full circle.). As an aside this heat equation equivalence allows Black -Scholes to be solved by finite element methods with financial constraints on the boundaries if the integral proves difficult or not in closed form. The authors recommend the text Probability with Martingales (Cambridge Mathematical Textbooks) for the measure theoretic probability as well as measure theory and martingales. This goes for me too. In this text the Lebesgue integral is first developed through construction of a probability distribution on the unit interval with the use of Caratheodory's Extension Theorem (Williams proves this in an appendix) then a trivial extension to the real line. Elegant-even easier! First rate guide to financial calculus!

Okay this is an intro, but you should have at least an understanding of Calculus. The purpose of this book is not to teach the fundamentals of the math, it teachs the financial pricing theorems, how they are applied to various assets and derivitives, and how to apply it to larger models.The authors provide a very clear foundation of both discrete and continous processes. From Binomial to Brownian motion, this book packs in alot of material.In the later chapters the authors cover the various derivative and asset pricing models, which really puts everything together in a context which will show you how to apply everything.There is clear instruction for the novice in finance.The only real issue I have with this book is that it does cover alot, but is not everything you will ever need to know. But it is a great intro which will enable you to move onto the advanced books on the subject.

I think this is one of the best introductions to mathematical finance around. Unfortunately, the book was out of print when I taught the subject, so I never got to test it as a textbook.In particular I really like chapter 2, where the authors introduce the key concepts in discrete time binomial processes. This allow them to introduce deep concepts like information and filtration in an understandable manner, while few students really understand measurability. (If you think that is a trivial idea from stochastic analysis, you may want to go for another textbook.) The binomial representation theorem is almost trivial, but show what the general version, the martingale representation theorem is all about, and why it is so useful. Similarly, the Cameron Martin Girsanov is heavy stuff in continuous time, but the idea is simple for binomial processes. I guess a lot of students will understand what the theorem i all about for the first time when they se the binomial version.The book then goes on to generalize all these ideas to continuous time and space, but with somewhat less mathematical formalism than many other books.

Baxter/Renie's book makes it easier to understand Shreve's texts on stochastic calculus (vol.1,2). In particular, ch 2 (discrete) & ch. 3(continuous) gives nice and simple descriptions of the essential concepts: filtration, measure, numeraire, drift, Ito formula. (These concepts can be difficult without a more detailed description of a stochastic process). The chapters 4,5,6 can be considered applying the concepts to SDE's in a number of cases, say, forex., equities, interest rates and multi-dimensional problems. These applications provide a good grasp of the mechanics to better understand the more detailed description of the same concepts in Shreve's texts.

This is a beautifully crafted introduction to Ito calculus and its use in financial markets suitable for undergraduates or graduate business majors whose memory of Freshman Calculus has largely faded. It is logically organized so as to bring the student clear step by clear step from simple expectations to the introduction of serious financial models. At each step there are simple examples showing what is going on.It lacks computer exercises, and a serious course will want to supplement it with a computer lab having financial software.It was a joy to find such a beautifully crafted textbook.