Financial Calculus: An Introduction to Derivative Pricing

As an Msc financial engineering student I underwent a fairly rigerous treatment of martingale pricing but at a pace which left little room to appreciate what was really going on. True, this does not go into PDE methods at all, neither does it discuss practicalities however for what it does cover it is very clear. Along with Nefcti this really provides understanding for those whose mathematical background may have some gaps.

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.

Very good a book. I am still going through it. It doesn't go through every proof step by step which is why it's good, it forces you to really think.

内容は、複製戦略を求め、金融商品の理論価格を求める、これに尽きます。最初に、ブラック・ショールズモデルから、ＣＭＧの定理やマルチンゲール表現定理を用いてオプションの理論価格を求めます。その後、対象を変え、モデルを拡張しても、やっていることは一緒です。この本の特徴としては、直感的な説明がなされていて、数学の定理の証明や冗長な数式変形は省略されていること、がまず挙げられます。もう一つの特徴は、この本がカバーしているのは、タイトルが示すようcalculus、すなわち、モデルから金融商品の理論価格を導くプロセスであって、市場の仕組み・金融商品の種類と定義といった「知識」や、理論価格に具体例をあてはめる、数値計算を行う、といったことは扱われていないこと、です。物事を琡?解する際、特に数学的に理解する際は、論理的厳密性と同時に、イメージ・アイデアが重要ですが、この本は後者による理解を助けるのいい本であると言えるでしょう。