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!