Steele's book is a sophisticated introduction to stochastic calculus with applications from basic Black-Scholes theory. There are other financially oriented introductions to stochastic calculus. What sets this one apart is its sophistication. Steele assumes a strong working knowledge of measure theory and a passing familiarity with function spaces. He delves into the theory of stochastic calculus more deeply than more accessible books by Malliaris and Brock (1982), Klebaner (1998) or Meyer (2001).

If you have the math background to follow Steele, I highly recommend the book. His style is wonderful, and concepts really build on one another. For example, in preliminary discussions of discrete time martingales, he introduces the notion of martingale transforms. Later, when he introduces stochastic integrals, he points out that they are merely generalizations of martingale transforms.

The first nine chapters introduce stochastic calculus, starting with random walks, proceeding to continuous time processes, stochastic integrals, Ito's lemma and stochastic differential equations. Next, two chapters turn to finance, introducing arbitrage arguments, developing the Black-Scholes argument, defining diffusions, presenting the Black-Scholes problem as a diffusion, and finding a solution.

The remaining four chapters explore more advanced topics in stochastic calculus, illustrating them with basic financial engineering applications. Topics include representation theorems, Girsanov theory, and Feynman-Kac theory.