New York University 3 credits Mathematics
The goal of the first half of the semester of the course is for students to develop an understanding of the techniques of stochastic processes and stochastic calculus as it is applied in financial applications. We begin by constructing the Brownian motion (BM) and the Ito integral, studying their properties. Then we turn to Ito’s lemma and Girsanov’s theorem, covering several practical applications. Towards the end of the course, we study the linkage between SDEs and PDEs through the Feynman-Kac equation. In the second half of the semester, we turn to asset pricing and the trading of derivative securities using stochastic calculus techniques. Using tools and techniques from stochastic calculus, we cover (a) Black-Scholes-Merton option pricing; (b) the martingale approach to arbitrage pricing; (c) incomplete markets; and (d) the general option pricing formula using the change of numeraire technique. As an important example of incomplete markets, we discuss bond markets, interest rates and basic term-structure models such as Vasicek and Hull-White.
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