Content: An introduction to those aspects of partial differential equations and optimal control
most relevant to finance. PDE’s naturally associated to diffusion processes: the forward and backward
Kolmogorov equations and their applications. Linear parabolic equations: fundamental solution,
boundary value problems, maximum principle, transform methods. Dynamic programming
and optimal control: Hamilton-Jacobi-Bellman equation, verification arguments, optimal stopping.
Applications to finance will be distributed throughout the course, including: barrier, Asian, and
other exotic options; pricing and hedging in incomplete markets; options on an underlying that can
jump; American options, portfolio optimization, and other examples of optimal decision-making.
Prerequisites: Working knowledge of stochastic calculus, and some familiarity with financial
models. The fall semester course Stochastic Calculus (G63.2902) is ideal; actually I’ll assume
somewhat less, roughly the material covered by Chapters 9, 10, and 11 of S. Neftci’s book. See my
handout “Stochastic Calculus Review” for a quick refresher.
In addition to stochastic calculus, students also need considerable scientific maturity, at a level
typically obtained through an undergraduate math or science major.
Course requirements: There will be several homework sets, one every couple of weeks, probably
6 in all. Collaboration on homework is encouraged (homeworks are not exams) but registered
students must write up and turn in their solutions individually. There will be one in-class final
exam.
Lecture notes: Lecture notes and homework sets will be handed out, and also posted on my
web-site as they become available.
Text: There is no textbook for this course – the right book just doesn’t exist. Lectures will
draw from many sources, including recent articles from the quantitative finance literature. See the
separate handout “Library Reserve” for a list of some books that may be useful. I may from time
to time make use of articles that are not easily downloaded electronically; if so then I’ll place copies
in the “Green Box” associated with my name, which you can request from the CIMS library staff.