This is the introductory nance theory course in the PhD program of Columbia Business School. The course is devoted to the theory of asset pricing and portfolio management. The course assumes that students have no prior knowledge in the theory of nance. We will start with "classical" results for single period models and move towards dynamic models in continuous-time. The goal of the course is to provide students with an understanding
and methodology to successfully master more advanced PhD courses.
The mathematical sophistication is kept to a minimum. The introductory nature of the course makes us focus on the intuition of results.
The course starts with the single period model. We cover utility functions of risk averse investors. Once we understand how we model the investor's objective function we analyze the portfolio choice problem in a single period. The portfolio choice problem will also introduce us to the notion of a stochastic discount factor (SDF). We will see why the SDF is
the asset pricers best friend.
We move on to talk about Pareto optimal equilibriums and eciency. We will learn that if the economy is in equilibrium there exists a SDF that ensures no-arbitrage in the securities market. We will learn how we can use the SDF to price assets under the risk-neutral probability measure or under the physical probability measure, depending on what is more convenient for the task at hand. We will also see an example of how to price options in an arbitrage-free market via binomial trees. Once we come to this point we will already have a good overview about key concepts in the theory of nance.
We continue to talk about the mean-variance approach to portfolio allocation. Our last topic in the single period world will the representative investor.
The second part of the course is devoted to dynamic models. State-of-the art asset pricing and portfolio allocation problems are written in continuous-time. This has several reasons. First, analytical results are more convenient to derive. Second, the notation is less messy. Third, we have great numerical tools at hand that allow us to discretize models and bring them to data. We therefore start to talk about how to model securities markets in continuous-time. This includes a primer on stochastic calculus. This primer will be very applied and hands on. Armed with the necessary techniques we will be able to understand continuous-time portfolio choice problems and asset pricing.
At the end of the course you will have the intuition and techniques to (i) master advanced nance PhD courses and (ii) push yourself to the frontier of nancial theory. Chances are low that we will have time to talk about option pricing, forwards and futures and term structure models in greater detail. These are extremely exciting topics for which our tools from this course are well suited. Do not be disappointed if we cannot cover this material. There are advanced PhD courses waiting which teach these topics.
Maxim Ulrich was a Columbia Business School faculty member from 2008 to 2010.