I’ve been meaning to learn the math behind stock trading for a while, but I’ve found it’s hard to find quality information. Most of the stuff online is (1) non-technical, (2) trying to sell you something, or (3) both. So I decided to collect my own notes on modern portfolio theory (MPT). Here’s the pdf: Notes on Itô calculus and quantitative trading.

The information comes from various lecture slides and articles. I didn’t put specific references in there, since it’s standard, textbook stuff. Just search for any piece you’d like more information about.

Here is a rough outline:

• How to select stocks, given their risk and return statistics
• How to model risk and return in the first place

The first part takes the “Minimum Variance” approach due to Markowitz. To model stock prices, I give an overview of Itô calculus (one form of stochastic calculus) and geometric Brownian motion (GBM). This is the model used by the Black-Scholes formula for pricing derivatives.

I suspect that a simple index fund might beat a portfolio selected with this recipe. In the future, I’d like to test on historical data and find out if there really is an advantage to picking your own stocks.

# Diversity at Harvard

I recently started a PhD program at Yale. To commemorate the occasion, I wrote a song about Harvard. It’s called “Rich Dumb People.” Here it is:

Email me if you want the chords.

# Interactive math textbooks

There is no excuse for Ron Larson‘s Calculus, the textbook I used in high school. It’s over 1,000 pages long and it’s printed on paper. This may have been acceptable in 1998 when it was first printed, but today I am amazed that it remains in print. That’s because textbooks should be electronic and interactive.

You have to be suspicious of exisitng “interactive” math content, because it may be interactive in a very shallow sense. For example, if you get a hard question wrong then it shows you an easier question. What I’d like to see are interactible examples and figures: Move a point around on the figure and see how things change in real-time.

Strom et al. have done a very good job of this with Immersive Math, an interactive linear algebra textbook. (A paper textbook cannot update its figures in real-time.) They also have hover-tips for referenced theorems and definitions, which is a nice touch. (In a paper textbook you’d have to flip back.)

I made my own demo for solids of revolution, a difficult topic for many students. You can move a slider to see the differential volume element change. But you can also hover over a term, like $dA$, in the equations to highlight the geometrical interpretation corresponding to the term. Click the animation to be redirected to the demo and try it yourself.