## Using Returns in Pairs Trading

This blog article is taken from our book [1]. In most entry-level materials on pairs trading such as in [2],  a mean reverting basket is usually constructed by this relationship: , where is the price of asset at time t, the price of asset at time t, and the price of the mean reverting asset to trade. One way to find is to use cointegration. There are numerous problems in this approach as detailed in [1]. To mention a few: the identified portfolios are dense; executions involve considerable transaction costs; the resultant portfolios behave like insignificant and non-tradable noise; cointegration is too stringent and often unnecessary a requirement to satisfy. This article highlights one important problem: it is much better to work in the space of (log) returns than in the space of prices. Therefore, we would like to build a mean reverting portfolio using a similar relationship to (eq. 1) but in returns rather than in prices. The Benefits of Using Log Returns When we compare the prices of two assets, [… TODO …]   A Model for a Mean Reverting Synthetic Asset Let’s assume prices are log-normally distributed, which is a popular assumption in quantitative finance, esp. in options pricing. Then, prices are always positive, satisfying the condition of “limited liability” of stocks. The upside is unlimited and may go to infinity. [5] We have: is the return for asset between times 0 and t; likewise for asset . Instead of applying a relationship, e.g., cointegration (possible but not a very good way), to the pair on prices, we can do it on returns. This is possible because, just like prices, the returns at...

## Change of Measure/Girsanov’s Theorem Explained

Change of Measure or Girsanov’s Theorem is such an important theorem in Real Analysis or Quantitative Finance. Unfortunately, I never really understood it until much later after having left school. I blamed it to the professors and the textbook authors, of course.  The textbook version usually goes like this. Given a probability space , and a non-negative random variable Z satisfying (why 1?). We then defined a new probability measure Q by the formula, for all . Any random variable X, a measurable process adapted to the natural filtration of the , now has two expectations, one under the original probability measure P, which denoted as , and the other under the new probability measure Q, denoted as . They are related to each other by the formula If , then P and Q agree on the null sets. We say Z is the Radon-Nikodym derivatives of Q with respect to P, and we write . To remove the mean, μ, of a Brownian motion, we define Then under the probability measure Q, the random variable Y = X + μ is standard normal. In particular, (so what?). This text made no sense to me when I first read it in school. It was very frustrated that the text was filled with unfamiliar terms like probability space and adaptation, and scary symbols like integration and . (I knew what meant when y was a function and x a variable. But what on earth were dQ over dP?) Now after I have become a professor to teach students in finance or financial math, I would get rid of all the jargon...