Normalize Returns in PCA Hedging?

2017-09-21 06:31:19

Lets say I want to form a portfolio of $N$ correlated instruments that minimizes the variance of my daily PnL, given by

$$

PnL = \sum_{i=1}^N h_i \Delta S_i

$$

where $h_i$ are the number of units of instrument $i$ and $\Delta S_i$ is the daily difference in price.

To minimize the variance, I find the covariance matrix of $\{\Delta S_i\}_i$ and perform PCA on this covariance matrix. I then take the eigenvector with smallest eigenvalue and use the entries as my hedging ratios.

My question is, if the daily difference in instruments are vastly different, say on average $\Delta S_1 = \$10$ while $\Delta S_2 = \$0.1$, does it make sense to normalize these before performing PCA?

I an inclined not to because they are in the same units, and I think it makes sense for instruments with a larger variance to contribute more; however, I have always thought I needed to normalize the data before performing PCA, so I am hesitant to do otherwise now.