Using Principal Components to Reconstruct Futures Price Movements
Summary
The discussion considers whether two principal components can recover theoretical prices for futures contracts when the components were computed from cumulative log differences. It explains that reconstruction depends on how much of the original variation those components capture: a low dimensional approximation will not reproduce all observed movements. Principal component signs are arbitrary, so their orientation can differ between runs without changing the underlying analysis.
The answers distinguish modeling price changes from estimating price levels. PCA can describe common factors in futures movements, but the resulting factors do not directly determine contract prices. Any reconstruction of log differences would need to be interpreted in light of the retained components and then related back to price levels. The exchange offers no worked calculation or quantitative example, and it leaves accuracy requirements to the researcher. The main practical lesson is to understand the decomposition and its explained variation before treating a small set of components as a price reconstruction method.
Key ideas
- PCA components can approximate futures log differences when they explain enough of their variation.
- Component signs are arbitrary and may flip across separate analyses.
- A factor model of price movements does not directly provide futures price levels.
- Reconstruction accuracy depends on how much information the selected components retain.
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# Interpretation of PCs # Interpretation of PCs I have computed PC1 and PC2 wts on future contracts derived from cumulative log differences. How can I use them to get back the theoretical price of each contract using those 2 pcs? Thanks in advance. ## Answer by Matt Wolf (score 3) https://quant.stackexchange.com/a/7275 Yes you can, how depends fully on your required accuracy and also whether PC1 and PC2 are sufficient in explanatory power of the log differences of your futures contract. Also, make sure you understand the signs of the eigenvalues (sign of the PC) can be different from one experiment to the next as they are arbitrary (the values are obviously not). Here some comments on that which I found when I tried to find supporting documents: https://stats.stackexchange.com/questions/30348/acceptable-to-reverse-score-a-principal-component The following describes in a somewhat theoretical way how to PCA-Reverse but they also bring up couple neat examples. You are not presented here with a off-the-shelf R code toolbox that gets you to your results in the next 5 minutes but I think anyone using PCA should actually first understand the related math and stats behind it first. I am sure you will be able to easily reverse the PCA post reading this: http://www.cs.columbia.edu/~stratos/research/pca_cca.pdf Edit: I was actually so intrigued by the above paper's examples that I currently play with PCA and the PCA-Reverse in regards to image manipulation. Sorry that comment is not quant finance related but just wanted to share my excitement with math that sometimes overcomes me when playing with some interesting stuff. ## Answer by Richi Wa (score 1) https://quant.stackexchange.com/a/7283 I found a link and I have to repeat: I don't think that PCA helps you to find a price ... it helps to model the movements of prices but not their values. You get something like a factor model ... this does not directly give you a price ... maybe you also want to have a look at this link where PCA is applied to the oil market.
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