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Using PCA I have the below PC1, first component weights, for 4 quarterly expiries of short term interest rate future. These are hypothetical values used to help my question.

March: 0.005542604, June: 0.028545363, Sep: 0.043833371, Dec: 0.049176453,

Assuming tick value for all 4 products is $12.50 and each price increment is 0.005 (half of a basis point), to construct a hedge between durations of say, June versus Dec, one would:

  1. Normalize the function to June such that (0.028545363 / 0.028545363) June * (0.028545363 / 0.049176453 ) * Dec, which is equal to: 1 June - 0.580468 Dec. This implies that for every 1 unit of June, we need less of Dec to the tune of approximately 0.58. Likewise, if a hedger is looking to neutralize risk described by the first component PCA, for every 100 June, he/she would sell 58 Dec. (tick value and price increment the same for June and Dec as mentioned above)

To display this spread function in normalized fashion , one would input June market price / 0.005 - 0.580468 * Dec market price / 0.005. This way, if we observe the function and saw a 50% change in the function as a whole, we know the total value change of the above spread for every 100 units is 0.50(spread change) * $12.50 (change per 1.00) * 100 units.

However, the above relates to 2 products hedging each other. How does this change if we have 3 products? Say to hedge 100 units of June, we will use a combination of Sep and Dec instead of just Dec only. We go through this process again first by creating a spread function to display price movement.

  1. standardizing Sep and Dec to June, we have 1 June - 0.651224 * Sep - 0.580468 * Dec
  2. Let's say for 100 units of June, we will use 30 units of Sep and 31.31 units of Dec to be risk neutral. This is derived from 30 units of Sep / 0.651224 = 46.06707, so the remaining units for Dec needs to be (100 - 46.06707) / 0.58046, which gets us our quantity of 31.31 units.

Like for two products in the June vs Dec example, one can easily calculate the value changed by the change in the function * 12.50 * 100 units. However, this is not the same for spread functions that has more than 2 product such as the June vs Sep and Dec. One cannot simply say 50% change observed in 1 June - 0.651224 * Sep - 0.580468 * Dec * 12.50 * 100 units is the total change in the spread function. This is due to the uneven weighting of quantities used for hedging for spread functions that has more than 2 symbols.

My question is: How does one formulaically represent the total value change for spread functions with more than 2 symbols in a way that accounts for the different quantities used in an elegant way similar to our spread function with just 2 symbols?

I am beginning to think that for spread functions with more than 1 product as a hedge, another coefficient needs to be applied for each term to describe the relative proportion of the hedge quantity or contribution...

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  • $\begingroup$ Perhaps this existing answer is a guide: quant.stackexchange.com/a/49626/29443, $\endgroup$ Commented Apr 20, 2024 at 23:40
  • $\begingroup$ Hey @Attack68 - thank you for your link to your post. It has a lot of relevant information to my question. Would agree with my thoughts on this topic: 1. PCA loading factors are individual weights meaning that to construct a hedge like above, the weights are used relative to each other as opposed to multiple regression where it is the combination of the x terms together to predict y. 2. Because of #1, to produce a price display of a multi symbol spread, one must not only account for the PCA coefficients but also need to take into account qty used for hedging as a multiplier as well. $\endgroup$ Commented Apr 21, 2024 at 14:36

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