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I'm trying to think about the right way to estimate the delta of a VX contract to the S&P 500. VX futures are on the VIX index, which is a basket of S&P 500 options. By extension, VX and ES (E-mini futures on S&P 500) returns have a strong contemporaneous relationship. Having a good estimate allows for construction of a portfolio where you isolate the return of the "pure volatility" move, stripping out the returns due to equities.

I have thought of some very rudimentary ways to estimate it:

  1. Simple statistical relationship (VX returns / ES returns over some historical window)
  2. Linear regression using expanding/rolling window (y = mu + beta(x) + error, y=VX returns, x=ES returns)

Both of these seem very naïve. There seems to be very little literature on this topic, are there any better estimators?

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  • $\begingroup$ I'm afraid I don't have a good answer but I would add that you need to consider the futures roll as it will affect both the futures return and the beta to the S&P 500. $\endgroup$
    – user42108
    Jan 26 '21 at 21:57
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It depends. If you believe in (fractional) stochastic volatility then the delta, in the strict sense of the word, is zero, since the VIX future is a volatility derivative.

A simple linear regression is probably not such a bad idea to estimate the "delta" of the VIX future wrt to SPX if you do not believe in / assume any particular model. It is more natural however to write $x = a + \beta y$ where $y$ is the SPX return. The "delta" in this case is correlation, which is not delta in the strict sense.

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    $\begingroup$ You are the vol expert: Shouldn't there be a relation between the IV skew and the response of ATM vol to a decline in S&P? (I flunked an interview question on this subject, so I don't know). $\endgroup$
    – noob2
    Jan 26 '21 at 20:55
  • $\begingroup$ @noob2 I'm far from being an expert really (and that's not false modesty), but thanks. Although under SV models vol derivs do not have delta exposure to spot movements, the implied vol wil have spot exposure as you point out. Isn't Bergomi's skew stickiness ratio related to this? Let me check / think about it. $\endgroup$ Jan 27 '21 at 6:53
  • $\begingroup$ Yes, there is. At least in equities, skew is positive, ie vols are much higher for lower strikes, so ATM vol increases into lower index values. But (in theory), VIX is not based on ATM vol but on all strikes across the "smile". In practice, beta of returns somewhere in a 500-1000% range; and equity vol not stochastic in the first place! . $\endgroup$
    – demully
    Jan 27 '21 at 6:59

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