Multiplicative Back-Adjustment for Volatility Futures

# Why do unleveraged VIX ETPs have large beta relative to VX futures, with much faster decay?


# Why do unleveraged VIX ETPs have large beta relative to VX futures, with much faster decay?












I hope the title explains it fairly adequately.

To add a little more detail, it's my understanding that VIX ETPs such as VXX and VIXY hold VX Futures as their underlying assets. I believe that this is usually achieved with a weighted average of the front two monthly contracts (weighted by number of days to expiry I think??).

However, by simply holding the front month contract and rolling it (adjusted roll i.e. like for like cash value) at expiry date, a very different result is obtained. The image shows the roll-adjusted front month contract chart for VX futures (candles) vs VXX VIX ETF (orange line) since VXX's inception. VIXY looks practically identical to VXX by the way. Logarithmic Y axis of course.

Evidently the ETP outcome has much greater (negative) beta, and consequently faster decay. Could somebody explain, rather precisely, why this is? Intuitively I would expect the 1 month rolling future to be more volatile (with greater resultant long term decay), since it is not a weighted average with a longer term (less volatile) contract. Evidently I'm missing something!

How would I replicate VXX or VIXY (fairly precisely) using just futures?

Conversely, why are there (I think!) no VIX ETPs which simply replicate VX futures with a single roll at expiry, obtaining the much lower volatility/decay observed in the chart.

PS. Please no need to explain why VX futures decay in the long run, I fully understand that (supply and demand builds in an insurance premium).

## Answer by barneypitt (score 1)

https://quant.stackexchange.com/a/78684

Answering my own question.

I've discovered why the "continuous VX futures" series above has vastly less decay than the nominally equivalent VIX ETF(s).

The standard way in which continuous futures series are generated is plain wrong! The chart above is from TradingView. It shows (the lower line) a "continuity adjusted" price series generated from the individual front-month contracts, but the way that the discontinuity at rollover is removed is by using an additive constant applied to all prior data. The constant is the difference between close of the expiring contract and open of the following month's contract; at each rollover such a constant is applied to all points preceding the rollover.

This is clearly a wrong methodology. But I've discovered that the continuity-adjusted futures data I have purchased from a vendor (apparently unrelated to TradingView's) uses the same methodology - indicating it appears to be a standard method.

The correct way to back-adjust prior data points to remove the discontinuity is with a multiplicative constant (with each rollover, multiply all prior data points by the ratio of prices at the instant of rollover).

(The "standard" methodology is in fact doubly wrong: it generates the additive constant by differencing close of the expired contract and following day's open price in the next month's contract, thus erroneously including any gapping from close to next day's open. Of course, one will ordinarily roll over from one contract to another in very quick succession.)

The multiplicative method is correct because to maintain continuous exposure (/leverage) to the new contract, one needs to buy n1 contracts where n1 = n0 * p1/p0 (where n0 the number of contracts in the expired future, and p0 and p1 are the respective prices of old/new contracts). In fact I cannot see any reasonable way one would replicate the exposure represented in the additively adjusted series, as to do so (for a decaying future like VX) would require that one continually increases one's exposure(/leverage) over time.

The incorrect methodology also explains why the so-generated continuous series for VX appear to be decaying exponentially faster and faster and getting exponentially more volatile.

I can only assume that this standard (and wrong) methodology has stood the test of time because most futures do not continually, reliably, and significantly decay in value from contract to contract. The additive and multiplicative methods will not produce massively different results for most commodities or underlying instruments, but for VX they produce utterly different results (over the long term).

The lesson: for volatility futures, do not use the continuity-adjusted historical series you are likely to get from most sources: they give a thoroughly misleading impression of the real decay characteristics of the futures, and of their internal volatility over time.

I am now generating my own multiplicatively-adjusted series both for front month rollover and further-away rollover. The generated front-month series looks much more like the ETF data that it does the misleadingly additively-adjusted data.