- observer84
**Posts:**195**Joined:**

Very interesting discussion. I am still stuck with the contango structure in "normal" times though. I get it that institutional investors want to hedge vega expsosure and therefore pay a premium (negative drift in future). But I don't see why longer-term futures are quoted higher than short term ones. I would understand it in terms of "hedging cost". If you buy the underyling vix (or a rough proxy) and "store" it (meaning daily rebalancing etc.) it would be more expensive to do so for a longer-term future. So, by cost-of-carry longer-term future would be quoted higher..?bestobs

QuoteOriginally posted by: AlanThanks, guys -- I am convinced.I spent the afternoon on this.Taking a long position in all available VIX futures each day from 2004, and computing the log returns to expiration, I findTS Obs Mean StdDevAny 17895 -0.11 0.36 Contango 14001 -0.13 0.35Backward 3876 -0.07 0.39Flat 18 0.04 0.38TS=Nominal Initial Term Structure, defined by the sign of the VIX spot - Future being consideredObs = number of observations Mean = Mean Log Return StdDev = Std Dev of Log ReturnsAlan, I think that nominal contango can be verified (or rejected) semi-analytically in SVJ model (perhaps with one numeric integration). In this model, spot vix is a a simple function of instantaneous variance and vix future has a distribution which depends on instantaneous variance (known in analytic form). If one was to compute expectation of the difference between spot vix and future using stationary distribution of instantaneous variance chances are it will always be negative (or not). Edit: Actually, expectation of the difference between spot and future vix is zero in SVJ model. However, realistic SVJ parameters that fit market SP skew are such that PDF of instantaneous variance is concentrated around zero. In other words, instantaneous variance is close to zero most of the time, and becomes large only rarely. Dependence of spot vix on the instantaneous variance is stronger than for futures, so it should have lower value most of the time. I think this explains why there is nominal contango most of the time, at least in this model.

Last edited by EBal on February 12th, 2014, 11:00 pm, edited 1 time in total.

QuoteOriginally posted by: gardener3QuoteOriginally posted by: AlanQuoteOriginally posted by: gardener3QuoteOriginally posted by: AlanGood point, EBal, re Keynes. It raises the question I touched on in this thread. Namely, in very stressed markets (like the Financial Crisis),we have nominal backwardation (the spot VIX is higher than the futures and their term structure is downwardsloping. But, to use your or acastaldo's language, I wonder if we still have Keynesian contango in thosesituations? (so the expected spot VIX at future time T is lower than the current VIX future maturing at T) Certainly with mean-reversion, the spot would be expected to fall. So it's potentially self-consistent.But with very pronounced nominal term structure inversion, it is somewhat counter-intuitive to me that the market expects thespot to fall that much. What is your take on that type of situation?It should be easy to test. If VIX(t) - VIXfuture(t,T) is highly positive then VIX(T) - VIXfuture(t,T) should be positive, if I understood you correctly.Good point re testing. But if EBal's Keynesian natural contango is right (in all VIX markets, stressed or not), then I think the hypothesis here is that the realized value of VIX(T) - VIXfuture(t,T) is, on average, negative, (suggesting VIX futures are always expected to fall), regardless of the sign of VIX(t) - VIXfuture(t,T). At least that is my interpretation of what EBal is saying about (the proper application of) Keynes' theory to VIX. And you're right: that should be easy to test.The average is very negative as ebal mentioned (although if you take the inverse and include the crisis, they are not as good as vxx would suggest). I was referring to the relationship you mentioned earlier that when the curve is inverted, the vix mean reversion would be less than expected by the market. In other words, would VIX(t) - VIXfuture(t,T) help predict VIX(T) - VIXfuture(t,T)I've run quite a few regressions concerning that question now.I find log VIX(t)/VIXfuture(t,T) is a very weak explanatory of the holding period return.Regardless of the holding period, the R^2 are low: (0.01 - 0.04) range. However the sign of its regression contribution is what you would expect, given my earlier table and/or common sense. The only t-stats > 2 I've seen are for the shortest possible holding period: buying the expiring future on the Tues close prior to its Weds am expiration. Even for that one, the R^2 is only 0.045 and the unconditional annualized mean log-return is -58%/yr

Last edited by Alan on February 11th, 2014, 11:00 pm, edited 1 time in total.

QuoteOriginally posted by: AlanQuoteOriginally posted by: gardener3QuoteOriginally posted by: AlanQuoteOriginally posted by: gardener3QuoteOriginally posted by: AlanGood point, EBal, re Keynes. It raises the question I touched on in this thread. Namely, in very stressed markets (like the Financial Crisis),we have nominal backwardation (the spot VIX is higher than the futures and their term structure is downwardsloping. But, to use your or acastaldo's language, I wonder if we still have Keynesian contango in thosesituations? (so the expected spot VIX at future time T is lower than the current VIX future maturing at T) Certainly with mean-reversion, the spot would be expected to fall. So it's potentially self-consistent.But with very pronounced nominal term structure inversion, it is somewhat counter-intuitive to me that the market expects thespot to fall that much. What is your take on that type of situation?It should be easy to test. If VIX(t) - VIXfuture(t,T) is highly positive then VIX(T) - VIXfuture(t,T) should be positive, if I understood you correctly.Good point re testing. But if EBal's Keynesian natural contango is right (in all VIX markets, stressed or not), then I think the hypothesis here is that the realized value of VIX(T) - VIXfuture(t,T) is, on average, negative, (suggesting VIX futures are always expected to fall), regardless of the sign of VIX(t) - VIXfuture(t,T). At least that is my interpretation of what EBal is saying about (the proper application of) Keynes' theory to VIX. And you're right: that should be easy to test.The average is very negative as ebal mentioned (although if you take the inverse and include the crisis, they are not as good as vxx would suggest). I was referring to the relationship you mentioned earlier that when the curve is inverted, the vix mean reversion would be less than expected by the market. In other words, would VIX(t) - VIXfuture(t,T) help predict VIX(T) - VIXfuture(t,T)I've run quite a few regressions concerning that question now.I find log VIX(t)/VIXfuture(t,T) is a very weak explanatory of the holding period return.Regardless of the holding period, the R^2 are low: (0.01 - 0.04) range. However the sign of its regression contribution is what you would expect, given my earlier table and/or common sense. The only t-stats > 2 I've seen are for the shortest possible holding period: buying the expiring future on the Tues close prior to its Weds am expiration. Even for that one, the R^2 is only 0.045 and the unconditional annualized mean log-return is -58%/yrIf your hypothesis is true, it would only show up in severe backwardation. So the R^2 would be low overall but maybe significant for those points.

It would be interesting to test your other hypothesis that the risk premiums in both markets are driven by the same dynamics. That is, whether VIX(t) - realized vol (t,T) is correlated with VixFuture(t,T) - VIX(T). If the part of the roll yield that is due to changes in VIX is eliminated you end up with just the risk premium (assuming you have good prediction of realized vol).

Well, I am done with what I wanted to check, namely that the expected return to the futures is negative even in backwardation markets.update:After a little more checking, a more correct statement seems to be:In backwardation markets, - the expected return to the VIX futures is negative if you hold to expiration, - but positive, if you only hold for one dayAnd, this also assumes that the historical averages for the last 10 years are a good guide to market expectations. Also, these results come with very large std errors, so take it with a grain of salt. Now I am done

Last edited by Alan on February 14th, 2014, 11:00 pm, edited 1 time in total.

Observer84,Re your question on term structure: my explanation would be to assume a "term risk" is priced in vol markets, like it is priced in rates markets.Assume the term structure of expected volatility is flat, i.e. market participants do not expect VIX to be on average different in 1 year or 2 years.As explained by others here the market is driven by investors who need to buy vol as protection (read insurance) and the price will reflect the premium sellers of vol will naturally add to the (flat in this case) future expectation. Let's also assume demand for 1-year and 2-year protection is the same for the sake of the argument. A seller of a 2-year VIX futures vol will be exposed to a much wider range of extreme scenarios compared to the seller of 1-year VIX (more time so a wider distribution of VIX values) so his risk aversion will make him add a bit of premium as the quoted maturity extends.Makes sense?