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Braindumping a thread on $SPX, $VIX and vol dynamics prompted by a really nice tweet by @JimmyJude13. Robinhoodies, we're going swimming in deep water so please wear armbands and let a grown up know you're out.
Here's the chart. SPX ~3050 twice with different $VIX curves. This shows you can't write VIX=VIX(SPX). Approaching 3050 slowly from below gives a different $VIX curve to coming quickly from above.
To be specific, the $VIX curve acts like it has a memory: so we actually have VIX(SPX live, SPX history). This is a problem.
There's another, related, problem. $SPX goes up, say, 8% a year, and $VIX doesnt seem to go down, it's stationary (if you squint at it, cover one eye, and work weekends).
Note this means that in the long run hedging $VIX futures with $SPX futures isn't going to work.
Now this is a big problem, because we know $VIX = $VIX(SPX spot, SPX vol) via the $VIX white paper etc.
And VIX futures are similarly forward variance plus an adjustment and mostly this works OK (until it doesn't, THIS water is deep enough for whole trading books to sink in).
The ONLY route through VIX = VIX(SPX live, SPX vol) and VIX(SPX live, SPX history) is SPX vol = SPX vol(SPX live, SPX history).
So I'm saying SPX option vols must be a function of SPX history. How, exactly, is deep enough water for a whole other thread, another career.
How do economists usually look at this?
Wrong, I think, but if you insist on delving into "Time Series Analysis" there's a nice intro at the blackarbs blog. It's got all the python you need. It's better than a textbook which will only send you to sleep.…
How do quants look at this? Also wrong - this stuff heads toward the stochastic volatility models that French banks use to lose 100-200M EUR every time the market collapses, so you may want to approach with caution. Or at least a sense of humor.
A gentle (and much more generous) look at modelling $VIX by Vance Harwood @6_Figure_Invest here, explaining how $VIX is tricky, it jumps and mean reverts etc. It's very good.…
So lets go a different way. There must be days where the vol doesn't do what you expect. Maybe option vols suddenly go up with $SPX going up, which puts a floor under $VIX, or option vols go down when $SPX goes down, effecively capping $VIX - either or both have to occur.
In both cases vols 'float', or go more 'sticky delta'. (SPX vols arent usually sticky delta they are usually super sticky strike, but sometimes the regime must change). So how do you know when option vols are going to move oddly?
First, when market makers say so. And when does that happen? When they want to own more/less of the vol.
Absent some large flow, the main part of that is the carry, the implied-realized spread (give or take vanna, spot-vol correlation, more deep water).
So if realized vol falls, market makers struggle to carry long options and start reducing the implied vol, and $VIX goes down more than you'd think. Or they get uncomfortable, say, marking calls 4 points below realized and start bidding vols back up, causing $VIX to hold firm.
Second the directional traders who aren't delta hedging just looking at at option premium, trading when they feel price is away from 'fair value'. This 'fair value' is the final Mariana Trench - if I knew enough, then I wouldn't be here tweeting I'd be on my yacht. Tweeting.
But what is clear is that when $SPX breaks a trend, then option premium traders have to adjust their view of fair value and that causes trading and vol moves.
Let's use some simple technical analysis term for trend following, like bollinger stuff.
So you end up with $VIX(SPX live, SPX realized vol, SPX bollinger %b)

Now - put that 3 factor model into a neural network. End.
OK postscript - after telling quants and econometricians they're wrong, only fair to say what i think could be right. I think vol modelling could be done with hysteresis loops like magnetism in physics.
There's *almost* no work been done on this. It's, not hard to think of agent based models that would look similar too.
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