Thread by @LilyWrites4

What is volatility beta and the vol surface?
If we go back to the prior thread on volatility, we can make two critical observations:
- Volatility is a function of variance (how much spot "moves") and time
- Volatility isn't constant over time or space (price range)

We should add two more important observations, which aren't necessarily theoretical, but have strong empirical support:
- Volatility tends to cluster in time - when volatility in one direction (e.g. down) is high, we expect in the near-term continued high volatility and vice vers

- Volatility has a strong correlation to how spot price moves. This is a key observation, and differs across assets. We call this the spot-vol correlation.
In equities it's fairly well supported that volatility increases on the way down, and decreases (or remains constant) on the

way up. This can sometimes break (we observe this in meme stocks, where spot-vol actually behaves the opposite way), but tends to reflect demand pain, rather than a fundamental law of assets. A good counter-example to this is oil and other commodities.

This part is probably wrong and better a question for @volmagorov or @moreproteinbars, but oil buyer pain is higher on the way up - the higher price moves, the more pain, and the more volatility. In oil, usually, spot-vol correlation is positive. In equities, it is negative.

It's fairly well supported that given the non-constancy of volatility, any 'proper' way to hedge an option contract cannot only rely on the black-scholes, given that BSM assumes constant volatility. So it fails to reproduce empirical truths like the spot-vol corr or clustering.

In general, this is best reflected by options market makers (and I guess largely other hedger market participants) is adjusting the delta (the hedge ratio) according to a modified form of vanna, which essentially reflects how we expect delta (our hedge) to change as IV does.

Now, let's tighten up our notation here. In general, when we're thinking about options, we're thinking about *implied* volatility, which as we talked about in the prior thread isn't just a forward reflection of volatility (the mirror of realized volatility, which is backwards

looking), but also a measure of supply/demand technical factors in the market! In general options are supplied by market makers (this is not necessarily the same as long/short), and demanded by market participants. However, market participants may sell or buy options.

This reflects in such jargon as the over/under supply of various Greeks (usually vega) or implied volatility. A great example of arguable implied volatility oversupply is the roll of the 3 month JPM collar (at the end of June), where market participants are both selling calls and

buying puts. As we discussed previously, IV in general goes up when options are being sold BY market-makers, and goes down when options are being sold TO market-makers. So we expect in this case an over-supply of IV on the call side, as market makers absorb and hedge out the long

calls, and we expect an over-demand of put IV. This creates a. fundamental property called the volatility skew, which traditionally is measured by looking at, for the same asset, the IV on the call and put options at 25 delta.

Interpreting the volatility skew is more the domain of the modeler rather than an empirical truth, and I've seen decent evidence that while it has some forecasting ability, it may also act in extreme events as a contrarian signal.

Last but not least, let's talk finally about volatility beta ("vol beta") and the vol surface. As we can surmise from our above examples, IV depends to a large degree on how the spot price moves. In fact, there are two old, somewhat wrong models here: sticky delta and sticky

strike. These models reflect on how we expect the implied volatility of an option to change as the market does. If we believe the asset/regime is sticky delta, what this means essentially is we expect the IV of a xx delta option to be the same regardless of how spot moved.

To give a brief example, if we start at spot = $100 and check the ATM (~50 delta) position, it might have 12% IV. If spot then drops to $90, under the sticky delta rule, we'd expect that the ATM (so $90) would now have 12% IV. Sticky strike is in many ways the converse of this.

In sticky strike, we expect the skew will be held constant on the strike, rather than the delta. So in the above example, if our spot starts at $100 with 12% IV and then drops to $90, we still expect the $100 option (our prior ATM position) to be 12% IV.

In general these models are important for the volatility trader to understand at least in high level. In practicality neither sticky strike or sticky delta is exactly correct, but various mixture and regime models and analogous formulations (e.g. Derman's Sticky Implied Tree)

exist to fix some deficiencies.
Finally, let's get back to volatility beta. In general, it is expected that the change in IV tends to be proportional to the spot move, especially on the way down (it's more a rule of thumb than a hard rule, but works empirically).

This is an important relationship, because it tells us *things* about how an asset is behaving. The most common and practical vol beta to observe is on SPX, and its relationship to the VIX. VIX, if you remember, is essentially composed of the 30 day average maturity option prices

for SPX options. This of course is largely dependent on how IV behaves, given the other factors of option price are well known (and wouldn't really vary much, except for some calendrical effects like VIX's weekend effect). IV as mentioned is both subject to forecast of

realized volatility, as well as demand/supply technical factors (e.g. our JPM collar). Because of this dependency, it's one of the strongest and easiest ways to compute our volatility beta (given it reflects SPX option IV directly). We expect over time a moderately constant

relationship between spot changes in SPX and spot changes in VIX, but i'll leave that exercise up to the reader. What's interesting of course, is when this breaks. A fantastic example of this breakage was Volmageddon in Feb 2018 - due to supply technical factors in the VX futures

market, VIX effectively doubled (around a 20 point move) with a much more muted SPX reaction. In general, understanding how volatility beta changes over time is integral to trading volatility. When a massive volatility beta shift occurs, it's often due to market structural

factors, and a much more safe bet to trade on as a temporary move, in the context of expecting realized volatility and implied volatility to converge (while these are different concepts, in general rvol vs ivol obeys stationarity, and mean reverts).

Vol beta however is an asset-based dependency, and an important skill in the vol modeler's portfolio is understanding how it behaves on an asset basis.
Finally, we'll get to our volatility surface.

Given the structure of option chains (segmented by days to expiry and strike price), we can trivially observe it forms a 3 dimensional grid space in conjunction with implied volatility - for example, x (strike), y (expiry), z (IV). This creates a surface in graph world:

Two things tend to be important of the vol surface:
- it tends to be smooth (because of some fancy arbitrage I won't go into) over time and space
- it needs to be interpolated to be smooth (because option strikes/days to expiry are NOT continuous)

This is usually done to obey certain constraints such as forbidding negative variance, but tends to be more the domain of the data scientist (and lots of packages like Quantlib make it easy now anyway).
What's useful, however, is more observing how the vol surface changes over

time and days to expiry. Often you can use it to observe key events (like earnings), where the vol surface might kink. Kinks however tend to be rapidly smoothed out if possible simply due to arbitrage.
That's all for today!