Thread by @OneHotCode1

Haven't written on anything in a while, so might be a little rusty, but let's give it a go...

Let's have a short thread about skew/smile delta and why the hell traders use it..

Back in the good ol' days of 1973 life was simpler, and delta (as B&S) modeled it was dP/dS

https://twitter.com/liquiditygoblin/status/1948297554543919559

or in other words, the change in the value of the option wrt the change in the underlying asset. That was mostly true because one of the other key assumptions of the B&S model is that implied volatility is constant across all strikes and maturities

Unfortunately (or fortunately, depending on how you look at it), we don't live in the eutopic 1973 B&S world, and the existence of the volatility smile is a given.

This introduces a new layer of complexity when calculating the option's delta because of the option's time value

If we break down an option's value (any vanilla option before expiry), we have two components:

The intrinsic value - how far the option is in/out of the money wrt to the strike

The time value - The vega (and theta)

the more time to expiry == greater effect of the vega

Which means that when we calculate the effect of spot change on the option value, we should account for the effect of the Vega P&L on our option's value

Ideally, to hedge the vega component of our option, we should use options to offset the vega exposure, but that is costly

Not to mention that hedging one option's vega with another option creates more exposure that needs another option, so we could end up with an infinite amount of strikes, which is costly and a massive headache.

So we try to hedge our instantaneous vega with the next best thing

The existence of skew/smile implies that there is a relation (or a beta) between spot move and vol move, so if we estimate the dVol/dSpot correctly, we can hedge the instantaneous effect of vol move with delta (i.e, spot)

In simple terms, skew delta adjusts the option's delta to account for gains/losses over vol (and vega) change to account for gains/losses over vol (and vega) change

or skew delta= BS delta + (VegaP&L)/dS

Now, how do we estimate the effect of dVol/dSpot?

This is where things become a little trickier, as we, at any given point in time, are merely guessing how vol will react to the change in spot (or how the spot-vol dynamic will unfold)

While spot-vol dynamic (or spot-vol beta, however you wanna call that) is a mysterious system driven by massive forces of supply/demand, by many agents in the market, we can simplify it down to two dynamics:

1. Sticky by delta
2. Sticky by strike

In a "sticky delta" world, the vol of the "floating" delta strikes remains unchanged, so effectively we "ride" the vol smile.

In other words, whenever the spot moves, the corresponding vol of the "delta strike" (let's say a strike that corresponds with 25delta Put) will change

Which means that what previously was the vol of the 25-delta Put strike will roll (either under or over, depending on the shape of the skew)

On the other side of the spectrum, we have the "sticky strike" world, where fixed strike volumes remain unchanged.

Under this dynamic, the implied vol smile as spot moves, implied vol rolls to where it was predicted earlier to roll into

While some markets follow some dynamics more closely than others, I'd say that in most cases, the "stickiness" oscillates on a spectrum between the two

Now that we understand (sort of) why we need to adjust our delta for skew/smile, let's discuss how to do it practically.

There are many ways to skin that cat (metaphorically speaking), but one method that is usually used by trading desks is a finite difference method.

While it might sound a bit intimidating, finite diff (or analytical) delta is:
bump spot up/down -> recalibrate the smile (under whatever dynamic you assume) -> calculate new option value -> divide by (2*dS) -> delta

This might be a slower method than parametric BS

But if one runs a large vega/vanna/volga position, the tracking error of the vega component could generate significant noise in the P&L, especially for desks that don't run options to expiry and constantly trade in/out of strikes.

That's the skew delta's rabbit hole

The big question that options desks (and an army of quants) try to figure out is how to model the surface evolution with spot/time accurately. How do we correctly estimate dynamics that will lead us to minimize the tracking error of our delta hedging

This is the essence inho