Friends,
If spot is up 1% and vol is up 1%, what delta lean do you need?
Say you’re long 1,000 vega and vol is 40%. Vol ticks up 0.4 (1% relative move), so P&L is +$400. To stay flat, you need to lean short $400 / 1% = $40,000 of underlying.
Compactly:
delta lean = −vega × σ × β
-$400 = -1000 x 40% x 1.0
σ is the vol in percent and β is the relative or percent vol move per 1% spot move.
(Note that short vega flips the sign.)
I introduced this idea in embedding spot-vol correlation in option deltas. It’s a response to a familiar problem where your option model says you’re long $50mm of SPY but acts like it’s only long $40mm.
But where does β actually come from?
In that post, we started with correlation as a given and built an adjustment. Later we used VXX/SPY beta as a proxy for spot/vol beta.
This is adequate if you’re trading SPY against VIX futures or using the general spot-vol relationship as an input to sizing in portfolio construction.
But if you’re in the daily weeds of running a market-neutral option book, those estimates are riddled with at least 2 glaring problems:
They update too slowly since it’s typical to use at least daily data for both correlations and volatilities.
They are too low-resolution. If you have options inventory, you don’t care what “vol” is doing broadly because composite measures like VIX and floating measures like “ATM vol” capture movement along the vol curve. Insofar as there’s a skew, those are already “baked in”. Option p/l depends on changes in option premium and the portion of that you are trading, the vega, cares about changes in the strike vol.
It’s one thing to say SPX vol increases as the index falls but that is expected from the skew. The question is “do the strike vols actually increase as the index falls and with what vol beta?”
That’s the sensitivity you need to inform your lean.
I must confess, this exact topic has a small audience. The market-neutral vol book jockey is definitely the type of job you would only find in a pretty advanced economy. We’re well past pineapples for coconut barter here. If that’s you, well, you don’t need convincing to read on. Nobody talks to you as it is, but here I am bringing flowers and chocolate.
But for the sake of the wider population, I’d say the absolute specifics are not the best reason to follow along. But there are reasons.
If you are an option market maker who doesn’t get their model handed to them from on high with marching orders to trust it, you’ve had to hack together mods like you’re Xzibit in Pimp My Ride. Being both a non-quant and surviving several turns of 10,000 hours of repetition, I’ve hacked together many solutions to option pricing and modeling. They ain’t gonna win any awards, but you can sketch ‘em on a napkin. They’re useful enough to work, but juuuust dumb enough to keep you paranoid.
Let’s step through this one and see if everyone can learn a little somethin somethin.
Pulling it from the data
We want to know how strike vol changes with the spot price, not the ATM vol.
Here’s the recipe:
