In this post, we will learn what it means for a position to be convex with respect to volatility.
In preparation for this post, you may want a refresher.
Refresher Post: Why Option Traders Focus On Vega (Link)
Refresher Post: Where Does Convexity Come From? (Link)
In Moontower style we will do this without anything more than middle school math. This 80/20 approach provides the intuition without the brain damage that only a relative handful of people need to know.
Most investors are looking to profit from the direction of stocks. Stated another way, most investors are taking active delta exposures. The size of their delta determines the slope of their P/L with respect to the market’s movement.
Directional Convexity
Some of these investors use options to make directional bets. This gives their positions convexity with respect to the changes in stock (also known as gamma). The convexity derives from the fact that their delta or P/L slope changes as the stock moves.
Now consider another, much small, class of investor. The option traders who try to keep delta-neutral portfolios. They are not seeking active delta exposure. They have no alpha in that game. Instead, they are taking active vega exposures. The size of their vega determines the slope of their P/L with respect to changes in implied volatility.
Volatility Convexity
Like the directional traders who use options, vol traders maintain convex exposures with respect to changes in the stock. Again, that’s gamma. But vol traders are much more focused on vol convexity. The reason vol traders focus on this more than directional traders is that vol traders typically run large portfolios of options across names, strikes, and tenors. These portfolios can include exotic and vanilla options. The presence of vol convexity means vol changes propagate through the entire portfolio in uneven ways. Risk managers model how vega exposures morph with vol changes.
For directional traders with just a few line items of options on their books, vol convexity is going to be much further down on the list of concerns. Somewhere in between “What’s for lunch?” and getting flamed by intern on Glassdoor.
Vol traders often think in terms of straddles. In fact, in many markets, brokers publish “straddle runs” every few hours. This is just a list of straddle prices and their implied vol per expiration.
A handy formula every novice trader learns is the at-the-money straddle approximation1:
Straddle = .8Sσ√T
where S = stock price
σ = implied volatility
T = time to expiry (in years)
So if there is 1 year until expiration, the 1 year ATM straddle on a 16% vol, $50 stock is $6.40 (.8 x 50 x .16).
So if implied volatility goes up 1 point to 17% how much does the straddle change?
.8 x 50 x.17 = $6.80
So the straddle increased by $.40 for a 1 point increase in vol. Recall that vega is the sensitivity of the option price with respect to vol. Voila, the straddle vega is $.40
More generally this can be seen from re-arranging the approximation formula.
Vega = Straddle/σ = .8S√T
Ok, so we have quickly found the ATM straddle price and ATM straddle vega. Look again at the expression for the straddle vega.
.8S√T
There are 2 big insights here. The first can be seen from the expression. The second cannot.
While we were able to compute the vega for the ATM straddle to be $.40 from the straddle approximation, how about the rest of the strikes?
For those, we need to rely on Black Scholes. You can find the formula for vega anywhere online. Let’s feed in a $50 stock, 0 carry, 16% vol, a 1-year tenor, and a strike into a vega formula. We will do this for a range of strikes.
Here’s the curve we get:
This chart assumes a single option per strike which is why the vega of the .50 delta strike is $.20 (not $.40 like the straddle vega).
The big takeaways:
If option traders’ profits are a function of vol changes, then their vega positions represent the slope of that exposure. If the vega of the position can change as vol moves around then their position sizes are changing as vol moves around. The changes in exposure or vega due to vol changes create a curved P/L.
Let’s see how changes in volatility affect vegas.
Here’s the vega by strike chart the same stock. The blue line assumes 16% vol across all strikes. The red line is 32% vol across all strikes.
In fact, imagine overnight, the stock’s vol doubled from 16% to 32%.
The maximum vega at any strike is still fixed at $.20, it just occurs at the new .50 delta strike. The .50 delta strike moved up $2 or about 4% but look how the vega of the options at nearly every strike increased. This is intuitive. If you double the vol then a strike that used to be 1 standard deviation away is now 1/2 a standard deviation away. All the OTM deltas are creeping closer to .50 while of course, the .50 delta option remains .50 delta.
You can start to see the reason why a position can be convex with respect to changes in vol. Imagine you were long the .50 delta option and short the way OTM 90 strike call.
Your vega which represents your slope of P/L with respect to vol has changed simply by the vol changing. The higher the vol goes, the short vol you become.
Observe:
This chart shows vega profiles across strikes over a wider range of vols. At extreme vols lots of strikes look like .50d options!
Starting conditions:
Stock price = $50
Implied vol = 16%
Portfolio
Long leg
Short leg
Summed as a vertical call spread
Now let’s change implied vol up and down.
It’s a busy picture. Let’s walk through the scenarios:
We start at 16% vol and increase vol
We start at 16% vol and decrease vol
Same starting conditions:
Stock price = $50
Implied vol = 16%
We are targeting a vega-neutral portfolio
New Portfolio
Long leg
Since the short leg has $.20 of vega and our long leg has $.11 of vega we need to buy 1.75 of the 60 strike OTM calls ($.20 / $.11) to have a net flat vega position.
Short leg
Summed as a ratioed vertical call spread
You know what’s coming. Let’s change the implied volatility and look at the structure price. Remember you shorted the ATM option at $2.90 and bought 1.75 OTM calls for a total premium of $.9625.
In other words, you shorted this structure for an upfront premium of $1.9375. Watch what happens to its value when you raise or lower the implied vol.
To understand why the structure behaves like this, look at the scenarios.
We start at 16% vol and increase vol
We start at 16% vol and decrease vol
One last chart to drive it home. The green line is your P/L as vol changes. Notice that your max P/L in the vol declining scenario is $1.9375, the entire value of the structure. It is unbounded on the upside. It looks like the more familiar picture of being long a straddle! The fact that the P/L chart is curved and not linear is convexity and as we know, results from the size of exposure changing with respect to vol.
The blue line shows exactly how that vega exposure changes with respect to vol. You started vega-neutral. As vol increased you got longer. As vol fell you got shorter.
This was intended to be an introduction. But here’s a non-exhaustive list of “gotchas”:
The main intuition I want you to get is that OTM options are sensitive to the vol of vol because their vegas can bounce around between 0 and the maximum vega. ATM options are already at their maximum vega. So structures that own extra options relative to be being short the ATM are convex in vol.
Vol convexity is important because changes in vol influence much of the greeks. Understanding the concept can be used for defense and offense. Vol directly impacts option prices according to their vega. But it also changes their vega.
Who should care?
When you combine the convexity of options with respect to vol (volga) with the convexity of options due to changes in the stock price (gamma) you get nitroglycerine.
Remember even when dealing with non-linear instruments, a snapshot of a portfolio at a single point in time might show it to be vega-neutral. But a photo of a car can make it look parked. Only the video can show how fast the car can move.
If you use options to hedge or invest, check out the moontower.ai option trading analytics platform
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Great education here: a complex topic patiently explained via the lens of a trading strategy that can be effective.