The Mechanics of Skew and Vanna: Part 2

[WITH CODE] Why is the Volatility Surface Skewed, and Can We Use It to Predict Returns?

Hello!

Welcome back to the second post in this short series. Last post, we discussed the origin of skew in the option chain and what drives this skew. Additionally, we covered analysis on the informational value of skew for future equity returns.

Today, we will be covering the option greek Vanna, which measures your exposure to the effects of skew, alongside some more academic papers. The code has been sent to your email. Let’s get into it.

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What is Vanna?

Vanna is a second-order option greeks that describes your exposure to changes in the volatility skew of the underlying asset. Here is the formal definition:

Vanna, also referred to as DvegaDspot and DdeltaDvol, is a second-order derivative of the option value, once to the underlying spot price and once to volatility. It is mathematically equivalent to DdeltaDvol, the sensitivity of the option delta with respect to change in volatility; or alternatively, the partial of vega with respect to the underlying instrument’s price.

To define this precisely, here is the mathematical relationship that shows Vanna and the equivalence between DvegaDspot and DdeltaDvol, which is due to the symmetry of second derivatives, which is also known as Schwarz’s Theorem.

Skew can also be visualized by looking at the vega of options across strikes, and then taking the derivative of this change (which is how we get DvegaDspot).

Graph of Vega by Spot for 150 Strike Call

Spot price is x axis, Vega is Y axis. Our current call option has a strike price of 150.

As you can see, Vega is the highest for when the call option is at-the-money. You can think of this chart as showing the Vega of the option for when it is OTM (when spot is at 125, and Vega is approx. 0.15), when the option is ATM (spot at 150, Vega is approx. 0.32), and when the option is ITM (spot at 200, Vega is approx. 0.08).

Therefore, as spot rises, the Vega also rises (positive Vanna) until the option is ATM, then Vega falls as spot continues to rise (negative Vanna).

This exact relationship is shown below.

Graph of Vanna by Spot for 100 Strike Call

This vanna graph is applicable to long calls and long puts. It is flipped for short calls and puts.

This can fuel losses for market makers who sold put options to institutions for downside protection. These market makers are long delta, short gamma, short vega, and have positive (long) vanna exposure from this short OTM put position.

Because they are long vanna (positive vanna), when volatility spikes, their delta becomes more positive (longer). Since the market is crashing, being "longer" is bad. They must sell the underlying to reduce that delta and remain delta-neutral, which drives the market down further.

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Risk Reversals

Now that we know what skew is, and that vanna describes our skew exposure, how do we trade skew?

The option structure used to trade skew is the delta-hedged Risk Reversal (RR)¹. Now, there is some nuance here depending on the market you trade, as some markets traditionally have negative skew (equities), and some have positive skew (some currency pairs and commodities). Once again, refer to footnote 1 as some markets define skew differently.

However, the structure is still the same, and today we will (mostly) be focusing on equities, as I am sure that is what most of you are most familiar with.

Long Risk Reversal (RR) = Long OTM Call & Short OTM Put²

Short Risk Reversal (RR) = Short OTM Call & Long OTM Put³

I like to understand the risk reversal in an intuitive and mathematical way. In the intuitive way, you can think of skew like a lever. In equities, skew is typically negative (OTM put IV > OTM call IV), so the lever would look something like this:

My own image

Now, if you are long skew, you would buy the OTM put and short the OTM call (short RR), hoping that the difference in volatility increases between your two options that have the same delta. Therefore, you are also pushing on this lever, hoping that this lever becomes steeper.

And if you are short skew (short OTM put, long OTM call) with a delta-hedged RR, then you would be betting that the difference in volatility between your options falls (pushing down on the left side of the lever).

A risk reversal position will have the same skew position (long or short) for a wide range around spot (but it will eventually flip if spot moves very far away from your strikes).

This is because of the Vega profile of the RR. Below, I show the Vega profile of a short RR (long 90 strike OTM put, short 113 strike OTM call).

Looks familiar!

Therefore, as spot moves higher, you are initially long vanna (vega higher as spot rises) until spot reaches the strike price of the long put (90).

Then, until the spot price reaches the short call strike (113), you are short vanna (vega lower as spot rises) and long skew. This would be your initial position, as you would enter into this trade when the underlying spot price is at 100.

Lastly, when the spot price rises above your short call strike, you enter into a long vanna position (vega higher as spot rises).

Hopefully, you have learned something. But, is there actual information power in the skew of various securities and assets? Spoiler, the answer is yes, but I’ll explain more below.