In today’s opening section, the Options Nerds will write about an essential options terminology i.e, Volatility Skew. Pundits talk a lot about it but don’t quite clarify it well for the general public. It is our turn to give it a shot. Let’s dive in!

**There are three main ways of measuring Skew.**

a) **Strike skew**

= Difference in vol between 2 strikes

= 90% - 100% of ATM ( for example)

Remember not to divide the strike skew by ATM volatility (this is a common mistake). You could argue that a stock with a 10% difference in a low-volatility stock is more significant than a 10% difference in a high-volatility stock, so the temptation to divide by ATM is understandable. However, this is wrong because it doesn’t account for the fact that the Difference in width of 2 strikes also increases with the vol.

b) **Delta skew**

This is very similar to strike skew with a very high R-square.

= Difference between constant delta put and call

= 25 delta put vol - 25 delta call vol (for example)

**c) CBOE skew index ( third moment)**

The CBOE has created a skew index on the S&P500 which is strike-independent.

Skew = 100-10*3rd moment.

I suggest checking here if you want a deep dive into how the third moment is evaluated.

We will move on to other essential features of volatility skew.

**Skew and other option greeks**

Implied Skew, in general, represents non-symmetry. In theory, it is measured by calculating:

**Vanna**, which is d vega/d spot (rate of change of vega with respect to spot level)

or,

d delta/d vol (rate of delta change with respect to implied volatility level).

or,

Vanna = d^2 PV/ d(spot)d(vol)

**Why is skew negative?**

Skew for equities is normally negative; i.e. volatility of puts is higher than calls. This makes sense because big market jumps tend to be more down than up (of course not always). Here is the keyword - ‘jump’. Markets could be ‘trending’ up but the direction of ‘jump’ ( or potential expectation of future jump) could be down to make the Skew negative.

Volatility is a measure of risk and leverage increases when equities decline. If we assume no change in a company’s number of shares outstanding or debt, a simple reduction in stock price increases company leverage (debt/equity ratio).

Investors typically invest in equities and want protection against price declines, thus are willing to pay a price (premium) for puts.

When the actual correlation of stock prices and volatility is negative, ‘realized skew’ is negative.

Implied Skew is almost always negative.

**Index skew vs single stock skew**

Imagine an index where all constituent stocks have a flat (or zero) skew i.e. Volatility of different strikes of the same maturity is the same. What will be the index skew? Flat as well—-right?

No, wrong.

The index skew will be negative. Low strike index volatility will be almost equal to single stock volatility as the implied correlation is close to 100%. However, ATM index volatility will be less than this value because the implied correlation for ATM strikes is less than 100% ( diversification is most effective when spot = strike).

Remember , σIndex² = implied correlation × average(σSingle stock)².

Therefore even if single stock constituents have zero Skew, the index can have a skew as **low strike index implied vol > index ATM implied vol**.

Furthermore, the magnitude of the index skew will be greater than the average single stock skew.

You can somewhat think of it this way.

**Index skew = Single stock skew + implied correlation skew**

A less diverse index will have a lower implied correlation skew and hence, a lower index skew. A more diverse index will have a higher implied correlation skew (ATM vol is lower but low strike vol is higher) and hence, a higher index skew.

**Relationship of Skew with time**

Consider buying a 1 year put option (long skew position). When it becomes a 3-month put option, its implied vol will have risen, assuming the same ATM vol and spot because

“skew increases with the passage of time”

In other words,

vol of 1y 99% put - vol of 1y 100% put < vol of 99% put - vol of 100% put

**How to trade Skew**

Remember, you can have a view on Skew based on expected market correlations or panics or trends, but your real PnL or delta will be determined by the volatility of the volatility surface that reflects the true regime and the model that maps the regime best.

What do I mean by this?

**There are 4 different ways to model a vol surface.**

STICKY delta

STICKY strike

STICKY local vol

JUMPY vol

**STICKY delta**

This model has the same vol for options of the same strike as the percentage of spot (or same delta). Sticky delta literally means if the delta of the option is the same, the vol is the same, irrespective of the spot move.

For instance, 90% put strike with SPX at 4800 is the same vol as 90% put strike with SPX at 4500.

Think about this.

Well, it’s sometimes possible when everyone is expecting the market to go lower and has already bought a lot of puts, and when the market actually goes lower it’s not surprising anymore and vol of lower strikes remains the same.

In such cases, if you are long Skew, you lose money.

**Repricing PnL from long Skew = negative**

**Skew theta = Always negative**. ( skew theta is always the price you pay for owning Skew)

You can define skew theta as →

theta per unit gamma of all strikes of a given maturity - theta per unit of gamma of ATM strike.

**Net PnL = negative.**

Spot-vol correlation = positive.

**STICKY strike**

As the name suggests, the volatility of a particular strike remains the same even if the spot market has moved.

For instance, consider SPX is at 4800 and 90% put strike has vol of 15.

Now, when SPX actually moves to the 90% put strike, the strike will turn into an ATM option.

This ATM vol in this case is now expected to be 15 (which was earlier predicted by Skew)

If the market keeps dropping, this ‘new’ ATM call volatility still remains at 15.

According to the sticky strike model, vols do reprice exactly as Skew suggests.

In this model, if you are long Skew,

Repricing PnL = 0

Skew theta = negative

Net PnL = negative.

Spot-vol correlation = 0

**Constant Local Vol Model**

If the local volatility surface stays constant, the amount volatility surfaces move for a change in spot is equal to the Skew (i.e., ATM volatility moves by twice the Skew, once for moving up the Skew and another by the movement of the volatility surface itself)

In this case,

Repricing PnL = positive

Skew theta = negative

Net PnL = 0

Spot vol correlation = negative.

You can argue this is the ‘only’ model that prices skew correctly.

**Jump Vol Model**

This is the only model where a **long skew position actually makes money** where repricing repricing PnL > skew theta. Best long skew profitable trades happen when market moves in panicky fashion in the direction the Skew is priced for ( for instance - a large rise in implied vol following a decline in spot)

Spot-vol correlation = Very negative

Repricing PnL = positive

Skew theta = negative

Net PnL = Positive

Ok, that is it for today. In a subsequent section, we will discuss practical examples of how to monetize skew.

Thanks for reading and feel free to leave questions in the comments!

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