Dude, where's my Skew?

I recently came across a presentation on LinkedIn which promised to reveal some secrets of the options implied volatility skew. Unfortunately, the presentation was very poorly written and contained nothing of substance. Further, the information was presented in a garbled and obfuscated way, which would only confuse non-expert readers and make the concept of skew appear even scarier than it actually is.

Poor exposition is a pet peeve of mine, but intentionally obfuscated exposition actively pisses me off. Here is my honest review of the presentation; My intent is not to put any one down, but to illustrate to others that one should exercise their critical thinking faculties while reading authorative sounding stuff on LinkedIn.

Title: How to monetize the relationship between skew and term structure. There is not a single trade mentioned in the entire presentation, so it fails to deliver on the one promise embedded in it’s title.

Title graphic: Plots skew (vol v/s strike) and term-structure (vol v/s time to maturity) using the same x-axis(!). It even marks their intersection point with a star for some reason. Meaningless graphs like this are a HUGE red flag.

Content: 4 of the slides inside, are the exact same slide with the words (left,right), (call, put) and (up, down) interchanged. So you can get 4 slides for the cost of 1. In an age where you can use LLMs to generate garbage slides, I almost admire how the authors stuck with the time-tested Cartesian product technique to artificially bloat their presentation.

OK, how about the 1 seed slide they used? Is it any good? Let’s review it line by line:

“Suppose that a stock is trading at $100, and the implied volatility of out-of-the-money put options with a strike price of $90 is higher than that of out-of-the-money call options with a strike price of $110. Additionally, the term structure is upward-sloping, meaning that longer-term options have higher implied volatility levels than shorter-term options.” This part sets up the situation (In a fairly verbose way)

“In this scenario, the skew is causing the left side of the term structure to be steeper than the right side. Options with lower strike prices (i.e., closer to the current market price) will have higher implied volatility levels than options with higher strike prices.“ This section is vacuously true, its just a restatement of the first paragraph.

“Furthermore, since the term structure is upward-sloping, this means that the skew effect is more pronounced for shorter-term options than for longer-term options.” This line is the real PROTEIN of the entire presentation. Term structure is upward sloping ? The skew effect is more pronounced for short term options than for long term options.

To avoid doubt, lets express this mathematically. You have a vol surface, which is a function that maps (strike, maturity) → implied volatility.

1. Left-skewed means: For all times T: vol(90, T) > vol(110, T)
2. Upward sloping termstructure means: For all strikes S, and T1 < T2: vol(S, T1) < vol(S, T2)

The claim of this presentation is

3. for T1 < T2: skew(T1) > skew(T2)

? vol(90, T1) - vol(110, T1) > vol(90, T2) - vol(110, T2).

Obviously, this is not a mathematical certainty - We can come up with vol surfaces which satisfy the first 2 conditions and not the third.

Maybe the authors are trying to say this relationship holds for the vol surfaces that we actually observe in the market. But why is that the case? What market mechanics cause this behaviour? What exactly does the upward term structure, and leftward skew of vol have to do with the (upward/downward) term structure of skew? None of these extremely important questions are asked, let alone answered.

If you recently read an article that doesn't seem to make much sense, try putting on your thinking cap and break it down. Maybe you aren't as stupid as you thought you were and the article was actually some cruel post modern nonsensical joke.