Trading term structure of skew

Question

Is there a way to trade IV skew between two maturities? For example, bull put in near maturity and bear put in far maturity.

Answer 1

You're on the right track.

Specifically, let $D(K,T)$ be the price of a digital put option of strike $K$ and maturity $T$. Then, as you might know $$ D(K,T) = N(-d_2(K,T)) + K n(-d_2(K,T)) \sqrt T \, \frac{\partial I(K,T) }{\partial K} $$ where $I(K,T)$ is the IV for $K$ and $T$, $N()$ is the standard normal distribution and $n()$ is the standard normal density, and $$ d_2 = \frac{ \log S_0/K }{I(K,T) \sqrt T} + \frac {I (K,T) \sqrt T}{2} $$

Let's introduce the log strike variable $k := \log K$, then $$ K \frac{ \partial }{ \partial K} = \frac{ \partial }{ \partial k} $$

So you see the slope of the IV appearing naturally in the price of a digital, which suggests that you might be able to trade it if you have a calendar spread of digitals.

The question is which strikes should you choose for the calendar spread? For the sake of theory/illustration suppose you can trade the strikes $K$ for maturity $T$ and $K'$ for maturity $T'$ such that for both maturities $$ d_2 = 0 $$ In theory these strikes almost always exist, except for some pathological cases.

For these strikes $N(-d_2) = 1/2$ and $n(-d_2) = 1/\sqrt{2\pi}$

Take a calendar spread of these digitals, and denote the strike where $d_2 = 0$ by $K_{d2}$, then

$$ \sqrt{2\pi} \left. \frac{ \partial D(K,T) }{\partial T} \right|_{K=K_{d2}} = \left. \frac{\partial^2 (I(K,T) \sqrt T) }{\partial T \partial k} \right|_{k = \log K_{d2}} $$ which is the term structure of the (de-annualized) skew at $K = K_{d2}$

Of course you can / might have to take other strikes, but then there will be some additional noise because the $N(-d_2)$s won't cancel quite so neatly.

Furthermore, in practice the digitals are implemented as tight put spreads, so that will be a source of additional noise too.