Trading the Term Structure of Implied Volatility Skew with Digital Calendars
Summary
The note describes a theoretical way to express changes in implied volatility skew across maturities using a calendar spread of digital puts. A digital option’s price depends on the strike slope of implied volatility, so comparing digitals across expiries can reveal how that slope changes with time. The derivation uses log strike and selects strikes at each maturity where the Black–Scholes d2 value is zero; under that condition, terms in the digital prices simplify and the spread corresponds to the maturity change in de-annualized skew.
This is a conceptual relationship rather than a tested trading strategy. The author notes that other strike choices introduce extra noise because the distribution terms no longer cancel neatly. In practice, digitals are approximated with narrow put spreads, adding further measurement noise. The document does not assess transaction costs, hedge behavior, or realized profitability.
Key ideas
- A digital put’s value contains a term linked to the strike derivative of implied volatility.
- A calendar spread of digitals can therefore provide exposure to changes in skew between maturities.
- Choosing strikes with d2 equal to zero at each expiry simplifies the theoretical relationship.
- Alternative strikes and put-spread approximations add noise to the skew exposure.
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Full text
# Trading term structure of skew
# Trading term structure of skew
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 by Frido (score 1)
https://quant.stackexchange.com/a/76878
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.