# 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.

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.

• Can this idea be extended to european vanilla options? Oct 4 at 3:13
• @smg_08 Not sure what exactly you mean, but as I wrote the digitals are implemented as vanilla option call/put spreads in practice. So you could use index options. Options on single stocks are usually American - then I'm not sure if the above works. Oct 5 at 6:17
• Thank you @Frido Oct 6 at 2:27