Eachway tunnel theta is the metric that describes the change in the fair value of a eachway tunnel (ET) due to a change in time to expiry. The ET theta is the first derivative of the ET fair value with respect to a change in time to expiry. It is depicted as:
where 
is theta and P is the tunnel price and t is the time to expiry.
Evaluating Eachway Tunnel Theta
The eachway tunnel theta can be constructed from the digital call theta.
Eachway Tunnel Theta = R1 x Digital Call Theta(K1) + R2 x Digital Call Theta(K2)
– R2 x Digital Call Theta(K3) – R1 x Digital Call Theta(K4)
where K1, K2, K3 & K4 are lowest strike to highest strike, and
where R1 + R2 = 1 and R1 < R2 and R2 = 1 – R1.
Eachway Tunnel Theta Over Time
The theta of a conventional call is extremely inaccurate for 0.1 days to expiry so the thetas of Figs. 1a & 1b with 0.1 days to expiry are worthless. If we were to accept that 0.1 days to expiry was inherently inaccurate then 0.5 days might just pass muster. Yet who needs to know the theta when there is half a day to expiry? It’s the premium or 1 less the premium.
But just say the maths behind the theta was accurate right up to expiry, would Figs. 1a & 1b be of use? One suspects that for any digital with less than 2 days to expiry the answer is “No”. Since it is highly unlikely that a future on the passage of time will ever be issued one can’t hedge theta with anything other than other options. In particular, if you tried to hedge away all your theta exposure in these examples one would simply exit the trade; there is no other efficient hedge.
All that can really be stated with any certainty is that between the inner strikes the theta is always positive. (Or zero once the eachway tunnel has reached the value of 1.0).
| European Digitals | Eachway Tunnels | Eachway Tunnel Delta | Eachway Tunnel Gamma | Eachway Tunnel Vega |
Eachway Tunnel Theta and Volatility
Figures 2a & 2b present the ultra low 2% volatility theta profiles. In this the 2% profiles have least absolute value compared with the higher volatilities. At 100.00 the theta is heading back down to zero as the maximum value of the eachway tunnel has already hit 1.00.
The only other point of note is how alike the two images are suggesting that the rebate level has very little effect on the theta.
Summary
A trader looking to use theta to establish when to buy or sell the eachway tunnel can glean a great deal of information from this illustration. For example, if the underlying is at 100.00 then buying the 10% profile gives you the greatest gain assuming the underlying remains at 100.00 . Unfortunately, this constraint is an issue since a quick look at Figure 2 of tunnel options makes obvious the downside of the asset moving away from 100.00.