Eachway Tunnel Delta

The eachway tunnel (ET) delta describes the change in the fair value of an eachway tunnel due to a change in the underlying price. The eachway tunnel delta is the first derivative of the eachway tunnel with respect to a change in underlying price and is depicted as:

where P is the eachway tunnel value and S is the underlying price. The delta is the gradient of the slope of the eachway tunnel price.

Evaluating ET Delta

The eachway tunnel delta can be constructed from the digital call delta.

Eachway Tunnel Delta = R1 x Digital Call Delta(K1) + R2 x Digital Call Delta(K2)

– R2 x Digital Call Delta(K3) – R1 x Digital Call Delta(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 Delta Over Time

The eachway tunnel delta are displayed against time to expiry in Figures 1a and 1b for rebates of 0.4 and 0.25 respectively.

The 0.1 day to expiry profiles show the two humps and two troughs of the deltas of the individual strikes. The ratio of the eachway tunnel for Figure 1a is R1 = 40% and R2 = 60%. This creates absolute deltas at the middle strikes (K2 & K3) considerably higher than at the outer strikes as expiry approaches.

The ratios of the eachway tunnel for Figure 1b are R1 = 25% and R2 = 75%. The disparity between R1 and R2 now provides two peaks of widely different heights and two troughs of widely different depth.

What is uniform between both illustrations is the delta profile, irrespective of days to expiry and rebate, travelling through zero midway between the central strikes. Furthermore, below this midway point between the central strikes the delta is always positive or zero, while above the central point the delta is always  negative  or zero.

As the time to expiry approaches zero should the asset price be pinned on one of the four strikes the delta tends to plus or minus infinity.

Eachway Tunnel Delta and Volatility

Figures 2a and 2b show the same eachway tunnel but the time to expiry is fixed to 2 days and the volatility is the variable. Figure 2a has a rebate of 0.40 while Figure 2b’s rebate is 0.25.

As one might expect the lowest volatility creates a comparable delta profile to the 0.1 days to expiry of Figs 1a and 1b.

With a relatively high volatility of 18% both 2a and 2b have shallow deltas as the deltas are now the weighted averages of four strikes.

An exceptionally low volatility makes the height disparity between the two positive delta humps far wider for Fig. 2b than 2a. The same can be observed for the troughs.

The 18% volatility profiles mean a shallower delta profile. For the market-maker this means a safer trade to put on so one can expect a more competitive, tighter bid/ask spread for the eachway tunnel.

Summary

• The eachway tunnel delta is positive or zero below the midway point of the central strikes, negative above.
• The lower the rebate the greater the disparity in the deltas.
• Lower absolute deltas than the tunnel deltas as four strikes now create a lower absolute weighted average of four, not two strikes.