# Eachway Tunnel Theta

Eachway  tunnel theta is the metric that describes the change in the fair value of an eachway tunnel owing to a change in time to expiry. The eachway tunnel theta is the first derivative of the eachway tunnel fair value with respect to a change in time to expiry. It depicted as:

#### $Θ=\frac{dP}{dt}$

where $\Theta$ is theta and P is the tunnel price and t is the time to expiry.

### Evaluating Eachway Tunnel Theta

Eachway Tunnel Theta = R1 x Binary Call Theta(K1) + R2 x Binary Call Theta(K2)

― R2 x Binary Call Option Theta(K3) ― R1 x Binary Call Option Theta(K4)

where the terms are the binary call theta with strikes K1, K2, K3 and K4 respectively.

K1, K2, K3 & K4 are lowest strike to highest strike, and

R1 + R2 = 1 and R1 < R2.

### Eachway Tunnel Theta Over Time

Eachway tunnel theta is displayed against time to expiry in Figures 1a & 1b. The longer to expiry profiles of Figure 1a create a more practical measurement of time decay (and appreciation) than the 0.1-day profile of Figure 1b. As usual the 0.1 day profile comes up with some pretty unhelpful numbers in terms of risk management.

At Figure 1a’s underlying level of 100.00 the profiles rise as time to expiry falls from 25 days. The level 100.00 is midway between the central two strikes and would generate a  settlement price of 100. So, as time recedes, the strategy gains most at this central strikes midpoint  of 100.00.

Figure 1b has a much wider ranged scale compared with Figure 1a. Midway between the central strikes there is a ‘V’ shaped valley in the profile. This is owing to the eachway tunnel already at 100 settlement price and can’t go any higher. A further reduction in time to expiry has no effect on the price  so the theta returns towards zero.

### Eachway Tunnel Theta and Volatility

Figure 2 provides eachway tunnel theta over a range of implied volatilities.

When the underlying is at 100.00 the black 2.0% profile reflects that at low volatility the strategy is already worth 100. Hence the dip in the profile at the asset price of 100.00 as a further drop in volatility can’t make the eachway tunnel value higher.

As the volatility rises to 6.0% the profile rises as the greater volatility means a lower value to the eachway tunnel.

At 10% the profile peaks before falling to the 14% level and then falling again to the 18% level. This is because at the volatility levels higher than 10% the eachway price profiles are getting flatter. Figure 3a of eachway tunnel shows a very shallow rise to 100.00 before falling back to zero.

To summarise, the positive and negative nature of the eachway tunnel theta can provide some interesting clues to the time trader. All traders will see different opportunities in Figure 1a & 2, with some eccentrics willing to trade with 0.1 days to expiry. These latter guys can genuinely wear the gambler’s hat as just a small move can change the trader’s P&L drastically.

By: Hamish Raw

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