# Tunnel Options Theta

Tunnel options theta is the metric that describes the change in the fair value of a tunnel option due to a change in time to expiry. The tunnel option is the first derivative of the tunnel fair value with respect to a change in time to expiry. It is depicted as:

#### $Θ=\frac{dP}{dt}$

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

### Evaluating Tunnel Theta

Tunnel Theta = Binary Call Theta(K1) ― Binary Call Theta(K2)

where the first term and second terms are the binary options call theta with strikes K1 and K2 respectively.

### Tunnel Theta Over Time

Tunnel theta is displayed against time to expiry in Figure 1.

At 10% volatility the 25-day and 8-day profiles are flat at zero or fractionally above zero at 100.00.

The red 0.5-day profile establishes a positive theta between the strikes and a negative theta outside the strikes. This is what one may expect on studying Figure 1 of tunnel options where it is apparent the price between the strikes rises and outside the strikes falls.

The 0.1-day to expiry profile shows that between the strikes the profile has fallen to zero. This is because between the peaks of the 0.1-day profile the tunnel price has already reached 100.00. The two peaks are yet reach 100.00 so are extremely positive.

Outside the strikes the asset price is out-of-the-money and the tunnel price is falling to zero hence the negative zero.

### Tunnel Options and Volatility

Figure 2 presents a very interesting set of profiles. The 2.00% profile shows that between the strikes the theta is still just positive meaning that the tunnel value is still approaching 100.00. Outside the strikes the 2.00% profile is negative as the tunnel becomes a lost cause to the holder.

The interpretation of the 6% to 18% profiles is fairly clear. The hump between the strikes reflects that the tunnel value of all volatilities is positive.

The 18% profile shows a positive theta across the underlying asset range. At 100.00 a reduction in time to expiry does not generate an increase in tunnel value as much as the 6%, 10% and 14% profiles.

The 6% and 14% profiles are about the same but are lower than the 10% profile. The 14% yellow profile is still rising while the 6% green profile is on its way back down.

### Summary

A trader looking to use theta to establish when to buy or sell the 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.

By: Hamish Raw

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