Eachway put theta measures the change in the fair value of an eachway put due to a change in time to expiry. Eachway put theta is the first derivative of the eachway put fair value with respect to a change in time to expiry. It is depicted as:
where P is the fair value of the eachway put and t is time to expiry.
Evaluating Eachway Put Theta
Eachway Put Theta = R1 x Digital Put Theta(K1) + R2 x Digital Put Theta(K2)
where the terms to the right are the digital put theta with strikes K1 and K2 respectively, K1 being the lower strike, and
where R1 + R2 = 1 and where R1 < R2.
Theta Over Time
The 100:25:0 ratio eachway put fair value and eachway put theta (Figures 1a & 1b) are displayed against time to expiry. By comparing the 8 day profile (yellow) with the 0.5 day profile (red) one can intuitively figure out the theta at any point of the underlying price.
For example: at 98.20 the 0.5 day profile is above the 8 day profile so as time passes the eachway put rises in value therefore establishing a positive theta.
Starting at an asset price of 102.00 (to the extreme right of the lower scale) and moving to the left the theta starts negative or zero as both individual digital put options are out-of-the-money. As the underlying approaches the upper strike of 101.00 theta rises to pass through the strike and immediately below the strike price is positive. As the asset price falls further, dependant on how much time to expiry is left, the theta increasingly becomes influenced by the lower strike of 98.50. With 0.2 days to expiry, midway between the strikes neither strike exerts influence with theta flat at zero.
As the asset price moves lower towards 98.50 the lower strike now dominates the profile of the theta, increasingly so the less time to expiry.
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Theta and Volatility
Figures 2a & 2b provide 100:50:0 theta over a range of implied volatilities.
Although the 2% implied volatility profile presents the same wave pattern as the Figure 1b, 0.5 day profile, the higher implied volatility profiles are a very smooth rise as the asset price falls from 102.00 to 98.00.
Since the ratio is 100:50:0 the thetas all intersect roughly midway between the strikes at 99.75. In effect the higher the implied volatility and the 100:50:0 ratio generates a very uniform theta profile. In fact if Figure 2b was revolved 180° the profiles would look almost exactly the same.