This post was published by Hamish Raw of https://hamishraw.com/.
Binary call options theta measures the change in the value of binary call options owing to the shortening of the time to expiry, so if theta is positive the option increases in value over time, if negative it decreases in value over time. If binary call options are out-of-the-money then binary call options theta will be negative. If binary call options are in-the-money then binary call options theta will be positive. This is because as time decreases there is less chance of the out-of-the-money option becoming in-the-money, and therefore a winner, while by the same token, the in-the-money option has less time to fall out-of-the-money and turn into a loser.
Figures 1 & 2 represents binary call options theta profiles with respect to time remaining to expiry (Fig.1) and implied volatility (Fig.2).
The binary call options theta shows that as time to expiry decreases the theta profiles concertina until at 0.1 days to expiry the absolute value of the minimum and maximum theta reaches unrealistic levels. In the instance of Fig.1 the trough bottoms out at an underlying price of $99.50 with a theta of -120.37 while the peak is at an underlying of $100.50 where the theta is +121.58. Clearly the option can not lose more than its maximum value of 100 so this particular example very much underlines the unreliability of the theta measure when there is less than one day to expiry.
Fig.2 represents the binary call options theta over a range of implied volatility from 20% to 40%.
Here we see that the minimum and maximum absolute values remain fairly constant but the peaks and troughs close in on the strike price as implied volatility decreases.
Another feature of these two illustrations revolves around the scale of the graphs; where time to expiry is less than one day the theta grossly exaggerates the decay of the option so that unhelpful numbers are chucked out which can be very misleading to the uninitiated. On the other hand, with just 5 days to expiry the theta produces not unreasonable forecasts of time decay.