Digital call theta measures the change in the value of the digital call option owing to the shortening of the time to expiry. It is represented as:
where C is digital call value and t is time to expiry.
If theta is positive the option increases in value over time while if negative it decreases in value over time: if
- the call option is out-of-the-money then the digital call option theta will be negative.
- the digital call option is in-the-money then the digital call option theta will be positive.
- when the option is at-the-money the theta is zero.
This is because as time decreases there is less chance of the out-of-the-money option becoming in-the-money. When the former it remains a loser, the latter a winner. By the same token, the in-the-money option has progressively less time to fall out-of-the-money and turn into a loser.
When the asset price equals the the strike price the theta is zero. This is because an at-the-money call option is always worth 50. This reflects that irrespective of time to expiry the option always has a 50:50 chance of being a winner or loser.
Figures 1 & 2 represent the digital call option theta profiles with respect to time remaining to expiry (Fig.1) and implied volatility (Fig.2).
Digital Call Theta Over Time
The digital call theta shows that as time to expiry decreases the theta profiles concertina to the strike.
When there are 25 days to expiry the high and low of the theta are at their most distant from strike. They also have the lowest absolute values.
The 0.1 day to expiry profile has the peak and trough closest to the strike. The absolute value of the minimum and maximum theta reaches unrealistically high levels.
The theta is lower when more time to expiry, a common rule irrespective of conventional or digital instrument. In the instance of the digital call the theta has its absolute value highest and closest to the strike the least time to expiry.
In the instance of Figure 1 the 0.1-day theta at 99.90 is -1.0025 (or -100.25 if pricing the winning option at 100). Clearly this number is unrealistic as it exceeds the 0-1(00) digital option price range and this inaccuracy is covered in more detail under the section on Greeks.
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Digital Call Theta and Volatility
Fig.2 represents the digital call theta over a range of implied volatility from 2% to 18%. 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.
As the time to expiry shortens the theta becomes an untrustworthy measure. Where time to expiry is less than one day the theta grossly exaggerates the decay of the option. This leads to 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.