Digital put theta is the metric that describes the change in the fair value of a digital put option due to a change in time to expiry. It is the first derivative of the digital put option fair value with respect to a change in time to expiry and is depicted as:
where P = value of the put and t is the incremental change in time to expiry.
Digital Put Theta Over Time
Digital put theta is displayed against time to expiry in Figure 1.
As with digital call theta, the digital put theta is negative when out-of-the-money and positive when in-the-money.
When out-of-the-money, above 100.00 in Figure 1, the premium decays as over time the probability that the asset price drops below the strike decreases.
When the asset price is below the strike the theta is positive. As time passes there is less probability of the asset price moving above the strike so the value of the option increases.
The amount of time to expiry has a major influence on the absolute value of the theta. Very short-term options often have a theta that far outweighs the amount of premium that can actually decay. This bizarre feature is discussed in greater detail in theta calculations [coming soon].
As time to expiry increases the absolute theta falls dramatically. When there are 25 days to expiry the digital put theta peaks at just 0.0023 at 97.80. How is this number arrived at?
Example: Set 25% vol and asset price to 97.80. When there are 25.5 and 24.5 days to expiry the option (based on a range of 0-1.0, not 0-100.0) is worth 0.644194 and 0.646482 respectively.
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Digital Put Options Theta and Volatility
The absolute maximum value of the digital put theta is fairly static over the range of implied volatility. As the implied volatility falls the peak and trough of the options close in on the strike. This reflects that lower volatility increases the probability of the digital put option settling at 0 or 100.
Digital put theta is zero when at-the-money. Therefore, as the underlying passes through the strike the position will change from short theta to long theta, or vice versa. This feature of vanilla digital options clearly does not make them ideal for taking in time decay by selling out-of-the-monies. A sale of an out-of the-money put would not only lose money on a fall through the strike, but the subsequent position would lose money as the premium now increased in value over time.