This post is published by Hamish Raw from https://hamishraw.com/

Binary put options theta is the metric that describes the change in the fair value of a binary put options due to a change in time to expiry, i.e. it is the first derivative of the binary put options fair value with respect to a change in time to expiry and is depicted as:

Θ=dP/dt

Binary put options theta is displayed against time to expiry in Figure 1. In the case of Gold $1700 binary put options the out-of-the-money options are to the right above the strike of $1700 while the in-the-money options are to the left below the strike.

As with binary call options theta, the binary put options theta is negative when out-of-the-money and positive when in-the-money. The amount of time to expiry has a major influence on the absolute value of the theta with very short-term options having theta that far outweighs the amount of premium that can actually decay.

As time to expiry increases the theta falls dramatically so that the 25-day binary put options theta peaks at just 0.5 ticks.

Figure 2 provides binary put options theta over a range of implied volatilities. The absolute value of the binary put options 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 reflecting that lower volatility increases the probability of the binary put option settling at 0 or 100.

Binary put options theta is zero when at-the-money so that as the underlying passes through the strike the position will change from short theta to long theta, or vice versa. This feature of vanilla binary options clearly does not make them ideal for taking in time decay by selling out-of-the-monies since 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.

Finite Theta

Figure 2of the Binary Put Options page shows a 5-day $1700 binary put option price profile. At the underlying gold price of $1725 this put is worth 31.4087. If 4.5-day and 5.5-day profiles were included then their values would be 30.4312 and 32.2627 respectively. Using the finite difference method:

**Binary Put Options Theta = ―(P _{1}―P_{2})/(T_{1}―T_{2})**

where:

T_{1} = The greater number of days to expiry

T_{2} = The lesser number of days to expiry

P_{1} = Binary put options fair value with greater number of days to expiry

P_{2} = Binary put options fair value with lesser number of days to expiry

so that the above numbers provide a 5-day binary put options theta of:

Binary Put Options Theta = ‒(32.2627‒30.4312)/(5.5‒4.5) = ‒1.8315

If the day increment was reduced from 0.5 to 0.00001 then:

T_{1} = 5.00001

T_{2} = 4.99999

P_{1} = 31.408715

P_{2} = 31.408679

so that the 5-day theta becomes:

Binary Put Options Theta = ‒(31.408715‒31.408679)/(5.00001‒4.99999) = ‒1.8221

Equations in financial engineering books will create a number which is:

1. based on the annual decay, and

2. base this number on a binary option price that ranges from 0 to 1. which in turn would provide a theta of:

Binary Put Option Theta = ‒1.8221×365/100 = ‒6.6506.

Arguably this number is about as useful as a chocolate teapot!

The Problem with Theta

The price of the 4-day and 5-day binary put options in the same price profile are 31.408697 and 29.296833 so that the actual 1-day time decay is ‒(31.408697‒29.296833)/(5‒4) = ‒2.1119, a price decay of 0.2898. In effect theta has underestimated the actual decay that will take place by 0.2898/1.8221 = 15.9%.

With 1-day to expiry the binary put option has a fair value of 13.3694 so the decay has to be 13.3694 at the gold price of $1725. In contrast the theta based on the first differential of price w.r.t. time, dP/dT , i.e. the one the text book equations trot out is 12.1013.

Why does the theta generated from the first derivative/finite difference method differ from the actual number? Binary and conventional options are calculated using an exponential factor e^{‒}^{rt} which in effect drives the price of the option over time to zero at an ever increasing rate.

In summary, if the theta is to be a useable number for practitioners it will need to be:

- Multiplied by 100 to reflect the range of prices 0-100 as opposed to 0-1, and
- Divided by 365 to get a daily rate.

But even then this number will be based 50% on time decay that has already taken place. If one is using the finite difference method then it might just possibly make more sense to simple evaluate the current option price, subtract a day from time to expiry and do a second calculation, and then take the second price from the first.