# Put Accumulator Theta

Put accumulator theta, like binary put strip theta, describes the change in the fair value of a put accumulator due to a change in time to expiry. It is the first derivative of the put accumulator fair value with respect to a change in time to expiry. It is depicted as:

#### $\Theta = \frac{dP}{dt}$

where P is the call accumulator fair value and t is time to expiry.

### Evaluating Put Accumulator Theta

Put Accumulator Theta = R1 x Binary Put Theta(K1) + R2 x Binary Put Theta(K2)

+ R3 x Binary Put Theta(K3) + R4 x Binary Put Theta(K4)

where the terms are the binary put options theta with strikes K1, K2, K3 & K4 respectively.

Strikes K1 < K2 < K3 < K4 and K1 + K2 + K3 + K4 = 1

The payouts in the below examples are:

R1 = 40%, R2 = 30%, R3 = 20% and R4 = 10%

In effect the binary put theta is the gradient of the theta profile of the binary put option.

### Put Accy Theta Over Time

Put accumulator theta is displayed against time to expiry in Figure 1. The black 0.05-day profile shows the theta increasingly rising and plunging around zero as the increasing payouts take effect. With 25-day to expiry the profile is almost flat. The averaging of all four strike’s binary put thetas smooths the manner in which the fair value decays and appreciates.