# 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.

### Put Accy Theta and Volatility

Here is the put accumulator theta against volatility. Far from being the profile that creates confusion in the Greek charts (as above), the theta now is the most  benign. The theta is almost flat at zero apart from the slope upwards either side of the lowest strike.

As the asset price rises the four theta profiles (apart from the 2%) uniformly descend to a negative reading. They are moving in tandem as the high 25 days to expiry has taken away the ‘get rick quick’ label.

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