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Call accumulator theta describes the change in the fair value of a call accumulator due to a change in time to expiry, i.e. it is the first derivative of the call accumulator fair value with respect to a change in time to expiry and is depicted as:

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

where ** P** is the call accumulator fair value and

*is time to expiry.*

**t**### Evaluating Call Accumulator Theta

Call accumulator Theta = Payout1 x Binary Call Option Theta(K_{1}) + Payout2 x Binary Call Option Theta(K_{2})

+ Payout3 x Binary Call Option Theta(K_{3}) + Payout4 x Binary Call Option Theta(K_{4})

where the terms to the right are the binary call option theta with strikes K_{1}, K_{2}, K_{3 }and K_{4 }respectively.

### Call Accumulator Theta Over Time

Call accumulator theta is displayed against time to expiry in Figure 1. The black 0.1-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 call thetas smooths the manner in which the fair value decays and appreciates.

### Call Accumulator Theta and Volatility

Implied volatility has a slightly different effect. The higher the ‘vol’ creates a more negative theta at low asset prices. It rises as it travels through all the strikes where the theta is now positive.

As volatility falls various strange effects take place on the theta until ‘vol’ is so low that the theta profile resembles the individual binary call options theta adjusted by the payout of each strike.