# Put Accumulator Delta

Put accumulator delta describes the change in the fair value of a put accumulator due to a change in the underlying price. The put accumulator delta is the first derivative of the put accumulator value with respect to a change in underlying price. It is depicted as:

#### $\Delta = \frac{dP}{dS}$

where P is the put accumulator value and S is the underlying price.

### Evaluating Put Accumulator Delta

Put Accumulator Delta = Payout1 x Binary Put Delta(K1) + Payout2 x Binary Put Delta(K2)

+ Payout3 x Binary Put Delta(K3) + Payout4 x Binary Put Delta(K4)

where the terms are the binary put options delta 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:

Payout1 = 40%, Payout2 = 30%, Payout3 = 20% and Payout4 = 10%

In effect the binary put delta is the gradient of the price profile of the binary put option.

### Put Accumulator Delta Over Time

The put accumulator delta is displayed against time to expiry in Figure 1. It is not until time to expiry falls to 0.05-days that the ‘stalactites’ of the deltas of the individual strikes is evident. As the underlying price falls the length of the stalactites increases. This absolute increase reflects the increasing payoff of the put accelerator, in this case, 10, 30, 60 and 100.

The 25-day put accumulator is nearly flat. This is because with 25-days to go the averaging of the individual deltas creates a very low risk strategy. The 25-day profile of put accumulator Figure 2 illustrates that this is an extremely conservative natured option.

### Put Accumulator Delta and Volatility

Figure 2 shows the delta over a range of implied volatilities and 25 days to expiry.