# Put Accumulator

The put accumulator is an extension of the eachway put. There are four strike prices with the settlement prices at 0, 10, 30, 60 and 100 as outlined in Figure 1. This is a strategy which, when bought out-of-the-money, can provide extremely high gearing, and increasingly rewards the speculator for an increasingly accurate forecast.

Put Accumulator Greeks | Below 1st Strike | Between 1st & 2nd Strikes | Between 2nd & 3rd Strikes | Between 3rd & 4th Strikes | Above 4th Strike |
---|---|---|---|---|---|

Delta | -ve | -ve | -ve | -ve | -ve |

Gamma | -ve | +ve -ve | +ve -ve | +ve -ve | +ve |

Theta | +ve | -ve +ve | -ve +ve | -ve +ve | -ve |

Vega | -ve | +ve -ve | +ve -ve | +ve -ve | +ve |

## Put Accumulator Expiry Price Profile

The put ‘accy’ has four strikes with ratio’d returns while Step options provide a special case.

The eachway put option has two strikes and hence consists of two vanilla binary put options. In the example used on the Eachway Put Options page the lower strike put contributed to 60% of the eachway put while the upper strike contributed the remaining 40%. As was discussed on that page, the ratios of 60% and 40% are not sacrosanct, with a final example of 75% and 25% being offered. But the total of the ratio must aggregate to 1.

The put ‘accy’ has four strikes and their aggregate must also sum to 1. In the following example the ratios 10%, 20%, 30% and 40%.

In the Fig.1 the Russian RTS index is used as the example with strike prices set at 650, 750, 850 and 950.

If the index settles above 950 the put accumulator settlement price is 0, between 950 and 850 the settlement prices is 10, between 850 and 750 it is 30, then 60 between 750 and 650 while anything below 650 has a settlement price of 100. If the index settles exactly on a strike then the settlement price is the arithmetic mean of the two adjacent settlement prices, i.e. 5, 20, 45 and 80.

This ‘strip’ of binary puts spreads the risk across a number of levels so that an increasingly correct forecast provides an increasingly incremental return.

## Put Accumulator Over Time

The put ‘accy’ of Figure 2 illustrates how time to expiry affects the value. Irrespective of the time to expiry the price profiles travel through four singular points at each strike, this point being the midpoint between the settlement values at each strike, i.e. at 9650 all profiles have a value almost exactly 5, at 850 20, at 750 45 and at 650 80. This constraint takes much of the vega and theta risk from the trade.

The 25-days put accumulator has a smooth profile creating extremely low gamma. With 0.001 days to expiry the black profile is almost vertical either side of the strike and horizontal elsewhere. This will generate unmanageable deltas around the strike although this should be of no concern since running these strategies naked immediately prior to expiry is not going to do huge material damage as the price swing for the lowest strike is a maximum of 40.

## Put Accumulator and Implied Volatility

Fig.3 plots the RTS put accumulator over a range of implied volatilities. When there are 5-days to expiry the wide range of implied volatility, i.e. 5% to 45%, does not have a major impact on the price profiles suggesting that vega will be low.

### Evaluating the Put Accumulator

The Put Accumulator is calculated as:

Put Accumulator = 0.1 x Binary Put(K_{1}) + 0.2 x Binary Put(K_{2}) + 0.3 x Binary Put(K_{3}) + 0.4 x Binary Put(K_{4})

where K_{1} is the uppermost strike and K_{4} the lowest strike.

The put accumulator does not fit into the standard all-or-nothing description of a binary option, yet this strategy is formed by binary options, the most dexterous of financial instruments.