# Up and Out Binary Puts K less than B

## Up and out binary puts K < B are binary put options with a strike (K) below the barrier (B) at which, if touched or traded above, the binary put option is knocked out, i.e. the strategy finishes immediately settling at zero.

This section covers up and out binary puts K < B (where the strike(K) is below the barrier).

### Up and Out Binary Put K < B w.r.t. Time to Expiry

Figure 1 illustrates the up and out binary put for gold, the barrier at $1850 pitched $150 above the strike at $1700. The barrier and strike are sufficiently distanced from each other for the barrier to have little effect on option when there three or less days to expiry. Only the 10- and 25-day profiles are in any way affected by the barrier as these price profiles approach the barrier at an acute angle.

With the down and out binary call (K>B) example, both the theta and vega between the barrier and strike were both positive and negative. In Figure 1 the strike and barrier are not generating this effect owing to a combination of the distance between the strike and barrier, time to expiry and changes in implied volatility which together lead to a uniformly negative theta above the strike, while positive below the strike. (The distance between the strike and barrier is referred to as the ‘stealth’ of the strategy.)

Unlike the down and out binary call (K>B) example which has the same time to expiry profiles as in Fig.1’s legend, all the price profiles of Fig.1 travel through the 50 price when at-the-money. Clearly this cannot be a function of time, so is a function of implied volatility and the difference between the barrier and the strike.

### Up and Out Binary Puts K < B w.r.t. Implied Volatility

Figure 2 shows a close resemblance to Figure 1 and it is clear that irrespective of the implied volatility the barrier has little or no effect on the at-the-money profiles as all profiles pass through 50. The barrier does have an effect on the profiles the higher the implied volatility which can be determined from the illustration by observing that the 10% and 20% profiles already have a value of zero at an underlying below the barrier.

The strategy would be of use if the trader wishes to go short gold and believes the price will be below the strike ($1700) at expiry. This knock out offers a cheaper entry but also comes with a built-in ‘stop-loss’ order at $1850, the cheaper entry being the premium one would have saved should the vanilla put travelled up to the barrier and been ‘stopped-out’.

**Formula**

where:

and: