# Down and Out Binary Call Options B less than K

Down and out binary call options B less than K are binary call options with the standard strike(K), but also have a second level, the barrier(B), which if touched by the underlying means that the instrument is immediately knocked out, settling at zero. In this section the level of the strike is always higher than that of the barrier.

Down and Out Binary Call (K > B)Below BarrierBetween Barrier & StrikeAbove Strike
Delta0+ve+ve
Gamma0-ve +ve-ve
Theta0+ve -ve+ve
Vega0+ve -ve-ve

## Down and Out Binary Call Options B less than K w.r.t. Time to Expiry

The following illustrations use a working example of crude oil with the barrier at \$100.00, strike at \$104.00, 25% implied volatility and a range of days running from 50 down to 0.2 days.

Fig.1 – Oil Down and Out Binary Call Option (K>B) FV w.r.t. Time to Expiry

When this knockout has just one day to expiry the barrier has no effect on the option price since, with an implied volatility of 25%, the \$104.00 binary call with the underlying at \$100, i.e. \$4.00 out-of-the-money, would be worthless regardless of whether there is a \$100.00 barrier.

With 50 days to expiry the price profile is almost a straight line with the fair value at the \$104.00 strike of  only 28.85 as opposed to 50, the price of the at-the-money binary call option. With 1 day to expiry the strategy is worth 49.74 when at-the-money so at the strike the down-and-out binary call therefore has a positive down and out binary call theta which reflects the decreasing effect over time of the probability of the barrier knocking the binary call out.

The following illustration shows the prices of the down-and-out binary call option with a common barrier of \$100.00 but a range of strikes from \$100.00 to \$108.00. The \$108.00 strike binary call option is unaffected by the barrier whereas the \$100.00 strike approaches the barrier from a relatively steep angle. The profile of the \$108.00 strike down and out binary call option follows the concave profile of an out-of-the-money binary call whereas the profile of the strategy where the strike equals the barrier is convex. Indeed this profile defines the down-and-out binary call for any strike that has the condition K<B.

Fig.2 – Oil Down and Out Binary Call Option (K>B) FV w.r.t. Strike Prices

It is apparent that with little time to expiry or with the strike at a remote distance from the barrier, the barrier has little effect on the binary call option. So when might this bet be worth trading?

1. The initial requirement when buying this knock-out option is the view that the barrier will not be hit during the life of the option.

2. The subsequent requirement is for the underlying to be above the strike at expiry.

The first requirement might be enhanced if there was a major level of support above the barrier which the barrier can hide behind. A knock-out would then be a cheaper alternative to the straight binary call for the chartist who believes the support will hold and wants to take a bullish stance on oil.

## Down and Out Binary Call Options B less than K w.r.t. Implied Volatility

The down-and-out binary call option with a range of implied volatility is illustrated in Fig.3.

Fig.3 – Oil Down and Out Binary Call Option (K>B) FV w.r.t. Implied Volatility

Figure 3 illustrates how the lower implied volatility decreases the impact of the barrier on the fair value of this option when in-the-money but also shows mixed results between the strike and the barrier. The profiles indicate that as implied volatility increases the down-and-out binary call vega above the strike is negative as the value of the option has decreased although immediately below the strike no clear rule concerning vega can is discernable.

A clearer deduction from the always upward-sloping profiles is that the down-and-out binary call delta is always positive with an increase in implied volatility lowering the delta when in-the-money. When the underlying is between the barrier and the strike yet again there are mixed signals when there is an increase in implied volatility from 5.0% which initially raises the delta, but then above 35% it falls.

Down-and-out binary call gamma properties below the strike are also mixed though above the strike the gamma is always negative.

share