Up-and-Out Binary Put Options K ≥ B
This knock out put, where the strike is higher or equal to the barrier, is shown in Figures 1 & 2 illustrating the price profiles with respect to time to expiry and implied volatility.
If these up-and-out binary put options K ≥ B profiles were inverted through the horizontal axis at 50 then the profiles will become those of one-touch calls, in effect meaning that this knockout becomes a short one-touch call.
Up-and-Out Binary Put Options K ≥ B w.r.t. Time to Expiry
When time to expiry is the variable (Fig.1) the lower the time to expiry leads to a higher fair value. The strike price of this strategy has become an irrelevance since the only criteria as to whether this is a winning bet or not is whether the barrier has been hit or not. The lower the time to expiry, the less the chance of the barrier being hit, therefore the higher the chance of a winner.
Up-and-Out Binary Put Options K ≥ B w.r.t. Implied Volatility
In the same manner, the lower volatility decreases the probability of the barrier being hit therefore the up and out binary put options (K ≥ B) fair value is higher.
The price profile of this knock-out is clearly that of a short one-touch call yet the equations are derived from completely different assumptions. The one-touch call is priced as twice the vanilla binary call while this knock-out, being a barrier option, is priced using the heat reflection theory. The price of a short one-touch call is therefore different from that of this up-and-out binary put. Work that one out!