Up and out Call (UAOC), Down and Out Put (DAOP) have digital risk. What about Up and In Call (UAIC) and Down and In Put (DAIP), do they also have digital risk?

remember that the vanilla option can be created from the knock-in and knock-outuoc + uic = vanilla calland also that vanilla options do not have risk-reversals embedded.so if you are short the UOC then you are short the vanilla and long the up and in and its the up and in behaviour which dominates at the barrier.

thanks davehow up and in behavior dominates at the barrier? and how digital risk is defined for UIC and DIP?

take an example of 100 strike call, knock out @ 110. As you approach the barrier the value of the option drops because the probability of getting koocked out increases. At the same time, the option's intrinsic value is increasing. At some point the option's delta switches from +ve to -ve so if you are short the UOC you short the underlying to hedge the fact that the option you are short is likely to expire worthless which is counter to the delta of a vanilla call which increases monotonically. The idea here is that if the spot goes through the barrier then as you are short the option a liability is extinguished so to hedge that you short the underlying - if the spot goes through then you lose on the delta hedge but the option disappears - you should then buy back your hedge. Obviously, there are a number of danger points here:1. As you get close to the barrier the vol of the stock dies and as time passes the option's delta switches back to positive in which case you buy back your hedges and lose money2. he spot gaps through the strike - the option extinguishes but you lose a lot on the short underlying on the gap move.So what youa re really hedging at the barrier is the up and in call - if you are short the UOC you are long the UIC which may or not come into life near the barrier.I don't know if this has answered your question ...

That was a good stuff, Dave.Please give suggestions on the following point."We have digital risk when there is a chance of loosing all the +ve money (it there is any) at the barrier."