Quote from riskarb:
The steepness of the ramp, due to simple proximity, has little to do with the gamma-convexity unless you're trading in the final moments before expiration. They work great if you've survived 80% of the holding period and can reach 80% of the max-gain on the vertical, but they're not a viable hedge if there is significant time remaining. Also, the gamma is reduced as you tighten strike width. Ok as a vol hedge, but gamma is the primary risk.
Well, my understanding is that a 1 tick wide vertical will replicate a European digital everywhere on the map of risk-neutral possible outcomes. Path-independence means only expiration risk profile replication is necessary from a static hedge point of view. This replication obviously doesn't hold for American digitals.
I'll never understand the need to replicate 100%... trading is about accepting risk. Any replication will entail significant edge-loss, correlation and risk-holes.
Agree, didn't mean to imply 100% replication was a good idea as that obviously negates any profit potential not to mention being wholly impractical!
However, from a purely academic point of view, it is useful to know that a 1 tick wide vertical is as close as you'll get to a European digital - and thus can help in terms of being an intuitive tool for pricing them, especially when taking skew into consideration.
To add to the discussion -- single and double barriers are +expectancy vanilla replications at fairval. Barrier-gamma is cheap, vanilla gamma is expensive.
This is clearly the reason to trade them!
The points I make above are all pertinent to European digitals not Americans. Bear with me, I'm not quite ready to have an intelligent conversation on those yet

MoMoney.
