John,


For a down and out put, I put the following as an approximation for our 
understanding purpose.

a down and out put with barrier B < strike K
is overestimated by

A put at struck at K - a put struck at B - (K-B)*digital put strike at B.

I ploted the barrier put (blue line)  versus the approximation (red line) as 
function of the 
barrier.  

The true barrier option has zero payoff whenever there is a touch down the 
barrier
during the lifetime of the option.  Therefore it goes to zero much faster than
above approximation when B -> K.

See the spreadsheet for more details.

Zimin