I was hoping you could clarify two points for me in chapter 10 (all references to page numbers are for 2nd edition, 4th printing):

1) For the static replication of continuous barriers you write (page 246):

The difference in price between the replicating portfolio and the down and out put must be less than than the maximum (discounted) value of the replicating portfolio on the line (B,t)...As N goes to infinity this difference will go to zero...

You seem to be assuming some kind of continuity of the replicating portfolio here. My concern is, if the payoff of the replicating portfolio behind the barrier is large, then it is possible perhaps that the value of the portfolio at (B,t), where t lies in between one of the barrier times (t_j), could be quite large and potentially ruin the convergence of the replicating portfolio price to the down and out option price. Why does this not happen (or if it does, can you recommend some literature which examines the convergence in more detail)?

2) For the put-call symmetry section (reflection arguments) where interest rate is 0 you write (page254):

An interesting fact about this replicating portfolio is that the composition of the portfolio does not depend on \sigma....In fact, all that really matters is that when spot is on the barrier B, the smile should be symmetric - a call option of strike K/B must trade with the same implied vol as a put option of strike B/K

I think this is probably a typo? It seems to me that we want the smile to be "invariant under geometric reflection". So a call option of strike K/B must trade with the same implied vol as a put of strike B^3 /K (since then the log of these strikes are the same distance from the log of the barrier).

Thanks.