Nope: Correct for binary (thank, that was the name I confused with barrier option!). But, as call Payoff (x-K)^+ have second order derivatives that are measure valued, there exists for any N, any probability measure [$]\mu[$] (with technical assumptions here) a sampling sequence [$]x^1,\ldots,x^N[$]such thatOK, so just so that I get it straight:

1. call payoff (S - K)+ isnotBV and there is NO sampling sequence for it which converges faster than 1/N

2. binary payoff Heaviside(S - K) is BV and there IS a sampling sequence for it which converges faster than 1/N

Correct?

[$] | \int_{R^D} (x_d-K)^+ d\mu(x) - \frac{1}{N} \sum_{n=1}^N (x_d^n-K)^+ | \le \frac{C}{N^2} [$]

for any strike K, d= 1..D.

More precisely, without proving this result, one can not pretend to break the curse of dimensionality, would he use deep tralala networks or not.