  We consider the pricing and the sensitivity calculation of continuously
monitored barrier options. Standard Monte Carlo algorithms work well for
pricing these options. Therefore they do not behave stable with respect to
numerical differentiation. One would generally resort to regularized
differentiation schemes or derive an algorithm for precise differentiation. For
barrier options the Brownian bridge approach leads to a precise, but
non-Lipschitz-continuous, first derivative.
  In this work, we will show a weak convergence of almost order one and a
variance bound for the Brownian bridge approach. Then, we generalize the idea
of one-step survival, first introduced by Glasserman and Staum, to general
scalar stochastic differential equations and combine it with the Brownian
bridge approach leading to a new one-step survival Brownian bridge
approximation. We show that the new technique can be adapted in such a way that
its results satisfies stable second order Greeks. Besides studying stability,
we will prove unbiasedness, leading to an uniform convergence property and
variance reduction.
  Furthermore, we derive the partial derivatives which allow to adapt a
pathwise sensitivity algorithm. Moreover, we develop an one-step survival
Brownian bridge Multilevel Monte Carlo algorithm to greatly reduce the
computational cost in practice.
