  Let D be a domain in the plane containing the origin. We are interested in
the ensemble of self-avoiding walks (SAW's) in D which start at the origin and
end on the boundary of the domain. We introduce an ensemble of SAW's that we
expect to have the same scaling limit. The advantage of our ensemble is that it
can be simulated using the pivot algorithm. Our ensemble makes it possible to
accurately study SLE predictions for the SAW in bounded simply connected
domains. One such prediction is the distribution along the boundary of the
endpoint of the SAW. We use the pivot algorithm to simulate our ensemble and
study this density. In particular the lattice effects in this density that
persist in the scaling limit are seen to be given by a purely local function.
