  In this article we discuss the problem of calculating optimal
model-independent (robust) bounds for the price of Asian options with discrete
and continuous averaging. We will give geometric characterisations of the
maximising and the minimising pricing model for certain types of Asian options
in discrete and continuous time. In discrete time the problem is reduced to
finding the optimal martingale transport for the cost function $|x+y|$. In the
continuous time case we consider the cases with one and two given marginals. We
describe the maximising models in both of these cases as well as the minimising
model in the one-marginal case and relate the two-marginals case to the
discrete time problem with two marginals.
