  This article addresses the problem of approximating the price of options on
discrete and continuous arithmetic average of the underlying, i.e. discretely
and continuously monitored Asian options, in local volatility models. A
path-integral-type expression for option prices is obtained using a Brownian
bridge representation for the transition density between consecutive sampling
times and a Laplace asymptotic formula. In the limit where the sampling time
window approaches zero, the option price is found to be approximated by a
constrained variational problem on paths in time-price space. We refer to the
optimizing path as the most-likely path (MLP). Approximation for the implied
normal volatility follows accordingly. The small-time asymptotics and the
existence of the MLP are also recovered rigorously using large deviation
theory.
