  The time average of geometric Brownian motion plays a crucial role in the
pricing of Asian options in mathematical finance. In this paper we consider the
asymptotics of the discrete-time average of a geometric Brownian motion sampled
on uniformly spaced times in the limit of a very large number of averaging time
steps. We derive almost sure limit, fluctuations, large deviations, and also
the asymptotics of the moment generating function of the average. Based on
these results, we derive the asymptotics for the price of Asian options with
discrete-time averaging in the Black-Scholes model, with both fixed and
floating strike.
