Abstract
This paper addresses the problem of approximating the price of options on discrete and continuous arithmetic averages 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). An approximation for the implied normal volatility follows accordingly. The small-time asymptotics and the existence of the MLP are also rigorously recovered using large deviation theory.
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