Path dependent options are options whose payoffs depend on the path dependent function F t = F (S t , t) defined specifically for the given nature of path dependence of the asset price process S t . The most well known examples are the lookback options and Asian options. In a lookback option, the payoff function is dependent on the realized maximum or minimum price of the asset over certain period within the life of the option. The Asian options are also called average options since the payoff

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... epends on a preset form of averaging of the asset price over certain period. Consider an arithmetic average Asian option that is issued at time 0 and expiring at T > 0, its terminal payoff is dependent on the arithmetic average A T of the asset price process S t over period [0, T ]. The running average value A t is defined by