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Path integral algorithms are developed for evaluating European-style Asian options within three new families of multi-parameter local volatility models. The forward price process is a martingale under an assumed risk-neutral measure and the transition probability densities for such models are given in analytically closed form. Implied volatility surfaces for these models exhibit a wide range of pronounced smiles and skews of the type observed in the option markets. By concatenating the pricing kernels in time, we obtain an exact multidimensional path integral representation for any discrete-time path-dependent option on these models. Nonlinear variable transformations and numeraire changes are exploited in simplifying the path integrals. Efficient weighted Monte Carlo schemes for accurate evaluation of the integrals are then presented. The algorithms combine variance reduction techniques such as the control variate method with distributed computing to significantly increase efficiency. Our numerical results show that distributed Monte Carlo simulations scale almost linearly with the increase in independent processors on a high-performance computer cluster. As a competitive approach we develop a multinomial lattice method that is based on the backward propagation of the option value function using an updating rule. The paper includes simulation results for two kinds of Asian options: average price and average strike options. [ABSTRACT FROM AUTHOR]