From the need for a more scientific approach to the problem of pricing and risk control has emerged the advantage of exploiting the availability of more advanced numerical techniques in Finance. The Asian options are finance options whose payoff function depends on the average price of the underlying asset, observed over the duration of the contract. The barrier options are options whose payoff depends on whether the underlying asset price reaches a predetermined value in a certain period of time. Both these types of options are conceived to protect investors from manipulations of the markets or adverse movements of the underlying asset values and are characterized by being path-dependent options. The pricing of Asian barrier options is traditionally based on Monte Carlo methods that are affected by high computational costs and inaccuracy, due to their slow convergence. A collocation Boundary Element Method (BEM) is particularly advantageous in this framework because the differential problem is often defined in an unbounded domain but the data are assigned on a limited boundary (the barrier). In particular, we have recently addressed the evaluation of Asian barrier options with continuous monitoring, but considering the unusual definition of geometric average for the underlying asset prices, for the greater simplicity of the related mathematical model and the absence of particular computational troubles in the implementation of the method. On the contrary, arithmetic Asian options are hard to price and several pricing methods have been put in place for their evaluation. To our knowledge, there are not many papers that address arithmetic Asian barrier options with continuous monitoring. Starting from the theoretical analysis, we begin investigating in deep an integral representation of the vanilla option price with still remarkable numerical issues and we develop a numerical strategy, addressing the highlighted computational problems, to price a barrier arithmetic Asian option.