In this work, the effect an asset price bubble has on optimal portfolio allocations is investigated. A price bubble is an economic phenomenon that occurs when the observed market price of an asset does not coincide with its value in an objective sense. Advancements have recently been made in the mathematical modeling of price bubbles and allow us to investigate the effect the presence of a bubble has on portfolio optimization. A duality viewpoint allows us to gain insight in our investigation and the tools from the Malliavin Calculus are used to characterize the investor's optimal holdings. A simulation framework is developed and the results are analyzed. From this investigation, it is concluded that the presence of asset price bubbles cause investors to reduce the number of shares they trade of the asset.