While it is commonly believed that an Asian option is always cheaper than its plain vanilla European counterpart, this paper shows that this notion is false. By deriving the lower bounds on Asian option prices as volatility goes to zero and comparing them to the lower bounds of vanilla European option prices, it can be proved that this notion may be violated for call options when the dividend yield of the underlying stock is higher than the interest rate, as well as for put options when the dividend yield of the underlying stock is lower than the interest rate.