Analytic bounds and approximations for annuities and Asian options

Next, we show how these results can be used to obtain fully analytic expressions for lower and upper bounds for the price of a continuously sampled European-style Asian option with fixed exercise price. These analytic lower bound prices are as sharp as those from [Rogers, L.C.G., Shi, Z., 1995. The value of an Asian option. J. Appl. Probab. 32, 1077-1088], if not sharper, but in contrast do not require any longer the evaluation of a two-dimensional or a one-dimensional integral.