Before revision:  We show that the logarithmic trading price process is a renewal reward process, and that, under a suitable scaling, this renewal reward process converges to a standard Brownian motion \to \to0 . Finally, we develop a GBM asymptotic model for trading prices, and derive a simple and effective prediction formula .
After revision:  We show that the logarithmic trading price process is a renewal reward process, and that, under a suitable scaling, this renewal reward process converges to a standard Brownian motion \to 0. We also prove that the modified ask and bid processes approach the original bouncing GBMs as \delta\to0 . Finally, we develop a GBM asymptotic model for trading prices, and derive a simple and effective prediction formula .