  We consider a limit order book, where buyers and sellers register to trade a
security at specific prices. The largest price buyers on the book are willing
to offer is called the market bid price, and the smallest price sellers on the
book are willing to accept is called the market ask price. Market ask price is
always greater than market bid price, and these prices move upwards and
downwards due to new arrivals, market trades, and cancellations. We model these
two price processes as "bouncing geometric Brownian motions (GBMs)", which are
defined as exponentials of two mutually reflected Brownian motions. We then
modify these bouncing GBMs to construct a discrete time stochastic process of
trading times and trading prices, which is parameterized by a positive
parameter $\delta$. Under this model, it is shown that the inter-trading times
are inverse Gaussian distributed, and the logarithmic returns between
consecutive trading times follow a normal inverse Gaussian distribution. Our
main results show that the logarithmic trading price process is a renewal
reward process, and under a suitable scaling, this process converges to a
standard Brownian motion as $\delta\to 0$. We also prove that the modified ask
and bid processes approach the original bouncing GBMs as $\delta\to0$. Finally,
we derive a simple and effective prediction formula for trading prices, and
illustrate the effectiveness of the prediction formula with an example using
real stock price data.
