  This paper develops a model for the bid and ask prices of a European type
asset by formulating a stochastic control problem. The state process is
governed by a modified geometric Brownian motion whose drift and diffusion
coefficients depend on a Markov chain. A Girsanov theorem for Markov chains is
implemented for the change of coefficients, including the diffusion coefficient
which cannot be changed by the usual Girsanov theorem for Brownian motion. The
price of a European type asset is then determined using an Esscher transform
and a system of partial differential equations. A dynamic programming principle
and a maximum/minimum principle associated with the stochastic control problem
are then derived to model bid and ask prices. These prices are not quotes of
traders or market makers but represent estimates in our model on which
reasonable quantities could be traded.
