In the last few years, martingales and stochastic
integration have been widely used to describe the behavior of
markets or to derive computingmethods. The theme of this paper
is discussing about how to price the contingent claim by using
martingale and stochastic integrals, in a perfectmarket. We will
price all integrable contingent claims without restrictprice
processes to be geometric Brownian motions. Our basic
assumption is that threr exists at least one equilibrium
pricemeasure. This assumption is equivalent to the nonexistence
of arbitrageopportunities or the viability of the market. Under
the same assumption,Harrison priced contingent claims by using
locally bounded trading strategies.We will extend trading
strategies to a larger set, and price all integrablecontingent
claims. The mathematic tool we use are martingales and
stochastic integrals.We begin with some definitions and basic
theorems by the following threeproperties. First, we may extend
the definition of stochastic integrals,at least for local
martingale intrgrator. Second, stochastic integralswill preserve
the martingale properties. Third, Girsanov theorem. When we
price a contingent claim, our method is duplicating the cash
flowsof the contingent claim. If we have a portfolio for which
its cash flows isthe same with the cash flows of the contingent
claim, and the portfolio isself-fiancing, then we may use the
initial value of the portfolio to price thecontingent claim.
Another question is the uniqueness of the price. For the
uniqueness of theprice, we restrict trading strategies to be
self-financing such that the valueprocess of the portfolio is a
martingale. Furthermore, the martingale propertyalso gives
formulations to price the contingent claim. This paper is ended
bygiving two special cases in the Black-Scholes model, one is a
European calloption and the other is an American put option.