Martingales and Stochastic Integrals in the Theory of Trading

This paper develops a general stochastic model of a frictionless security market with continuous trading. The vector price process is given by a semimartingale of a certain class, and the general stochastic integral is used to represent capital gains. Within the framework of this model, we discuss the modern theory of contingent claim valuation, including the celebrated option pricing formula of Black and Scholes. It is shown that the security market is complete if and only if its vector price process has a certain martingale representation property. A multidimensional generalization of the Black-Scholes model is examined in some detail, and some other examples are discussed briefly.