A Brownian Model of Financial Markets

Abstract

Throughout this monograph we deal with a financial market consisting of N + 1 financial assets. One of these assets is instantaneously risk-free, and will be called a money market. Assets 1 through N are risky, and will be called stocks (although in applications of this model they are often commodities or currencies, rather than common stocks). These financial assets have continuous prices evolving continuously in time and driven by a D-dimensional Brownian motion. The continuity of the time parameter and the accompanying capacity for continuous trading permit an elegance of formulation and analysis not unlike that obtained when passing from difference to differential equations. If asset prices do not vary continuously, at least they vary frequently, and the model we propose to study has proved its usefulness as an approximation to reality. Our assumption that asset prices have no jumps is a significant one. It is tantamount to the assertion that there are no "surprises" in the market: the price of a stock at time t can be perfectly predicted from knowledge of its price at times strictly prior to t. We adopt this assumption in order to simplify the mathematics; the additional assumption that asset prices are driven by a Brownian motion is little more than a convenient way of phrasing this condition. Some literature on continuous-time markets with discontinuous asset prices is cited in the notes at the end of this chapter. The extent to which the results of this monograph can be extended to such models has not yet been fully explored.