Mathematics of Speculative Price

A variety of mathematical methods are applied to economists’ analyses of speculative pricing: general-equilibrium implicit equations akin to solutions for constrained-programming problems; difference equations perturbed by stochastic disturbances; the absolute Brownian motion of Bachelier of 1900, which anticipated and went beyond Einstein’us 1905 paper in deducing and analyzing the Fourier partial-differential equations of probability diffusion; the economic relative or geometric Brownian motion, in which the logarithms of ratios of successive prices are independently additive in the Wiener–Gauss manner, adduced to avoid the anomalies of Bachelier’s unlimited liability, and whose log-normal asymptotes lead to rational pricing functions for warrants and options which satisfy complicated boundary conditions; elucidation of the senses in which speculators’ anticipations cause price movements to be fair-game martingales; the theory of portfolio optimization in terms of maximizing expected total utility of all outcomes, in contrast to mean-variance approximations, and utilizing dynamic stochastic programming of Bellman–Pontryagin type; a molecular model of independent profit centers that rationalizes spontaneous buy-and-hold for the securities that exist to be held; a model of commodity pricing over time when harvests are a random variable, which does reproduce many observed patterns in futures markets and which leads to an ergodic probability distribution. Robert C. Merton provides a mathematical appendix on generalized Wiener processes in continuous time, making use of Ito formalisms and deducing Black-Scholes warrant-pricing functions dependent only on the certain interest rate and the common stock’s relative variance.