Dynamic equilibrium and volatility in financial asset markets (Q1379917)

Microeconomic theory is essentially concerned with the study of market equilibria. Agents make plans, in general as a result of solving individual optimization problems. Then certain variables, typically prices, are assumed to take the values required for those plans to be mutually consistent. The resulting prices are called equilibrium prices. The author models the market's microstructure as a stochastic differential game. The investor takes the price adjustment rule as given and determines his optimal holding of the stock, i.e., his best response, by maximizing his expected utility. Knowing the demand function of the investor, i.e., the investor's best response to his choice of price adjustment, the specialist determines the price adjustment by maximizing his expected profit. This is an equilibrium in strategies (not prices or quantities): each player's choice of an optimal strategy is a control problem. The author first solves the investor's optimization problem, obtaining his best response function to every possible choice of volatility by the specialist. Then he finds the optimal volatility choice by the specialist given the investor's best response function. Secondly, the objective of the paper is to estimate jointly the system of stochastic differential equations (SDEs) specifying the evolution of the price and volume variables. To filter the fundamental price at time \(t\) from observations of the market price up to that time, assuming full knowledge of the above mentioned joint dynamics of the system of SDEs, the author uses an extended Kalman-Bucy filter. This filter is obtained by expanding the dynamics through a Taylor series expansion of their drift and diffusion. Also, a consistent and asymptotically normal estimator of the unknown parameter values in the SDEs is constructed. Despite the fact that the model is written in continuous time and the data are sampled at discrete time intervals, the estimator is free of discretization bias. Some extensions of the results are considered: 1) the theoretical model could be extended to incorporate asymmetric information; 2) the model could incorporate a bid-ask spread, as a source of profit to the specialist; 3) the model predicts that market price changes exhibit additional volatility compared to those of the fundamental value of the stock.