Price dynamics on a stock market with asymmetric information

Abstract

International audienceWhen two asymmetrically informed risk-neutral agents repeatedly exchange a risky asset for numéraire, they are essentially playing an n-times repeated zero-sum game of incomplete information. In this setting, the price Lq at period q can be defined as the expected liquidation value of the risky asset given players' past moves. This paper indicates that the asymptotics of this price process at equilibrium, as n goes to ∞, is completely independent of the "natural" trading mechanism used at each round: it converges, as n increases, to a Continuous Martingale of Maximal Variation. This martingale class thus provides natural dynamics that could be used in financial econometrics. It contains in particular Black and Scholes' dynamics. We also prove here a mathematical theorem on the asymptotics of martingales of maximal M-variation, extending Mertens and Zamir's paper on the maximal L1-variation of a bounded martingale

Covariance Control Problems over Martingales with Fixed Terminal Distribution Arising from Game Theory

“…The main result in [9] is two-fold. At first, a characterization of the limit V ∞ = lim n V n as a maximal covariance function is given but without the corresponding continuous-time control formulation introduced in the present work.…”

“…Our aim is to characterize the solutions of this problem and to relate them with the limits of the maximizers of discrete-time functionals Ψ n defined below arising from the study of repeated games with incomplete information. The functionals Ψ n have been introduced in De Meyer [9] in order to solve the problem of optimal revelation over time for an informed agent in financial exchange games (see also Gensbittel [13] for the multi-dimensional extension). The maximizers of these discrete-time optimization problems are equilibrium price processes in these games.…”

“…Our main convergence results (Theorems 1.3 and 1.4 below) identify the continuous-time limits of these price processes as solutions of (1). Moreover, our motivation for studying both the continuous-time and the discretetime problems in the same work is motivated by the fact that the control problem cannot be directly interpreted as a continuous-time game of the same type as the games introduced in [9]. Indeed, convergence involves a Central Limit Theorem, and thus a loss of information on the data of the discrete-time problem.…”

“…The asymptotic behavior of such discrete-time functionals has been recently studied in De Meyer [9] for the case d = 1. The main result in [9] is two-fold.…”

“…We introduce five assumptions denoted A1-A5 on the function V . A1-A4 are the natural generalizations of the assumptions given in [9], while A5 is specific to the multi-dimensional case. Let ∆ 2 denote the set of probabilities with finite second-order moments over R d and ∆ 2 0 the subset of centered probabilities.…”