Convergence of Arbitrage-Free Discrete Time Markovian Market Models

Abstract

We consider two sequences of Markov chains inducing equivalent measures on the discrete path space. We establish conditions under which these two measures converge weakly to measures induced on the Wiener space by weak solutions of two SEDs, which are unique in the sense of probability law. We are going to look at the relation between these two limits and at the convergence and limits of a wide class of bounded functionals of the Markov chains. The limit measures turn out not to be equivalent in general. The results are applied to a sequence of discrete time market models given by an objective probability measure, describing the stochastic dynamics of the state of the market, and an equivalent martingale measure determining prices of contigent claims. The relation between equivalent martingale measure, state prices, market price of risk and the term structure of interest rates is examined. The results lead to a modification of the Black-Scholes formula and an explanation for the surprising fact that continuous-time arbitrage-free markets are complete under weak technical conditions.