Japan J. Indust. Appl. Math. (2011) 28:301–313 DOI 10.1007/s13160-011-0040-2 ORIGINAL PAPER Area 1 On the probability of completeness for large markets John A. Wright · Phillip S. C. Yam · Hailiang Yang Received: 19 July 2010 / Revised: 3 February 2011 / Published online: 15 May 2011 © The JJIAM Publishing Committee and Springer 2011 Abstract We consider a family of discrete multiperiod multinomial market models F n , each of which contains n - 1 stocks and one bond. All the securities are allowed to be risky and we assume that the number of states in each period is finite. We let the securities’ prices follow probability distributions that reflect the traders’ view of the market. Under mild restrictions on the probability structure of F n , we show that the probability that a market, chosen at random from F n , is complete tends to one as n approaches infinity. Keywords Market completeness · Large markets · Single period model · Multiperiod model · General linear groups over a finite field Mathematics Subject Classification (2000) 91G99 · 60B20 1 Introduction The completeness of real-world security markets has been the subject of much the- oretical and empirical study over the years. Several papers (such as [1, 3, 5]) reject the hypothesis of real-world market completeness statistically. Indeed, the assump- tion of market completeness has been blamed for the poor predictive power of many J. A. Wright Department of Mathematics, The University of Hong Kong, Hong Kong, China P. S. C. Yam Department of Statistics, The Chinese University of Hong Kong, Hong Kong, China H. Yang (B ) Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong, China e-mail: hlyang@hkusua.hku.hk 123