JMFJournal of Mathematical Finance2162-2434Scientific Research Publishing10.4236/jmf.2014.42008JMF-43032ArticlesBusiness&Economics Physics&Mathematics On Local Times: Application to Pricing Using Bid-Ask aulC. Kettler1*OlivierMenoukeu-Pamen2*FrankProske1*Institute for Financial and Actuarial Mathematics, Department of Mathematics, University of Liverpool, Liverpool, UKCentre of Mathematics for Applications, Department of Mathematics, University of Oslo, Oslo, Norway* E-mail:paulck@math.uio.no(ACK);Menoukeu@liverpool.ac.uk(OM);proske@math.uio.no(FP);1402201404028494October 12, 2013December 3, 2013 January 4, 2014© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

In this paper, we derive the evolution of a stock price from the dynamics of the “best bid” and “best ask”. Under the assumption that the bid and ask prices are described by semimartingales, we study the completeness and the possibility for arbitrage on such a market. Further, we discuss (insider) hedging for contingent claims with respect to the stock price process.

Order Statistics; Semimartingales; Local Times; ArbitrageNOTES

1In fact since S is continuous and since all continuous sigma martingales are in fact local martingales, we only need to concern ourselves with local martingales

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