On Local Times: Application to Pricing Using Bid-Ask

Affiliation(s)

Centre of Mathematics for Applications, Department of Mathematics, University of Oslo, Oslo, Norway.

Institute for Financial and Actuarial Mathematics, Department of Mathematics, University of Liverpool, Liverpool, UK.

Centre of Mathematics for Applications, Department of Mathematics, University of Oslo, Oslo, Norway.

Institute for Financial and Actuarial Mathematics, Department of Mathematics, University of Liverpool, Liverpool, UK.

ABSTRACT

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.

Cite this paper

P. Kettler, O. Menoukeu-Pamen and F. Proske, "On Local Times: Application to Pricing Using Bid-Ask,"*Journal of Mathematical Finance*, Vol. 4 No. 2, 2014, pp. 84-94. doi: 10.4236/jmf.2014.42008.

P. Kettler, O. Menoukeu-Pamen and F. Proske, "On Local Times: Application to Pricing Using Bid-Ask,"

References

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http://dx.doi.org/10.1016/0022-0531(79)90043-7

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http://dx.doi.org/10.1016/0304-4149(81)90026-0

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http://dx.doi.org/10.1016/0304-4068(81)90010-0

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http://dx.doi.org/10.1016/j.jmateco.2009.05.004

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http://dx.doi.org/10.1006/jeth.1995.1037

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http://dx.doi.org/10.1016/S0304-405X(01)00075-7

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http://dx.doi.org/10.1137/S0363012992232579

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http://dx.doi.org/10.1111/1467-9965.00022

[11] S. Peng, “Non Linear Expectation, Nonlinear Evaluation and Risk Measure, Volume 1856 of Lecture Notes in Mathematics,” Springer, Berlin, Heidelberg, 2004, pp.165-253.

[12] S. Peng, “Modelling Derivatives Pricing Mechanisms with Their Generating Functions,” 2006.

http://arxiv.org/pdf/math/0605599v1.pdf

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http://dx.doi.org/10.1016/j.jbankfin.2005.02.005

[14] E. B. Dynkin, “Markov Processes, Volume 1,” Springer-Verlag, Berlin, Gottigen, Heidelberg, 1965.

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http://dx.doi.org/10.1007/978-1-4757-3699-1

[16] A. D. Banner and R. Ghomrasni, “Local Times of Ranked Continuous Semimartingales,” Stochastic Processes and their Applications, Vol. 118, No. 7, 2008, pp. 1244-1253.

http://dx.doi.org/10.1016/j.spa.2007.08.001

[17] R.Ghomrasni and O. Menoukeu-Pamen, “Decomposition of Order Statistics of Semimartingales Using Local Times,” Stochastic Analysis and Applications, Vol. 28, No. 3, 2010, pp. 467-479.

http://dx.doi.org/10.1080/07362991003708341

[18] R. J. Chitashvili and M. G. Mania, “Decomposition of the Maximum of Semimartingales and Generalized It’s Formula,” In: V. V. Sazonov and T. L. Shervashidze, Eds., New Trends in Probability and Statistics, Volume 1, Proceedings of the Bakuriani Colloquium in Honour of Yu. V. Prohorov, USSR, Bakuriani, 24 February-4 March 1990, pp. 301-350.

[19] P. Protter and K. Shimbo, “No Arbitrage and General Semimartingales,” In: Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz, Volume 4, Institute of Mathematical Statistics, Beachwood, 2008, pp. 267-283.

[20] P. Protter, “Stochastic Integration and Differential Equations,” 2nd Edition, Springer-Verlag, Berlin, 2005.

http://dx.doi.org/10.1007/978-3-662-10061-5

[21] R. Coviello and F. Russo, “Modeling Financial Assets without Semimartingale,” 2006.

http://arxiv.org/pdf/math/0606642.pdf

[22] D. Nualart and E. Pardoux, “Stochastic Calculus with Anticipating Integrands,” Probability Theory and Related Fields, Vol. 78, 1988, pp. 535-581.

http://dx.doi.org/10.1007/BF00353876

[23] F. Russo and P. Vallois, “Forward, Backward and Symmetric Stochastic Integration,” Probability Theory and Related Fields, Vol. 97, No. 3, 1993, pp. 403-421.

http://dx.doi.org/10.1007/BF01195073

[1] J. Harrison and D. Kreps, “Martingales and Arbitrage in Multiperiod Securities Markets,” Journal of Economic Theory, Vol. 20, No. 3, 1979, pp. 381-408.

http://dx.doi.org/10.1016/0022-0531(79)90043-7

[2] J. Harrison and S. Pliska, “Martingales and Stochastic Integrals in the Theory of Continuous Trading,” Stochastic Processes and Their Applications, Vol. 11, No. 3, 1979, pp. 215-260.

http://dx.doi.org/10.1016/0304-4149(81)90026-0

[3] D. M. Kreps, “Arbitrage and Equilibrium in Economics with Infinitely Many Commmodities,” Journal of Mathematical Economics, Vol. 8, No. 1, 1981, pp. 15-35.

http://dx.doi.org/10.1016/0304-4068(81)90010-0

[4] J. Bion-Nadal, “Bid-Ask Dynamic Pricing in Financial Markets with Transaction Costs and Liquidity Risk,” Journal of Mathematical Economics, Vol. 45, No. 11, 2009, pp. 738-750.

http://dx.doi.org/10.1016/j.jmateco.2009.05.004

[5] E. Jouini and H. Kallal, “Martingales and Arbitrage in Security Market with Transaction Coast,” Journal of Economic Theory, Vol. 66, No. 1, 1995, pp. 178-197.

http://dx.doi.org/10.1006/jeth.1995.1037

[6] A. Cherny, “General Arbitrage Pricing Model. II. Transaction Costs,” In C. Donati-Martin, M. Emery, A. Rouault and C. Stricker, Eds., Séminaire de Probabilités XL, Volume 1899 of Lecture Note in Mathematics, Springer, Berlin, Heidelberg, 2007, pp. 447-461.

[7] P. Carr, H. Geman and D. Madan, “Pricing and Hedging in Incomplete Markets,” Journal of Financial Economics, Vol. 62, No. 1, 2001, pp. 131-167.

http://dx.doi.org/10.1016/S0304-405X(01)00075-7

[8] H. Fullmer and A. Schield, “Stochastic Finance: An Introduction in Discrete Time,” 2nd Edition, De Gruyter Studies in Mathematics, Berlin, New York, 2004.

[9] N. El Karoui and M. C. Quenez, “Dynamic Programming and the Pricing of Contingent Claims in Incomplete Markets,” SIAM Journal on Control and Optimization, Vol. 33, No. 1, 1995, pp. 29-66.

http://dx.doi.org/10.1137/S0363012992232579

[10] N. El Karoui, S. Peng and M. C. Quenez, “Backward Stochastic Differential Equations in Finance,” Mathematical Finance, Vol. 7, No. 1, 1997, pp. 1-77.

http://dx.doi.org/10.1111/1467-9965.00022

[11] S. Peng, “Non Linear Expectation, Nonlinear Evaluation and Risk Measure, Volume 1856 of Lecture Notes in Mathematics,” Springer, Berlin, Heidelberg, 2004, pp.165-253.

[12] S. Peng, “Modelling Derivatives Pricing Mechanisms with Their Generating Functions,” 2006.

http://arxiv.org/pdf/math/0605599v1.pdf

[13] R. Jarrow and P. Protter, “Large Traders, Hidden Arbitrage, and Complete Markets,” Journal of Banking & Finance, Vol. 29, No. 11, 2005, pp. 2803-2820.

http://dx.doi.org/10.1016/j.jbankfin.2005.02.005

[14] E. B. Dynkin, “Markov Processes, Volume 1,” Springer-Verlag, Berlin, Gottigen, Heidelberg, 1965.

[15] R. Fernholz, “Stochastic Portfolio Theory,” Springer-Verlag, Berlin, 2002.

http://dx.doi.org/10.1007/978-1-4757-3699-1

[16] A. D. Banner and R. Ghomrasni, “Local Times of Ranked Continuous Semimartingales,” Stochastic Processes and their Applications, Vol. 118, No. 7, 2008, pp. 1244-1253.

http://dx.doi.org/10.1016/j.spa.2007.08.001

[17] R.Ghomrasni and O. Menoukeu-Pamen, “Decomposition of Order Statistics of Semimartingales Using Local Times,” Stochastic Analysis and Applications, Vol. 28, No. 3, 2010, pp. 467-479.

http://dx.doi.org/10.1080/07362991003708341

[18] R. J. Chitashvili and M. G. Mania, “Decomposition of the Maximum of Semimartingales and Generalized It’s Formula,” In: V. V. Sazonov and T. L. Shervashidze, Eds., New Trends in Probability and Statistics, Volume 1, Proceedings of the Bakuriani Colloquium in Honour of Yu. V. Prohorov, USSR, Bakuriani, 24 February-4 March 1990, pp. 301-350.

[19] P. Protter and K. Shimbo, “No Arbitrage and General Semimartingales,” In: Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz, Volume 4, Institute of Mathematical Statistics, Beachwood, 2008, pp. 267-283.

[20] P. Protter, “Stochastic Integration and Differential Equations,” 2nd Edition, Springer-Verlag, Berlin, 2005.

http://dx.doi.org/10.1007/978-3-662-10061-5

[21] R. Coviello and F. Russo, “Modeling Financial Assets without Semimartingale,” 2006.

http://arxiv.org/pdf/math/0606642.pdf

[22] D. Nualart and E. Pardoux, “Stochastic Calculus with Anticipating Integrands,” Probability Theory and Related Fields, Vol. 78, 1988, pp. 535-581.

http://dx.doi.org/10.1007/BF00353876

[23] F. Russo and P. Vallois, “Forward, Backward and Symmetric Stochastic Integration,” Probability Theory and Related Fields, Vol. 97, No. 3, 1993, pp. 403-421.

http://dx.doi.org/10.1007/BF01195073