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No arbitrage and multiplicative special semimartingales

Part of: Mathematical economics Stochastic processes

Published online by Cambridge University Press:  09 June 2023

Eckhard Platen  and
Stefan Tappe Open the ORCID record for Stefan Tappe [Opens in a new window]
Eckhard Platen*
Affiliation:
University of Technology Sydney
Stefan Tappe*
Affiliation:
Albert Ludwig University of Freiburg
*
*Postal address: University of Technology Sydney, School of Mathematical and Physical Sciences, Finance Discipline Group, PO Box 123, Broadway, NSW 2007, Australia. Email address: eckhard.platen@uts.edu.au
**Postal address: Albert Ludwig University of Freiburg, Department of Mathematical Stochastics, Ernst-Zermelo-Straße 1, D-79104 Freiburg, Germany. Email address: stefan.tappe@math.uni-freiburg.de
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Abstract

Consider a financial market with nonnegative semimartingales which does not need to have a numéraire. We are interested in the absence of arbitrage in the sense that no self-financing portfolio gives rise to arbitrage opportunities, where we are allowed to add a savings account to the market. We will prove that in this sense the market is free of arbitrage if and only if there exists an equivalent local martingale deflator which is a multiplicative special semimartingale. In this case, the additional savings account relates to the finite-variation part of the multiplicative decomposition of the deflator.

Keywords

Fundamental theorem of asset pricingmarket without a numéraireself-financing portfoliono-arbitrage conceptmarket price of riskshort rate

MSC classification

Primary: 91B02: Fundamental topics (basic mathematics, methodology; applicable to economics in general)
Secondary: 91B70: Stochastic models 60G48: Generalizations of martingales
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 3 , September 2023 , pp. 1033 - 1074
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Acciaio, B., Fontana, C. and Kardaras, C. (2016). Arbitrage of the first kind and filtration enlargements in semimartingale financial models. Stoch. Process. Appl. 126, 1761–1784.10.1016/j.spa.2015.12.004CrossRefGoogle Scholar
Aksamit, A., Choulli, T., Deng, J. and Jeanblanc. M. (2018). No-arbitrage under a class of honest times. Finance Stoch. 22, 127–159.10.1007/s00780-017-0345-3CrossRefGoogle Scholar
Aksamit, A., Choulli, T., Deng, J. and Jeanblanc. M. (2019). No-arbitrage under additional information for thin semimartingale models. Stoch. Process. Appl. 129, 3080–3115.10.1016/j.spa.2018.09.005CrossRefGoogle Scholar
Aksamit, A. and Jeanblanc, M. (2017). Enlargement of Filtration with Finance in View. Springer, Cham.10.1007/978-3-319-41255-9CrossRefGoogle Scholar
Ansel, J. P. and Stricker, C. (1994). Couverture des actifs contingents et prix maximum. Ann. Inst. H. Poincaré Prob. Statist. 30, 303–315.Google Scholar
Baldeaux, J., Grasselli, M. and Platen, E. (2015). Pricing currency derivatives under the benchmark approach. J. Bank. Finance 53, 34–48.10.1016/j.jbankfin.2014.11.018CrossRefGoogle Scholar
Baldeaux, J., Ignatieva, K. and Platen, E. (2018). Detecting money market bubbles. J. Bank. Finance 87, 369–379.10.1016/j.jbankfin.2017.10.017CrossRefGoogle Scholar
Bálint, D. Á. and Schweizer, M. (2020). Large financial markets, discounting, and no asymptotic arbitrage. Theory Prob. Appl. 65, 191–223.10.1137/S0040585X97T98991XCrossRefGoogle Scholar
Bálint, D. Á. and Schweizer, M. (2020). Properly discounted asset prices are semimartingales. Math. Financial Econom. 14, 661–674.10.1007/s11579-020-00269-8CrossRefGoogle Scholar
Bálint, D. Á. and Schweizer, M. (2022). Making no-arbitrage discounting-invariant: a new FTAP version beyond NFLVR and NUPBR. Frontiers Math. Finance 1, 249–286.10.3934/fmf.2021010CrossRefGoogle Scholar
Choulli, T. and Stricker, C. (1996). Deux applications de la décomposition de Galtchouck–Kunita–Watanabe. In Séminaire de Probabilités XXX, ed. J. Azéma, Springer, Berlin, pp. 12–23.10.1007/BFb0094638CrossRefGoogle Scholar
Christensen, M. M. and Larsen, K. (2007). No arbitrage and the growth optimal portfolio. Stoch. Anal. Appl. 25, 255–280.10.1080/07362990600870488CrossRefGoogle Scholar
Christopeit, N. and Musiela, M. (1994). On the existence and characterization of arbitrage-free measure in contingent claim valuation. Stoch. Anal. Appl. 12, 41–63.10.1080/07362999408809337CrossRefGoogle Scholar
Criens, D. (2018). Deterministic criteria for the absence and existence of arbitrage in multi-dimensional diffusion markets. Internat. J. Theoret. Appl. Finance 21, 1850002.10.1142/S0219024918500024CrossRefGoogle Scholar
Criens, D. (2020). No arbitrage in continuous financial markets. Math. Financial Econom. 14, 461–506.10.1007/s11579-020-00262-1CrossRefGoogle Scholar
Cuchiero, C. and Teichmann, J. (2015). A convergence result for the Emery topology and a variant of the proof of the fundamental theorem of asset pricing. Finance Stoch. 19, 743–761.10.1007/s00780-015-0276-9CrossRefGoogle Scholar
Dalang, R. C., Morton, A. and Willinger, W. (1990). Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch. Stoch. Reports 29, 185–201.10.1080/17442509008833613CrossRefGoogle Scholar
Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520.10.1007/BF01450498CrossRefGoogle Scholar
Delbaen, F. and Schachermayer, W. (1995). The no-arbitrage property under a change of numéraire. Stoch. Stoch. Reports 53, 213–226.Google Scholar
Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250.10.1007/s002080050220CrossRefGoogle Scholar
Delbaen, F. and Schachermayer, W. (2008). The Mathematics of Arbitrage. Springer, Berlin.Google Scholar
Döberlein, F. and Schweizer, M. (2001). On savings accounts in semimartingale term structure models. Stoch. Anal. Appl. 19, 605–626.10.1081/SAP-100002104CrossRefGoogle Scholar
Döberlein, F., Schweizer, M. and Stricker, C. (2000). Implied savings accounts are unique. Finance Stoch. 4, 431–442.10.1007/PL00013524CrossRefGoogle Scholar
Du, K. and Platen, E. (2016). Benchmarked risk minimization. Math. Finance 26, 617–637.10.1111/mafi.12065CrossRefGoogle Scholar
Föllmer, H. and Schied, A. (2016). Stochastic Finance, 4th edn. De Gruyter, Berlin.10.1515/9783110463453CrossRefGoogle Scholar
Fontana, C. (2014). No-arbitrage conditions and absolutely continuous changes of measure. In Arbitrage, Credit and Informational Risks, eds C. Hillairet, M. Jeanblanc and Y. Jiao, World Scientific, Hackensack, NJ, pp. 3–18.10.1142/9789814602075_0001CrossRefGoogle Scholar
Fontana, C. (2015). Weak and strong no-arbitrage conditions for continuous financial markets. Internat. J. Theoret. Appl. Finance 18, article no. 1550005.10.1142/S0219024915500053CrossRefGoogle Scholar
Fontana, C., Jeanblanc, M. and Song, S. (2014). On arbitrages arising with honest times. Finance Stoch. 18, 515–543.10.1007/s00780-014-0231-1CrossRefGoogle Scholar
Fontana, C. and Runggaldier, W. J. (2013). Diffusion-based models for financial markets without martingale measures. In Risk Measures and Attitudes, eds F. Biagini, A. Richter and H. Schlesinger, Springer, London, pp. 45–81.10.1007/978-1-4471-4926-2_4CrossRefGoogle Scholar
Harms, P., Liu, C. and Neufeld, A. (2021). Supermartingale deflators in the absence of a numéraire. Math. Financial Econom. 15, 885–915.10.1007/s11579-021-00299-wCrossRefGoogle Scholar
Harrison, J. M. and Kreps, D. M. (1979). Martingales and arbitrage in multiperiod securities markets. J. Econom. Theory 20, 381–408.10.1016/0022-0531(79)90043-7CrossRefGoogle Scholar
Harrison, J. M. and Pliska, S. R. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11, 215–260.10.1016/0304-4149(81)90026-0CrossRefGoogle Scholar
He, S., Wang, J. and Yan, J. (1992). Semimartingale Theory and Stochastic Calculus. Science Press, Beijing.Google Scholar
Herdegen, M. (2017). No-arbitrage in a numéraire-independent modeling framework. Math. Finance 27, 568–603.10.1111/mafi.12088CrossRefGoogle Scholar
Herdegen, M. and Schweizer, M. (2018). Semi-efficient valuations and put-call parity. Math. Finance 28, 1061–1106.10.1111/mafi.12162CrossRefGoogle Scholar
Hulley, H. and Schweizer, M. (2010). M $^6$ —On minimal market models and minimal martingale measures. In Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, eds C. Chiarella and A. Novikov, Springer, Heidelberg, pp. 35–51.10.1007/978-3-642-03479-4_3CrossRefGoogle Scholar
Imkeller, P. and Perkowski, N. (2015). The existence of dominating local martingale measures. Finance Stoch. 19, 685–717.10.1007/s00780-015-0264-0CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (1998). Local martingales and the fundamental asset pricing theorems in the discrete-time case. Finance Stoch. 2, 259–273.10.1007/s007800050040CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.10.1007/978-3-662-05265-5CrossRefGoogle Scholar
Jarrow, R. A. (2018). Continuous-time Asset Pricing Theory. Springer, Cham.10.1007/978-3-319-77821-1CrossRefGoogle Scholar
Kabanov, Y. M. (1997). On the FTAP of Kreps–Delbaen–Schachermayer. In Statistics and Control of Stochastic Processes: The Liptser Festschrift, Steklov Mathematical Institute, pp. 191–203.Google Scholar
Kabanov, Y. M., Kardaras, C. and Song, S. (2016). No arbitrage of the first kind and local martingale numéraires. Finance Stoch. 20, 1097–1108.10.1007/s00780-016-0310-6CrossRefGoogle Scholar
Kabanov, Y. M. and Stricker, C. (2002). A teacher’s note on no-arbitrage criteria. In Séminaire de Probabilités XXXV, ed. J. Azéma, Springer, Berlin, pp. 149–152.Google Scholar
Karatzas, I. and Kardaras, C. (2007). The numéraire portfolio in semimartingale financial models. Finance Stoch. 11, 447–493.10.1007/s00780-007-0047-3CrossRefGoogle Scholar
Karatzas, I. and Kardaras, C. (2021). Portfolio Theory and Arbitrage: A Course in Mathematical Finance. American Mathematical Society, Providence, RI.10.1090/gsm/214CrossRefGoogle Scholar
Kardaras, C. (2010). Finitely additive probabilities and the fundamental theorem of asset pricing. In Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, eds C. Chiarella and A. Novikov, Springer, Heidelberg, pp. 19–34.10.1007/978-3-642-03479-4_2CrossRefGoogle Scholar
Kardaras, C. (2012). Market viability via absence of arbitrage of the first kind. Finance Stoch. 16, 651–667.10.1007/s00780-012-0172-5CrossRefGoogle Scholar
Kardaras, C. and Platen, E. (2011). On the semimartingale property of discounted asset-price processes. Stoch. Process. Appl. 121, 2678–2691.10.1016/j.spa.2011.06.010CrossRefGoogle Scholar
Loewenstein, M. and Willard, G. A. (2000). Local martingales, arbitrage, and viability: free snacks and cheap thrills. Econom. Theory 16, 135–161.10.1007/s001990050330CrossRefGoogle Scholar
Mancin, J. and Runggaldier, W. J. (2014). On the existence of martingale measures in jump diffusion market models. In Arbitrage, Credit and Informational Risks, eds C. Hillairet, M. Jeanblanc and Y. Jiao, World Scientific, Hackensack, NJ, pp. 29–51.10.1142/9789814602075_0003CrossRefGoogle Scholar
Melnikov, A. V. and Shiryaev, A. N. (1996). Criteria for the absence of arbitrage in the financial market. In Frontiers in Pure and Applied Probability, eds A. N. Shiryaev et al., TVP Science Publishers, Moscow, pp. 121–134.Google Scholar
Platen, E. (2002). Arbitrage in continuous complete markets. Adv. Appl. Prob. 34, 540–558.10.1239/aap/1033662165CrossRefGoogle Scholar
Platen, E. and Heath, D. (2010). A Benchmark Approach to Quantitative Finance. Springer, Berlin.Google Scholar
Platen, E. and Rendek, R. (2020). Approximating the growth optimal portfolio and stock price bubbles. Internat. J. Theoret. Appl. Finance 23, article no. 2050048.10.1142/S021902492050048XCrossRefGoogle Scholar
Platen, E. and Tappe, S. (2021). No-arbitrage concepts in topological vector lattices. Positivity 25, 1853–1898.10.1007/s11117-021-00848-zCrossRefGoogle Scholar
Protter, P. and Shimbo, K. (2008). No arbitrage and general semimartingales. In Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz, eds S. N. Ethier, J. Feng and R. H. Stockbridge, Institute of Mathematical Statistics, Beachwood, OH, pp. 267–283.10.1214/074921708000000426CrossRefGoogle Scholar
Schachermayer, W. (1992). A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insurance Math. Econom. 11, 249–257.10.1016/0167-6687(92)90013-2CrossRefGoogle Scholar
Schachermayer, W. (1994). Martingale measures for discrete time processes with infinite horizon. Math. Finance 4, 25–56.10.1111/j.1467-9965.1994.tb00048.xCrossRefGoogle Scholar
Schweizer, M. (1991). Option hedging for semimartingales. Stoch. Process. Appl. 37, 339–363.10.1016/0304-4149(91)90053-FCrossRefGoogle Scholar
Shiryaev, A. N. and Cherny, A. S. (2002). Vector stochastic integrals and the fundamental theorem of asset pricing. Proc. Steklov Inst. Math. 237, 12–56.Google Scholar
Song, S. (2013). An alternative proof of a result of Takaoka. Preprint. Available at https://arxiv.org/ abs/1306.1062v1.Google Scholar
Stricker, C. (1990). Arbitrage et lois de martingale. Ann. Inst. H. Poincaré Prob. Statist. 26, 451–460.Google Scholar
Sun, J., Zhu, D. and Platen, E. (2021). Dynamic asset allocation for target date funds under the benchmark approach. ASTIN Bulletin 51, 449–474.10.1017/asb.2021.6CrossRefGoogle Scholar
Takaoka, K. and Schweizer, M. (2014). A note on the condition of no unbounded profit with bounded risk. Finance Stoch. 18, 393–405.10.1007/s00780-014-0229-8CrossRefGoogle Scholar
Tehranchi, M. (2015). Arbitrage theory without a numéraire. Preprint. Available at https://arxiv.org/ abs/1410.2976v2.Google Scholar