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Finance and Stochastics

, Volume 12, Issue 4, pp 441–468 | Cite as

Pricing by hedging and no-arbitrage beyond semimartingales

  • Christian Bender
  • Tommi Sottinen
  • Esko Valkeila
Article

Abstract

We show that pricing a big class of relevant options by hedging and no-arbitrage can be extended beyond semimartingale models. To this end we construct a subclass of self-financing portfolios that contains hedges for these options, but does not contain arbitrage opportunities, even if the stock price process is a non-semimartingale of some special type. Moreover, we show that the option prices depend essentially only on a path property of the stock price process, viz. on the quadratic variation. We end the paper by giving no-arbitrage results even with stopping times for our model class.

Keywords

Arbitrage Pricing Quadratic variation Robust hedging 

JEL Classification

G10 G13 

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References

  1. 1.
    Androshchuk, T., Mishura, Yu.: Mixed Brownian-fractional Brownian model: absence of arbitrage and related topics. Stoch. Int. J. Probab. Stoch. Process. 78, 281–300 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cheridito, P.: Mixed fractional Brownian motion. Bernoulli 7, 913–934 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cheridito, P.: Arbitrage in fractional Brownian motion models. Finance Stoch. 7, 533–553 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dasgupta, A., Kallianpur, G.: Arbitrage opportunities for a class of Gladyshev processes. Appl. Math. Optim. 41, 377–385 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dupire, B.: Pricing with a smile. Risk January, 18–20 (1994) Google Scholar
  7. 7.
    Föllmer, H.: Calcul d’Itô sans probabilités. In: Séminaire de Probabilités, XV. Lecture Notes in Mathematics, vol. 850, pp. 143–150. Springer, Berlin (1981) Google Scholar
  8. 8.
    Guasoni, P.: No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math. Finance 16, 569–582 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Guasoni, P., Rásonyi, M., Schachermayer, W.: Consistent price systems and face-lifting pricing under transaction costs. Ann. Appl. Probab. 18(2), 491–520 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Karlin, S., Taylor, H.M.: A First Course in Stochastic Processes, 2nd edn. Academic Press, San Diego (1975) zbMATHGoogle Scholar
  11. 11.
    Lin, S.J.: Stochastic analysis of fractional Brownian motions. Stoch. Stoch. Rep. 55, 121–140 (1995) zbMATHGoogle Scholar
  12. 12.
    Mishura, Yu., Valkeila, E.: On arbitrage and replication in the fractional Black–Scholes pricing model. Proc. Steklov Inst. Math. 237, 215–224 (2002) MathSciNetGoogle Scholar
  13. 13.
    Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2004) zbMATHGoogle Scholar
  14. 14.
    Russo, F., Vallois, P.: Forward, backward and symmetric stochastic integration. Probab. Theory Relat. Fields 97, 403–421 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Schoenmakers, J., Kloeden, P.: Robust option replication for a Black–Scholes model extended with nondeterministic trends. J. Appl. Math. Stoch. Anal. 12, 113–120 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sondermann, D.: Introduction to Stochastic Calculus for Finance: A New Didactic Approach. Springer, Berlin (2006) zbMATHGoogle Scholar
  17. 17.
    Shiryaev, A.: On arbitrage and replication for fractal models. Research Report, vol. 20. MaPhySto, Department of Mathematical Sciences, University of Aarhus (1998) Google Scholar
  18. 18.
    Shiryaev, A.: Essentials of Stochastic Finance. Facts, Models, Theory. World Scientific, Singapore (1999) Google Scholar
  19. 19.
    Wilmott, P.: Derivatives: The Theory and Practice of Financial Engineering. Wiley, New York (1998) Google Scholar
  20. 20.
    Zähle, M.: Long range dependence, no arbitrage and the Black–Scholes formula. Stoch. Dyn. 2, 265–280 (2002) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Christian Bender
    • 1
  • Tommi Sottinen
    • 2
    • 3
  • Esko Valkeila
    • 4
  1. 1.Institute for Mathematical StochasticsTU BraunschweigBraunschweigGermany
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  3. 3.School of Science and Engineering and School of BusinessReykjavík UniversityReykjavíkIceland
  4. 4.Department of Mathematics and Systems AnalysisHelsinki University of TechnologyHelsinkiFinland

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