Fractional Processes as Models in Stochastic Finance

Abstract

We survey some new progress on the pricing models driven by fractional Brownian motion or mixed fractional Brownian motion. In particular, we give results on arbitrage opportunities, hedging, and option pricing in these models. We summarize some recent results on fractional Black & Scholes pricing model with transaction costs. We end the paper by giving some approximation results and indicating some open problems related to the paper.

Keywords

Fractional Brownian motion Arbitrage Hedging in fractional models Approximation of geometric fractional Brownian motion 

Mathematics Subject Classification (2010)

91Gxx 91B70 60G15 60H05 

Notes

Acknowledgements

T.S. and E.V. acknowledge the support from Saarland University, and E.V. is grateful to the Academy of Finland, grant 127634, for partial support. We are grateful to Peter Parczewski and an anonymous referee for useful comments.

References

  1. 1.
    T. Androshchuk, Yu. Mishura, Mixed Brownian–fractional Brownian model: absence of arbitrage and related topics. Stochastics 78(5), 281–300 (2006) MATHMathSciNetGoogle Scholar
  2. 2.
    M.A. Arcones, On the law of the iterated logarithm for Gaussian processes. J. Theor. Probab. 8(4), 877–903 (1995) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    E. Azmoodeh, Geometric fractional Brownian motion market model with transaction costs. Preprint (2009), 12 p. Google Scholar
  4. 4.
    E. Azmoodeh, Yu. Mishura, E. Valkeila, On hedging European options in geometric fractional Brownian motion market model. Stat. Decis. 27(2), 129–144 (2009). doi: 10.1524/stnd.2009.1021 CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    E. Bayraktar, U. Horst, R. Sircar, A limit theorem for financial markets with inert investors. Math. Oper. Res. 31(4), 789–810 (2006) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    C. Bender, An S-transform approach to integration with respect to a fractional Brownian motion. Bernoulli 9(6), 955–983 (2003) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    C. Bender, R.J. Elliott, Arbitrage in a discrete version of the Wick-fractional Black–Scholes market. Math. Oper. Res. 29(4), 935–945 (2004) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    C. Bender, P. Parczewski, Approximating a geometric fractional Brownian motion and related processes via discrete Wick calculus. Bernoulli 16(2), 389–417 (2010). doi: 10.3150/09-BEJ223 CrossRefMathSciNetGoogle Scholar
  9. 9.
    C. Bender, T. Sottinen, E. Valkeila, Pricing by hedging and no-arbitrage beyond semimartingales. Finance Stoch. 12(4), 441–468 (2008) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    F. Biagini, Y. Hu, B. Øksendal, T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications. Probability and Its Applications (New York) (Springer, London, 2008) CrossRefMATHGoogle Scholar
  11. 11.
    T. Björk, H. Hult, A note on Wick products and the fractional Black–Scholes model. Finance Stoch. 9(2), 197–209 (2005) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    M. Bratyk, Y. Mishura, The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions. Theory Stoch. Process. 14(3–4), 27–38 (2008) MATHMathSciNetGoogle Scholar
  13. 13.
    P. Cheridito, Mixed fractional Brownian motion. Bernoulli 7(6), 913–934 (2001) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    P. Cheridito, Arbitrage in fractional Brownian motion models. Finance Stoch. 7(4), 533–553 (2003) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    A. Cherny, Brownian moving averages have conditional full support. Ann. Appl. Probab. 18(5), 1825–1830 (2008) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    A. Dasgupta, G. Kallianpur, Arbitrage opportunities for a class of Gladyshev processes. Appl. Math. Optim. 41(3), 377–385 (2000) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    F. Delbaen, W. Schachermayer, A general version of the fundamental theorem of asset pricing. Math. Ann. 300(3), 463–520 (1994) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    F. Delbaen, W. Schachermayer, The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312(2), 215–250 (1998) CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    T.E. Duncan, Y. Hu, B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion. I: Theory. SIAM J. Control Optim. 38(2), 582–612 (2000) CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    H. Föllmer, Calcul d’Itô sans probabilités. Séminaire de Probabilités, vol. XV (Springer, Berlin, 1981), pp. 143–150 Google Scholar
  21. 21.
    R. Gaigalas, I. Kaj, Convergence of scaled renewal processes and a packet arrival model. Bernoulli 9(4), 671–703 (2003) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    D. Gasbarra, T. Sottinen, H. van Zanten, Conditional full support of Gaussian processes with stationary increments. Preprint (2008) Google Scholar
  23. 23.
    P. Guasoni, No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math. Finance 16(3), 569–582 (2006) CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    P. Guasoni, M. Rásonyi, W. Schachermayer, Consistent price systems and face-lifting pricing under transaction costs. Ann. Appl. Probab. 18(2), 491–520 (2008) CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    P. Guasoni, M. Rásonyi, W. Schachermayer, The fundamental theorem of asset pricing for continuous processes under small transaction costs. Ann. Finance 6(2), 157–191 (2010) CrossRefGoogle Scholar
  26. 26.
    H. Holden, T. Lindstrøm, B. Øksendal, J. Ubøe, Discrete Wick calculus and stochastic functional equations. Potential Anal. 1(3), 291–306 (1992) CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    R. Jarrow, P. Protter, H. Sayit, No arbitrage without semimartingales. Ann. Appl. Probab. 19(2), 596–616 (2009) CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    E. Jouini, H. Kallal, Martingales and arbitrage in securities markets with transaction costs. J. Econ. Theory 66, 178–197 (1995) CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    S. Karlin, H.M. Taylor, A First Course in Stochastic Processes, 2nd edn. (Academic Press, San Diego, 1975) MATHGoogle Scholar
  30. 30.
    C. Klüppelberg, C. Kühn, Fractional Brownian motion as a weak limit of Poisson shot noise processes—with applications to finance. Stoch. Process. Appl. 113(2), 333–351 (2004) CrossRefMATHGoogle Scholar
  31. 31.
    Yu.S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics, vol. 1929 (Springer, Berlin, 2008) CrossRefMATHGoogle Scholar
  32. 32.
    Yu.S. Mishura, S.G. Rode, Weak convergence of integral functionals of random walks that weakly converge to fractional Brownian motion (Ukrainian). Ukr. Mat. Zh. 59(8), 1040–1046 (2007). Translation in Ukr. Math. J. 59(8), 1155–1162 (2007) CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    A. Nieminen, Fractional Brownian motion and martingale-differences. Stat. Probab. Lett. 70(1), 1–10 (2004) CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    M.S. Pakkanen, Stochastic integrals and conditional full support. Preprint (2008, Revised Jul. 24, 2009). Available as arXiv:0811.1847
  35. 35.
    P. Protter, Stochastic Integration and Differential Equations, 2nd edn. Stochastic Modelling and Applied Probability, vol. 21 (Springer, Berlin, 2004) MATHGoogle Scholar
  36. 36.
    L.C.G. Rogers, Arbitrage with fractional Brownian motion. Math. Finance 7(1), 95–105 (1997) CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    F. Russo, P. Vallois, P. Forward, backward and symmetric stochastic integration. Probab. Theory Relat. Fields 97(3), 403–421 (1993) CrossRefMATHGoogle Scholar
  38. 38.
    J.G.M. Schoenmakers, P.E. Kloeden, Robust option replication for a Black–Scholes model extended with nondeterministic trends. J. Appl. Math. Stoch. Anal. 12(2), 113–120 (1999) CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    A. Shiryaev, On arbitrage and replication for fractal models. Research Report 30 (1998), MaPhySto, Department of Mathematical Sciences, University of Aarhus Google Scholar
  40. 40.
    T. Sottinen, Fractional Brownian motion, random walks and binary market models. Finance Stoch. 5(3), 343–355 (2001) CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    T. Sottinen, E. Valkeila, On arbitrage and replication in the fractional Black–Scholes pricing model. Stat. Decis. 21(2), 93–107 (2003) CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    E. Valkeila, On the approximation of geometric fractional Brownian motion, in Optimality and Risk—Modern Trends in Mathematical Finance the Kabanov Festschrift, ed. by F. Delbaen, M. Rásonyi, C. Stricker (2009), pp. 251–265 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Christian Bender
    • 1
  • Tommi Sottinen
    • 2
  • Esko Valkeila
    • 3
  1. 1.Department of MathematicsSaarland UniversitySaarbrückenGermany
  2. 2.Department of Mathematics and StatisticsUniversity of VaasaVaasaFinland
  3. 3.Department of Mathematics and Systems AnalysisAalto UniversityAaltoFinland

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