Skip to main content
SpringerLink
Log in

The scaling limit of superreplication prices with small transaction costs in the multivariate case

  • Published:
Finance and Stochastics Aims and scope Submit manuscript

Abstract

Kusuoka (Ann. Appl. Probab. 5:198–221, 1995) showed how to obtain non-trivial scaling limits of superreplication prices in discrete-time models of a single risky asset which is traded at properly scaled proportional transaction costs. This article extends the result to a multivariate setup where the investor can trade in several risky assets. The \(G\)-expectation describing the limiting price involves models with a volatility range around the frictionless scaling limit that depends not only on the transaction costs coefficients, but also on the chosen complete discrete-time reference model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Our construction is motivated by the arguments we used to prove (3.2) in the previous section, which indicate which volatilities can be generated by which price systems close to \(S^{(n)}\) as in (3.1); see in particular (3.19) and compare the definition of \(A^{(n)}\) in (3.11).

References

  1. Akahori, J.: A discrete Itô calculus approach to He’s framework for multi-factor discrete markets. Asia-Pac. Financ. Mark. 12, 273–287 (2005)

    Article  MATH  Google Scholar 

  2. Bogachev, V.I.: Measure Theory, vol. II. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  3. Blum, B.: The face-lifting theorem for proportional transaction costs in multiasset models. Stat. Decis. 27, 357–369 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Bouchard, B., Touzi, N.: Explicit solution of the multivariate super-replication problem under transaction costs. Ann. Appl. Probab. 10, 685–708 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Cvitanić, J., Pham, H., Touzi, N.: A closed-form solution to the problem of super-replication under transaction costs. Finance Stoch. 3, 35–54 (1999)

    Article  MATH  Google Scholar 

  6. Cox, J.C., Ross, A.R., Rubinstein, M.: Option pricing: a simplified approach. J. Financ. Econom. 7, 229–263 (1976)

    Article  MATH  Google Scholar 

  7. Dudley, R.M.: Distances of probability measures and random variables. Ann. Math. Stat. 39, 1563–1572 (1968)

    MathSciNet  MATH  Google Scholar 

  8. Davis, M.H.A., Clark, J.M.C.: A note on super-replicating strategies. Trans. R. Soc. Lond., Ser. A 347, 485–494 (1994)

    Article  MATH  Google Scholar 

  9. Dolinsky, Y., Nutz, M., Soner, H.M.: Weak approximations of \(G\)-expectations. Stoch. Process. Appl. 2, 664–675 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Duffie, D., Protter, P.: From discrete to continuous time finance: weak convergence of the financial gain process. Math. Finance 2, 1–15 (1992)

    Article  MATH  Google Scholar 

  11. Grépat, J.: On a multi-asset version of the Kusuoka limit theorem of option replication under transaction costs. Working paper, 2014. Available online at https://sites.google.com/site/juliengrepat/papers

  12. Guasoni, P., Rásonyi, M., Schachermayer, W.: Consistent price systems and face-lifting pricing under transaction costs. Ann. Appl. Probab. 18, 491–520 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. He, H.: Convergence from discrete to continuous time contingent claim prices. Rev. Financ. Stud. 3, 523–546 (1990)

    Article  Google Scholar 

  14. Jouini, E., Kallal, H.: Martingales and arbitrage in securities markets with transaction costs. J. Econ. Theory 66, 178–197 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jakubėnas, P., Levental, S., Ryznar, M.: The super-replication problem via probabilistic methods. Ann. Appl. Probab. 13, 742–773 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kusuoka, S.: Limit theorem on option replication cost with transaction costs. Ann. Appl. Probab. 5, 198–221 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Luenberger, D.G.: Products of trees for investment analysis. J. Econ. Dyn. Control 22, 1403–1417 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Levental, S., Skorohod, A.V.: On the possibility of hedging options in the presence of transaction costs. Ann. Appl. Probab. 7, 410–443 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Peng, S.: \(G\)-expectation, \(G\)-Brownian motion and related stochastic calculus of Itô type. In: Benth, F.E., et al. (eds.) Stochastic Analysis and Applications. Abel Symp., vol. 2, pp. 541–567. Springer, Berlin (2007)

    Chapter  Google Scholar 

  20. Peng, S.: Multi-dimensional \(G\)-Brownian motion and related stochastic calculus under \(G\)-expectation. Stoch. Process. Appl. 12, 2223–2253 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Romagnoli, S., Vargiolu, T.: Robustness of the Black–Scholes approach in the case of options on several assets. Finance Stoch. 4, 325–341 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Soner, H.M., Shreve, S.E., Cvitanić, J.: There is no nontrivial hedging portfolio for option pricing with transaction costs. Ann. Appl. Probab. 5, 327–355 (1995)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the Einstein Foundation for the financial support through its research project on “Game options and markets with frictions”.

Author information

Authors and Affiliations

  1. Dept. of Mathematics, TU Berlin, Berlin, Germany

    Peter Bank & Ari-Pekka Perkkiö

  2. Dept. of Statistics, Hebrew University, Jerusalem, Israel

    Yan Dolinsky

Authors
  1. Peter Bank
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan Dolinsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ari-Pekka Perkkiö
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Bank.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bank, P., Dolinsky, Y. & Perkkiö, AP. The scaling limit of superreplication prices with small transaction costs in the multivariate case. Finance Stoch 21, 487–508 (2017). https://doi.org/10.1007/s00780-016-0320-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00780-016-0320-4

Keywords

Mathematics Subject Classification (2010)

JEL Classification

Search

Navigation