Skip to main content
SpringerLink
Log in

Optimal Algorithms for k-Search with Application in Option Pricing

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

In the k-search problem, a player is searching for the k highest (respectively, lowest) prices in a sequence, which is revealed to her sequentially. At each quotation, the player has to decide immediately whether to accept the price or not. Using the competitive ratio as a performance measure, we give optimal deterministic and randomized algorithms for both the maximization and minimization problems, and discover that the problems behave substantially different in the worst-case. As an application of our results, we use these algorithms to price “lookback options”, a particular class of financial derivatives. We derive bounds for the price of these securities under a no-arbitrage assumption, and compare this to classical option pricing.

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

References

  1. Ajtai, M., Megiddo, N., Waarts, O.: Improved algorithms and analysis for secretary problems and generalizations. SIAM J. Discrete Math. 14(1), 1–27 (2001)

    Article  MathSciNet  Google Scholar 

  2. Black, F., Scholes, M.S.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–54 (1973)

    Article  Google Scholar 

  3. Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, New York (1998)

    MATH  Google Scholar 

  4. Cont, R.: Empirical properties of asset returns: stylized facts and statistical issues. Quant. Finance 1, 223–236 (2001)

    Article  Google Scholar 

  5. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. CRC Press, Boca Raton (2004)

    MATH  Google Scholar 

  6. DeMarzo, P., Kremer, I., Mansour, Y.: Online trading algorithms and robust option pricing. In: Proceedings of the ACM Symposium on Theory of Computing, STOC, pp. 477–486 (2006)

  7. El-Yaniv, R., Fiat, A., Karp, R.M., Turpin, G.: Optimal search and one-way trading online algorithms. Algorithmica 30(1), 101–139 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Epstein, D., Wilmott, P.: A new model for interest rates. Int. J. Theor. Appl. Finance 1(2), 195–226 (1998)

    Article  MATH  Google Scholar 

  9. Goldman, M.B., Sosin, H.B., Gatto, M.A.: Path dependent options: “buy at the low, sell at the high”. J. Finance 34(5), 1111–1127 (1979)

    Article  Google Scholar 

  10. Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327–343 (1993)

    Article  Google Scholar 

  11. Hull, J.C.: Options, Futures, and Other Derivatives. Prentice Hall, New York (2002)

    Google Scholar 

  12. Kleinberg, R.D.: A multiple-choice secretary algorithm with applications to online auctions. In: SODA, pp. 630–631. SIAM (2005)

  13. Korn, R.: Worst-case scenario investment for insurers. Insur. Math. Econ. 36(1), 1–11 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lippmann, S.A., McCall, J.J.: The economics of job search: a survey. Econ. Inq. XIV, 155–189 (1976)

    Article  Google Scholar 

  15. Lippmann, S.A., McCall, J.J.: The economics of uncertainty: selected topics and probabilistic methods. In: Handbook of Mathematical Economics, vol. 1, pp. 211–284 (1981)

  16. Merton, R.C.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3(1–2), 125–144 (1976)

    Article  MATH  Google Scholar 

  17. Rosenfield, D.B., Shapiro, R.D.: Optimal adaptive price search. J. Econ. Theory 25(1), 1–20 (1981)

    Article  MATH  Google Scholar 

  18. Shreve, S.E.: Continuous-time models. In: Stochastic Calculus for Finance. II. Springer Finance. Springer, New York (2004)

    Google Scholar 

  19. Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Commun. ACM 28(2), 202–208 (1985)

    Article  MathSciNet  Google Scholar 

  20. Yao, A.C.C.: Probabilistic computations: toward a unified measure of complexity. In: 18th Symposium on Foundations of Comput. Sci., pp. 222–227 (1977)

Download references

Author information

Authors and Affiliations

  1. Institute of Theoretical Computer Science, ETH Zurich, 8092, Zurich, Switzerland

    Julian Lorenz, Konstantinos Panagiotou & Angelika Steger

Authors
  1. Julian Lorenz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Konstantinos Panagiotou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Angelika Steger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Julian Lorenz.

Additional information

J. Lorenz is partially supported by UBS AG.

K. Panagiotou is partially supported by the SNF, grant number: 200021-107880/1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lorenz, J., Panagiotou, K. & Steger, A. Optimal Algorithms for k-Search with Application in Option Pricing. Algorithmica 55, 311–328 (2009). https://doi.org/10.1007/s00453-008-9217-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-008-9217-8

Keywords

Search

Navigation