Skip to main content

Hyperfinite Mathematical Finance

  • Chapter
Nonstandard Analysis

Part of the book series: NATO ASI Series ((ASIC,volume 493))

Abstract

Financial markets have provided one of the most remarkable growth industries in the past two decades, and now constitute a major source of employment for graduates with high levels of mathematical expertise. The principal reason for this phenomenon lies in the explosive growth of the market in derivatives, whose levels of activity now frequently exceed the underlying markets on which their products are based. The variety and complexity of new financial instruments is often bewildering, and much effort goes into the analysis of the mathematical models on which their existence is predicated.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Albeverio, S., Fenstad, J-E., Høegh-Krohn, R., Lindstrøm, T. (1986) Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York.

    MATH  Google Scholar 

  2. Anderson, R.M. (1978) A nonstandard representation for Brownian Motion and Itô Integration, Israel Math. J. 25, pp. 15–46.

    Article  Google Scholar 

  3. Anderson, R.M., Rashid, S. (1978) A nonstandard characterization of weak convergence, Proc. Amer. Math. Soc. 69, pp. 327–332.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bick, A., Willinger, W. (1994) Dynamic spanning without probabilities, Stoch. Proc. Appl. 50, pp. 349–374.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bielecki, T.R. (1994) On integration with respect to fractional Brownian motion, to appear in Statistics and Probability Letters.

    Google Scholar 

  6. Black, F., Scholes, M. (1973) The pricing of options and corporate liabilities, J. Polit. Econom. 81, pp. 637–654.

    Article  Google Scholar 

  7. Cox, J., Ross, S., Rubinstein, M. (1979) Option pricing: a simplified approach, J. Financial Econom. 7, pp. 229–263.

    Article  MATH  Google Scholar 

  8. Cox, J., Rubinstein, M. (1985) Options Markets. Prentice-Hall, Englewood Cliffs, NJ.

    Google Scholar 

  9. Cutland, N.J., Kopp, P.E., Willinger, W. (1991) A nonstandard approach to option pricing, Math. Finance 1(4), pp. 1–38.

    Article  MathSciNet  MATH  Google Scholar 

  10. Cutland, N.J., Kopp, P.E., Willinger, W. (1993) From discrete to continuous financial models: new convergence results for option pricing, Math. Finance 3(2), pp. 101–123.

    Article  MathSciNet  MATH  Google Scholar 

  11. Cutland, N.J., Kopp, P.E., Willinger, W. (1993) A nonstandard treatment of options driven by Poisson processes, Stochastics and Stoch. Reports 42, pp. 115–133.

    MathSciNet  MATH  Google Scholar 

  12. Cutland, N.J., Kopp, P.E., Willinger, W. (1995) From discrete to continuous stochastic calculus, Stochastics and Stoch. Reports 52, pp. 173–192.

    MathSciNet  MATH  Google Scholar 

  13. Cutland, N.J., Kopp, P.E., Willinger, W., Stock price returns and the Joseph effect: a fractional version of the Black-Scholes model, in Progress in Probability 36. Birkhaeuser, Basel.

    Google Scholar 

  14. Cutland, N.J., Kopp, P.E., Willinger, W., Wyman, M.C. Convergence of Snell envelopes and critical prices in the American put, to appear in Mathematics of Derivative Securities,Eds. M.H.A. Dempster and S.R. Pliska. CUP, Cambridge.

    Google Scholar 

  15. Duffie, D. (1988) Security Markets: Stochastic models. Academic Press, Boston.

    MATH  Google Scholar 

  16. Duffie, D. (1992) Dynamic Asset Pricing Theory. Princeton University Press, Princeton, NJ.

    Google Scholar 

  17. Duffie, D., Protter, P. (1989) From discrete to continuous finance: weak convergence of the financial gain process, Technical Report #89/02, Department of Statistics, Purdue University.

    Google Scholar 

  18. Gripenberg, G., Norros, I. (1994) On the prediction of fractional Brownian motion, preprint, University of Helsinki, 11pp.

    Google Scholar 

  19. Harrison, J.M., Pliska, S.R. (1981) Martingales, stochastic integrals and continuous trading, Stoch. Proc. Appl., 11, pp. 215–260.

    Article  MathSciNet  MATH  Google Scholar 

  20. He, H. (1990) Convergence from discrete-to continuous-time contingent claims prices, Rev. Fin. Stud., 3, pp. 523–546.

    Article  Google Scholar 

  21. Kopp, P.E. (1984) Martingales and Stochastic Integrals. CUP, Cambridge.

    Book  MATH  Google Scholar 

  22. Lamberton, D., Lapeyre, B. (1996) Introduction to Stochastic Calculus Applied to Finance. Chapman and Hall, London.

    Google Scholar 

  23. Lin, S. J. (1995) Stochastic analysis of fractional Brownian motion, Stochastics and Stoch. Reports 55, pp. 121–140.

    MATH  Google Scholar 

  24. Lindstrøm, T. (1980), Hyperfinite stochastic integration I-III, Math. Scand. 46, pp. 265–333.

    MathSciNet  MATH  Google Scholar 

  25. Lindstrøm, T. (1996) Internal martingales and stochastic integration, this volume.

    Google Scholar 

  26. Loeb, P.A. (1979) Weak limits of measures and the standard part map, Proc. Amer. Math. Soc. 77, pp. 128–135.

    Article  MathSciNet  MATH  Google Scholar 

  27. Merton, R.C. (1973) Theory of rational option pricing, Bell J. Econ. Man. Sci. 4, pp. 141–183.

    Article  MathSciNet  Google Scholar 

  28. Myneni, R. (1992) The pricing of the American option. Ann. Appl. Prob. 2, pp. 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  29. Ross, D. (1996) Loeb measure and probability, this volume.

    Google Scholar 

  30. Taqqu, M.S., Willinger, W. (1987) The analysis of finite security markets using martingales, Adv. Appl. Prob. 18, pp. 1–25.

    Article  MathSciNet  Google Scholar 

  31. Wellmann, V. (1996) Stochastic models for the term structure of interest rates, MSc thesis, Hull University.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Kopp, P.E. (1997). Hyperfinite Mathematical Finance. In: Arkeryd, L.O., Cutland, N.J., Henson, C.W. (eds) Nonstandard Analysis. NATO ASI Series, vol 493. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5544-1_10

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-5544-1_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6335-7

  • Online ISBN: 978-94-011-5544-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics