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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Albeverio, S., Fenstad, J-E., Høegh-Krohn, R., Lindstrøm, T. (1986) Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York.
Anderson, R.M. (1978) A nonstandard representation for Brownian Motion and Itô Integration, Israel Math. J. 25, pp. 15–46.
Anderson, R.M., Rashid, S. (1978) A nonstandard characterization of weak convergence, Proc. Amer. Math. Soc. 69, pp. 327–332.
Bick, A., Willinger, W. (1994) Dynamic spanning without probabilities, Stoch. Proc. Appl. 50, pp. 349–374.
Bielecki, T.R. (1994) On integration with respect to fractional Brownian motion, to appear in Statistics and Probability Letters.
Black, F., Scholes, M. (1973) The pricing of options and corporate liabilities, J. Polit. Econom. 81, pp. 637–654.
Cox, J., Ross, S., Rubinstein, M. (1979) Option pricing: a simplified approach, J. Financial Econom. 7, pp. 229–263.
Cox, J., Rubinstein, M. (1985) Options Markets. Prentice-Hall, Englewood Cliffs, NJ.
Cutland, N.J., Kopp, P.E., Willinger, W. (1991) A nonstandard approach to option pricing, Math. Finance 1(4), pp. 1–38.
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.
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.
Cutland, N.J., Kopp, P.E., Willinger, W. (1995) From discrete to continuous stochastic calculus, Stochastics and Stoch. Reports 52, pp. 173–192.
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.
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.
Duffie, D. (1988) Security Markets: Stochastic models. Academic Press, Boston.
Duffie, D. (1992) Dynamic Asset Pricing Theory. Princeton University Press, Princeton, NJ.
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.
Gripenberg, G., Norros, I. (1994) On the prediction of fractional Brownian motion, preprint, University of Helsinki, 11pp.
Harrison, J.M., Pliska, S.R. (1981) Martingales, stochastic integrals and continuous trading, Stoch. Proc. Appl., 11, pp. 215–260.
He, H. (1990) Convergence from discrete-to continuous-time contingent claims prices, Rev. Fin. Stud., 3, pp. 523–546.
Kopp, P.E. (1984) Martingales and Stochastic Integrals. CUP, Cambridge.
Lamberton, D., Lapeyre, B. (1996) Introduction to Stochastic Calculus Applied to Finance. Chapman and Hall, London.
Lin, S. J. (1995) Stochastic analysis of fractional Brownian motion, Stochastics and Stoch. Reports 55, pp. 121–140.
Lindstrøm, T. (1980), Hyperfinite stochastic integration I-III, Math. Scand. 46, pp. 265–333.
Lindstrøm, T. (1996) Internal martingales and stochastic integration, this volume.
Loeb, P.A. (1979) Weak limits of measures and the standard part map, Proc. Amer. Math. Soc. 77, pp. 128–135.
Merton, R.C. (1973) Theory of rational option pricing, Bell J. Econ. Man. Sci. 4, pp. 141–183.
Myneni, R. (1992) The pricing of the American option. Ann. Appl. Prob. 2, pp. 1–23.
Ross, D. (1996) Loeb measure and probability, this volume.
Taqqu, M.S., Willinger, W. (1987) The analysis of finite security markets using martingales, Adv. Appl. Prob. 18, pp. 1–25.
Wellmann, V. (1996) Stochastic models for the term structure of interest rates, MSc thesis, Hull University.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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