Skip to main content
SpringerLink
Log in

The Stochastically Subordinated Poisson Normal Process for Modelling Financial Assets

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

The method of stochastic subordination, or random time indexing, has been recently applied to Wiener process price processes to model financial returns. Previous emphasis in stochastic subordination models has involved explicitly identifying the subordinating process with an observable quantity such as number of trades. In contrast, the approach taken here does not depend on the specific identification of the subordinated time variable, but rather assumes a class of time models and estimates parameters from data. In addition, a simple Markov process is proposed for the characteristic parameter of the subordinating distribution to explain the significant autocorrelation of the squared returns. It is shown, in particular, that the proposed model, while containing only a few more parameters than the commonly used Wiener process models, fits selected financial time series particularly well, characterising the autocorrelation structure and heavy tails, as well as preserving the desirable self-similarity structure, and the existence of risk-neutral measures necessary for objective derivative valuation. Also, it will be shown that the model proposed fits financial times series data better than the popular generalised autoregressive conditional heteroscedasticity (GARCH) models. Additionally, this paper will develop a skew model by replacing the normal variates with Lévy stable variates.

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.

Similar content being viewed by others

References

  1. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions(Dover, New York, 1972).

    Google Scholar 

  2. V. Akgiray and G.G. Booth, The stable-law model of stock returns, Journal of Business & Economic Statistics 6(1) (1988) 51–57.

    Google Scholar 

  3. E. Bachelier, Théorie de la spe´culation, Annales Scientifiques se L'École Normale Supérieure 17 (1900) 21–86. Also reproduced in Cootner [11].

    Google Scholar 

  4. G. Bakshi, C. Cao and Z. Chen, Empirical performance of alternative option pricing models, Journal of Finance 52(5) (1997) 2003–2049.

    Google Scholar 

  5. D.S. Bates, Testing option pricing models, in: Handbook of Statistics, Vol. 14, Statistical Methods in Finance, eds. G.S. Maddala and C.R. Rao (North-Holland, New York, 1996) pp. 567–611.

    Google Scholar 

  6. F. Black and M. Scholes, The pricing of options on corporate liabilities, Journal of Political Economy 81(3) (1973) 637–654.

    Google Scholar 

  7. R.C. Blattberg and N.J. Gonedes, A comparison of the stable and student distributions as statistical models for stock prices, Journal of Business 47(2) (1974) 244–280.

    Google Scholar 

  8. T. Bollerslev, Generalized autoregressive conditional heteroscedasticity, Journal of Econometrics 31(3) (1986) 307–327.

    Google Scholar 

  9. J.M. Chambers, C.L. Mallows and B.W. Stuck, A method for simulating stable random variables, Journal of the American Statistical Association 71 (1976) (354), Theory and Methods Section, 340– 344. Corrections in 82(398) (1987) 704 and 83(402) (1988) 581.

    Google Scholar 

  10. P.K. Clark, A subordinated stochastic process model with finite variance for speculative prices, Econometrica 41(1) (1973) 135–155.

    Google Scholar 

  11. P. Cootner, ed., The Random Character of Stock Market Prices(MIT Press, Cambridge, MA, 1964).

    Google Scholar 

  12. J.C. Cox, J.E. Ingersoll and S.A. Ross, An intertemporal general equilibrium model of asset prices, Econometrica 53(2) (1985) 363–384.

    Google Scholar 

  13. M.M. Dacorogna, U.A. Müller, R.J. Nagler, R.B. Olsen and O.V. Pictet, A geographical model for the daily and weekly seasonal volatility in the FX market, Journal of International Money and Finance 12(4) (1993) 413–438.

    Google Scholar 

  14. J.C. Duan, The GARCH option pricing model, Mathematical Finance 5(1) (1995) 13–32.

    Google Scholar 

  15. J.C. Duan, Cracking the smile, Risk 9(12) (1996) 55–59.

    Google Scholar 

  16. W.H. DuMouchel, On the asymptotic normality of the maximum likelihood estimate when sampling from a stable distribution, The Annals of Statistics 1(5) (1973) 948–957.

    Google Scholar 

  17. E. Eberlein E. and U. Keller, Hyperbolic distributions in finance, Bernoulli 1(3) (1995) 281–299.

    Google Scholar 

  18. D. Edelman, A note: natural generalization of Black–Scholes in the presence of skewness using stable processes, ABACUS 31(1) (1995) 113–119.

    Google Scholar 

  19. R.F. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation, Econometrica 50(4) (1982) 987–1008.

    Google Scholar 

  20. R.F. Engle and J. Mezrich, Grappling with GARCH, Risk 8(9) (1995) 112–117.

    Google Scholar 

  21. E. Fama, The behaviour of stock market prices, Journal of Business 38 (1965) 34–105.

    Google Scholar 

  22. B.D. Fielitz and J.P. Rozelle, Stable distributions and the mixtures of distributions hypotheses for common stock returns, Journal of the American Statistical Association 78(381) (1983) 28–36.

    Google Scholar 

  23. H. Geman and T. Ané, Stochastic subordination, Risk 9(9) (1996) 146–149.

    Google Scholar 

  24. C. Gouriéroux, ARCH Models and Financial Applications(Springer, New York, 1997).

    Google Scholar 

  25. R.L. Hagerman, More evidence on the distribution of security returns, Journal of Finance 33(4) (1978) 1213–1221.

    Google Scholar 

  26. S.L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies 6(2) (1993) 327–343.

    Google Scholar 

  27. N. Hofmann, E. Platen and M. Schweizer, Option pricing under incompleteness and stochastic volatility, Mathematical Finance 2 (1992) 153–187.

    Google Scholar 

  28. J. Hull, Options, Futures, and Other Derivative Securities, 3rd edn. (Prentice-Hall, 1997).

  29. J. Hull and A. White, The pricing of options on assets with stochastic volatilities, Journal of Finance 42(2) (1987) 281–300.

    Google Scholar 

  30. R. Jarrow and A. Rudd, Approximate option valuation for arbitrary stochastic processes, Journal of Financial Economics, 10(3) (1982) 347–369.

    Google Scholar 

  31. R. Jarrow and A. Rudd, Tests of an approximate option valuation formula, in: Option Pricing,ed. M. Brenner (Heath, Lexington MA, 1983) pp. 81–100.

    Google Scholar 

  32. R. Jarrow and A. Rudd, Option Pricing(Irwin, 1983).

  33. M.G. Kendall, The analysis of economic time series: Part 1, Prices, Journal of the Royal Statistical Society 96 (1953) 11–25.

    Google Scholar 

  34. M. Kijima and T. Yoshida, A simple option pricing model with markovian volatilities, Journal of the Operations Research Society of Japan 36(3) (1993) 149–166.

    Google Scholar 

  35. S.J. Kon, Models of stock returns: A comparison, Journal of Finance 39(1) (1984) 147–165.

    Google Scholar 

  36. W.K. Li and T.K. Mak, On the squared autocorrelations in non-linear time series with conditional heteroscedasticity, Journal of Time Series Analysis 15(6) (1994) 627–635.

    Google Scholar 

  37. J.D. MacBeth and L.J. Merville, An empirical examination of the Black–Scholes call option pricing model, Journal of Finance 34(4) (1979) 1173–1186.

    Google Scholar 

  38. I.L. MacDonald and W. Zucchini, Hidden Markov and Other Models for Discrete-Valued Time Series(Chapman & Hall, London, 1997).

    Google Scholar 

  39. D.B. Madan and E. Seneta, (1990), The variance gamma (V.G.) model for share market returns, Journal of Business 63(4), 511–524.

    Google Scholar 

  40. B. Mandelbrot, The variations of certain speculative prices, Journal of Business 36 (1963) 394–419.

    Google Scholar 

  41. J.H. McCulloch, Continuous time process with stable increments, Journal of Business 51(4) (1978) 601–618.

    Google Scholar 

  42. J.H. McCulloch, Financial applications of stable distributions, in: Handbook of Statistics, Vol. 14, Statistical Methods in Finance, eds. G.S. Maddala and C.R. Rao (North-Holland, New York, 1996) pp. 393–425

    Google Scholar 

  43. R.C. Merton, Option pricing when the underlying stock returns are discontinuous, Journal of Financial Economics 3(1/2) (1976) 125–144.

    Google Scholar 

  44. S. Mittnik and S.T. Rachev, Modeling asset returns with alternative stable distributions, Econometric Reviews 12(3) (1993) 261–330.

    Google Scholar 

  45. A.S. Paulson, E.W. Halcomb and R.A. Leitch, The estimation of the parameters of the stable laws, Biometrika 62(1) (1975) 163–170.

    Google Scholar 

  46. P.R. Perry, More evidence on the nature of the distribution of security returns, Journal of Financial and Quantitative Analysis 18(2) (1983) 211–221.

    Google Scholar 

  47. E.E. Peters, Chaos and Order in Capital Markets(Wiley, New York, 1991).

    Google Scholar 

  48. E.E. Peters, Fractal Market Analysis(Wiley, New York, 1994).

    Google Scholar 

  49. O.V. Pictet, M.M. Dacorogna and U.A. Müller, Hill, bootstrap and jackknife estimators for heavy tails, Olsen & Associates Working Paper, OVP.1996–12–10 (1996). Available from http://www.olsen.ch

  50. W. Press, B. Flannery, S. Teukolsky and W. Vetterling, Numerical Recipes in C, 2nd edn. (Cambridge University Press, 1992).

  51. S.T. Rachev and G. Samorodnitsky, Option pricing formulae for speculative prices modelled by subordinate stochastic processes, Serdica Mathematics Journal 19(2–3) (1993) 175–190.

    Google Scholar 

  52. M. Rubinstein M., Non-parametric tests of alternative options pricing models using all reported trades and quotes on the 30 most active CBOE options classes from August 23, 1976 through to August 31, 1978, Journal of Finance 40 (1985) 455–480.

    Google Scholar 

  53. M. Rubinstein, Presidential address: Implied binomial trees, Journal of Finance 49(3) (1994) 771–818.

    Google Scholar 

  54. G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes(Chapman & Hall, New York, 1994).

    Google Scholar 

  55. L.O. Scott, Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates: Applications of Fourier inversion methods, Mathematical Finance 7(4) (1997) 413–426.

    Google Scholar 

  56. N. Shephard, Statistical aspects of ARCH and stochastic volatility, in: Time Series Models in Econometrics, Finance and Other Fields, eds. D. Cox, D. Hinkley and O. Barndorff-Nielsen (Chapman & Hall, London, 1996) pp. 1–67.

    Google Scholar 

  57. J.H. Stock, Estimating continuous-time processes subject to time deformation, Journal of the American Statistical Association 83(401) (1988) 77–85.

    Google Scholar 

  58. S. Taylor, Modelling Financial Time Series(Wiley, Chichester, 1986).

    Google Scholar 

  59. J.R. Thompson and R.A. Tapia, Nonparametric Probability Density Estimation(Johns Hopkins University Press, 1978). Also reproduced in Thompson and Tapia [60].

  60. J.R. Thompson J.R. and R.A. Tapia, Nonparametric Function Estimation, Modelling, and Simulation, (SIAM, Philadelphia, 1990).

    Google Scholar 

  61. P. Zangari, Risk Metrics—Technical Document(Morgan Guaranty Trust Company, New York, December 1996).

    Google Scholar 

  62. V.M. Zolotarev, One Dimensional Stable Laws, Translations of Mathematical Monographs, Vol. 65 (American Mathematical Society, 1986). Translation from the original 1983 Russian edition.

  63. W.T. Ziemba, Calculating investment portfolios when the returns have stable distributions, in: Mathematical Programming in Theory and Practice, eds. P.L. Hammer and G. Zoutendijk (North-Holland) pp. 443–482.

Download references

Author information

Authors and Affiliations

  1. Department of Accounting and Finance, University of Wollongong, Northfields Avenue, Wollongong NSW, 2522, Australia

    David Edelman

  2. Salomon Smith Barney Australia Securities Pty Limited, PO Box 557, Sydney NSW, 2001, Australia

    Thomas Gillespie

Authors
  1. David Edelman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Gillespie
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Edelman, D., Gillespie, T. The Stochastically Subordinated Poisson Normal Process for Modelling Financial Assets. Annals of Operations Research 100, 133–164 (2000). https://doi.org/10.1023/A:1019219217900

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019219217900

Search

Navigation