Asia-Pacific Financial Markets

, Volume 8, Issue 1, pp 45–60 | Cite as

[Geometric Lévy Process & MEMM] Pricing Model and Related Estimation Problems

  • Yoshio Miyahara


In this article the [Geometric Lévy Process & MEMM] pricingmodel is proposed. This model is an option pricing model for theincomplete markets, and this model is based on the assumptions that theprice processes are geometric Lévy processes and that the pricesof the options are determined by the minimal relative entropy methods.This model has many good points. For example, the theoretical part ofthe model is contained in the framework of the theory of Lévyprocess (additive process). In fact the price process is also aLévy process (with changed Lévy measure) under the minimalrelative entropy martingale measure (MEMM), and so the calculation ofthe prices of options are reduced to the computation of functionals ofLévy process. In previous papers, we have investigated thesemodels in the case of jump type geometric Lévy processes. In thispaper we extend the previous results for more general type of geometricLévy processes. In order to apply this model to real optionpricing problems, we have to estimate the price process of theunderlying asset. This problem is reduced to the estimation problem ofthe characteristic triplet of Lévy processes. We investigate thisproblem in the latter half of the paper.

estimation of stochastic process geometric Lévy process incomplete markets minimal entropy martingale measure pricing model 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bühlmann, H., Delbaen, F., Embrechts, P., and Shiryaev, A. N. (1996) No-arbitrage, change of measure and conditional Esscher transforms, CWI Quarterly 9 (4), 291-317.Google Scholar
  2. Chan, T. (1999) Pricing contingent claims on stocks derived by Lévy processes, The Ann. Appl. Probab. 9 (2), 504-528.Google Scholar
  3. Delbaen, F. and Schachermayer, W. (1996) The variance-optimal martingale measure for continuous processes, Bernoulli 2, 81-106.Google Scholar
  4. Eberlein, E. and Keller, U. (1995) Hyperbolic distributions in finance, Bernoulli 1, 281-299.Google Scholar
  5. Fama, E. F. (1963) Mandelbrot and the stable paretian hypothesis, J. Busin. 36, 420-429.Google Scholar
  6. Föllmer, H. and Schweizer, M. (1991) Hedging of contingent claims under incomplete information. In M. H. A. Davis and R. J. Elliot (ed.), Applied Stochastic Analysis, Gordon and Breach, pp. 389-414.Google Scholar
  7. Frittelli, M. (2000) The minimal entropy martingale measures and the valuation problem in incomplete markets, Mathematical Finance 10, 39-52.Google Scholar
  8. Hurst, S. R., Platen, E., and Rachev, T. (1996) Subordinated Markov index models: A comparison.Google Scholar
  9. Ikeda, N. and Watanabe, S. (1989) Stochastic Differential Equations and Diffusion Processes, 2nd edn, North-Holland.Google Scholar
  10. Kunita, H. and Watanabe, S. (1967) On square-integrable martingales, Nagoya Math. J. 30, 209-245.Google Scholar
  11. Mandelbrot, B. (1963) The variation of certain speculative prices, J. Busin. 36, 394-419.Google Scholar
  12. Merton, R. C. (1976) Option pricing when underlying stock returns are discontinuous, J. Financ. Econom. 3, 125-144.Google Scholar
  13. Mittnik, S., Paolella, M. S., and Rachev, S. T. (1997) A tail estimator for the index of the stable paretian distribution.Google Scholar
  14. Miyahara, Y. (1996a) Canonical martingale measures of incomplete assets markets. In S. Watanabe et al. (eds), Probability Theory and Mathematical Statistics: Proceedings of the Seventh Japan-Russia Symposium, Tokyo 1995, pp. 343-352.Google Scholar
  15. Miyahara, Y. (1996b) Canonical martingale measures and minimal martingale measures of incomplete assets markets, The Australian National University Research Report, No. FMRR 007-96, pp. 95-100.Google Scholar
  16. Miyahara, Y. (1999a) Minimal entropy martingale measures of jump type price processes in incomplete assets markets, Asian-Pacific Financial Markets 6 (2), 97-113.Google Scholar
  17. Miyahara, Y. (1999b) Minimal relative entropy martingale measures of geometric Lévy processes and option pricing models in incomplete markets, Discussion Papers in Economics, Nagoya City University, No. 249, pp. 1-8.Google Scholar
  18. Miyahara, Y. (1999c) Minimal relative entropy martingale measures and their applications to option pricing theory. In Proceedings of JIC99, The 5-th JAFEE International Conference, pp. 316-323.Google Scholar
  19. Miyahara, Y. (2000a) A theorem related to LogLévy processes and its application to option pricing problems in incomplete markets. In L. Accardi, H.-H. Kuo, N. Obata, K. Saito, Si Si, and L. Streit (eds), Trends in Contemporary Infinite Dimensional Analysis and Quantum Probability, Italian School of East Asian Studies, Natural and Mathematical Sciences Series 3, Instituto Italiano di Cultura (Kyoto).Google Scholar
  20. Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press.Google Scholar
  21. Shiryaev, A. N. (1999) Essentials of Stochastic Finance: Facts, Models, Theory, World Scientific.Google Scholar
  22. Xiao, K., Miyahara, Y., and Misawa, T. (1999) Computer simulation of [Geometric Lévy Process & MEMM] pricing model, preprint.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Yoshio Miyahara
    • 1
  1. 1.Faculty of EconomicsNagoya City University, Mizuhochou, MizuhokuNagoyaJapan

Personalised recommendations