[Geometric Lévy Process & MEMM] Pricing Model and Related Estimation Problems
 Yoshio Miyahara
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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.
 Bühlmann, H., Delbaen, F., Embrechts, P., and Shiryaev, A. N. (1996) Noarbitrage, change of measure and conditional Esscher transforms, CWI Quarterly 9 (4), 291317.
 Chan, T. (1999) Pricing contingent claims on stocks derived by Lévy processes, The Ann. Appl. Probab. 9 (2), 504528.
 Delbaen, F. and Schachermayer, W. (1996) The varianceoptimal martingale measure for continuous processes, Bernoulli 2, 81106.
 Eberlein, E. and Keller, U. (1995) Hyperbolic distributions in finance, Bernoulli 1, 281299.
 Fama, E. F. (1963) Mandelbrot and the stable paretian hypothesis, J. Busin. 36, 420429.
 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. 389414.
 Frittelli, M. (2000) The minimal entropy martingale measures and the valuation problem in incomplete markets, Mathematical Finance 10, 3952.
 Hurst, S. R., Platen, E., and Rachev, T. (1996) Subordinated Markov index models: A comparison.
 Ikeda, N. and Watanabe, S. (1989) Stochastic Differential Equations and Diffusion Processes, 2nd edn, NorthHolland.
 Kunita, H. and Watanabe, S. (1967) On squareintegrable martingales, Nagoya Math. J. 30, 209245.
 Mandelbrot, B. (1963) The variation of certain speculative prices, J. Busin. 36, 394419.
 Merton, R. C. (1976) Option pricing when underlying stock returns are discontinuous, J. Financ. Econom. 3, 125144.
 Mittnik, S., Paolella, M. S., and Rachev, S. T. (1997) A tail estimator for the index of the stable paretian distribution.
 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 JapanRussia Symposium, Tokyo 1995, pp. 343352.
 Miyahara, Y. (1996b) Canonical martingale measures and minimal martingale measures of incomplete assets markets, The Australian National University Research Report, No. FMRR 00796, pp. 95100.
 Miyahara, Y. (1999a) Minimal entropy martingale measures of jump type price processes in incomplete assets markets, AsianPacific Financial Markets 6 (2), 97113.
 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. 18.
 Miyahara, Y. (1999c) Minimal relative entropy martingale measures and their applications to option pricing theory. In Proceedings of JIC99, The 5th JAFEE International Conference, pp. 316323.
 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).
 Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press.
 Shiryaev, A. N. (1999) Essentials of Stochastic Finance: Facts, Models, Theory, World Scientific.
 Xiao, K., Miyahara, Y., and Misawa, T. (1999) Computer simulation of [Geometric Lévy Process & MEMM] pricing model, preprint.
 Title
 [Geometric Lévy Process & MEMM] Pricing Model and Related Estimation Problems
 Journal

AsiaPacific Financial Markets
Volume 8, Issue 1 , pp 4560
 Cover Date
 20010301
 DOI
 10.1023/A:1011445109763
 Print ISSN
 13872834
 Online ISSN
 15736946
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

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

 Yoshio Miyahara ^{(1)}
 Author Affiliations

 1. Faculty of Economics, Nagoya City University, Mizuhochou, Mizuhoku, Nagoya, 4678501, Japan