Martingale methods in pricing contingent claim in a Lévy market

• Autorzy: Long Nguyen-Thanh

Warianty tytułu

Martyngałowe metody na rynku Levy'ego

• Języki publikacji: EN

Abstrakty

EN

The pricing and hedging of contingent claim in an incomplete market is one of the major question in modern financial mathematics. In this paper we will present and discuss the martingale approach to value the contingent claim of the asset, whose value is modeled by a geometric Levy process.

Praca przedstawia problematykę wyceny contingent claim przy wykorzystaniu minimalnej martygałowej miary, minimalnej entropy madrygałowej miary oraz Esscher transformacji, tzw. Levy rynku, na którym ceny aktywów ewolują się według procesu geometrycznego Levy'ego. Wprowadzona została nie tylko formuła wyżej wymienionych miar, ale i także zanalizowano reprezentację lokal-madrygałowych miar dla danego procesu cen zdyskontowanego przez stopę oprocentową, oraz relacje zachodzące między badanymi miarami.

Słowa kluczowe

EN

Options pricing; Financial mathematics; Levy processes

PL

Wycena opcji; Matematyka finansowa; Procesy Levy'ego

• Czasopismo: Przegląd Statystyczny
• Rocznik: 2003
• Tom: 50
• Numer: z. 4
• Strony: 59--79

Twórcy

autor

Long Nguyen-Thanh

Bibliografia

• [1] Barndorff-Nielsen O., Processes of Normal Inverse Gaussian type, Finance and Stochastic 2, p. 41-68, 1998.
• [2] Chan T., Pricing Contingent Claims on Stock Driven by Levy Process, The Annals of Applied Probability 1999, Vol. 9, Nr 2, p. 504-528.
• [3] Choulli T. Krawczyk L., Stricker Ch., E-Martingales and their Applications in Mathematical Finance, The Annals of Probability 1998, Vol. 26, Nr 2, p. 853-876.
• [4] Choulli T., Hurd T.R., The Role of Hellinger Process in Mathematical Finance, Entropy 3, p. 141-152, 2001.
• [5] Csiszar L., I-Divergence geometry of Probability Distributions and Minimization Problems, The Annals of Probability 1975, Vol. 3, No 1, p. 146-158.
• [6] Delbaen F., Schachermayer W., A General Version of the Fundamental Theorem of Asset Pricing, Mathematische Annalen 300, p. 463-520, 1994.
• [7] Delbaen F., Schachermayer W., The Variance-Optimal martingale measure for continuous processes, Bernoulli 2, p. 81-105, 1996.
• [8] Delbaen F., Schachermayer W" The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes, Mathematische Annalen 312, p. 215-250, 1998.
• [9] Eberlein E., Keller U., Hyperbolic distributions in Finance, Bernoulli 1, p. 289-299, 1995.
• [10] Follmer H., Schweizer M., Hedging of Contingent Claims under Incomplete Information, in M.H.A. Davis and R.J. Elliot (eds.) "Applied Stochastic Analysis", Stochastic Monographs, Vol. 5, Gordon and Bearch, London/NewYork 1991, p. 389-414.
• [11] Fritelli M., The Minimal entropy martingale measure and the valuation problem in incomplete markts, Mathematical Finance, Vol. 10, No. 1, 2000, p. 39-52.
• [12] Fujiwara T. and Miyahara Y., The minimal entropy martingale measures for geometric Levy process, Working Paper 2001.
• [13] Goll T. and Ruschendorf L., Minimax and Minimal distance martingale measures and their relationship to portfolio optimization, Finance and Stochastics 5, p. 557-581, 2001.
• [14] Ikeda N. and Watanabe S., Stochastic Differential Equations and Diffusion Processes, 2nd Edition, North-Holland/Kodansha 1989.
• [15] Jacod J., Shiryaev A., Limit Theorem for Stochastic Processes, Springer-Verlag 1987.
• [16] Miyahara Y., Pricing model and Related Estimation Problem, Working paper 2000, Nagoya City University.
• [17] Kallsen J. and Shiryaev A., The Cumulant Process and Esscher's Change of Measure, Working Paper 2001, University Freiburg.
• [18] Kunita H., Watanabe S., On Square Integrable Martingales, Nagoya Math. J. 30, 1967, p. 209-245.
• [19] Harrison J., Kreps D.M., Martingales and Arbitrage in multiperiod securities markets, J. Economy Theory 20, 1979, p. 381-408.
• [20] Harrison J., Pliska S., Martingales and Stochastic Integrals in the theory of continuous trading, Stochastic Processes and their Applications 11, 1981, p. 215-260.
• [21] Protter P., Stochastic Integration and Differential Equation - A new Approach, Springer-Verlag 1990.
• [22] Raibe S., Levy processes in Finance: Theory, Numerics and Empirical Facts, Dissertation 2000.
• [23] Sato K., Levy Processes and Infinitely Divisible Distributions, Cambridge University Press 1999.
• [24] Schweizer M., Option hedging for semimartingales, Stochastic Processes and their Applications 37, p. 339-363, 1991.
• [25] Schweizer M., On the Minimal Martingale Measure and the Follmer-Schweizer Decomposition, Stochastic Analysis and Applications 13(5), p. 537-599, 1995.
• [26] Schweizer M., Approximating pricing and the variance-optimal martingale measure, Ann. Probab.24, p. 20b-236, 1996.
• [27] Schweizer M., A Guided Tour through Quadratic Hedging Approaches, Working Paper 1999, Technische Universität Berlin.