Meixner process: theory and applications to financial Brazilian market
DOI:
https://doi.org/10.1590/S0101-41612011000200007Keywords:
Meixner process, option pricing, heavy tailsAbstract
Well-known models that are extensively used by market traders, such as the Black-Scholes model, assume that the daily log-returns of assets follow a Normal distribution. Empirical evidences, however, show that return rates are frequently asymmetric and have fatter tails. Hence, this work aims to investigate if the Meixner distribution would be more appropriate to fit daily log-return. Additionally, it will be explored if the Lévy process risen from this distribution, the Meixner process, is efficient to price financial derivatives. Therefore, this study proposes the replacement of the Brownian motion by the Meixner process in Black-Scholes.
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References
Bakshi , G.; CaoAO, C. & Chen , Z. Empirical performance of alternative option pricing models. Journal of Finance, v. 52, n. 5, 1997, p. 2003-49,
Barndorff -Nielsen, O.E. Processes of normal inverse Gaussian type. Finance and Stochastics, v. 2, n. 1, 1998, p. 41-68.
Barndorff -Nielsen, O.E & SHEPPARD, N. Non-gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics. Journal of the Royal Statistical Society, v. 63, n. 2, 2001, p. 167-241.
BATES, D. The crash of ‘87: was it expected? The evidence from options markets. Journal of Finance, v. 46, n. 3,p., 1991, 1009-44.
BATES, D. Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options. Review of Financial Studies, v. 9, n. 1, 1996, p. 69-108.
BATES, D. Post ‘87 crash fears in S&P 500 futures options. NBER Working Paper, n. 5894, jan. 1997.
BertoinERTOIN, J. Lévy processes. Cambridge University Press, 1996.
BlackLACK, F. & Scholes , M. The pricing of options and corporate liabilities. Journal of Political Economy, v. 81, 1973, p. 637–654.
Carr , P.; Geman , H.; MadanADAN, D.H. & Yor , M. The fi ne structure of asset returns: an empirical investigation. Journal of Business, v. 75, n. 2, 2002.
Delbaen , F. & SchachermayerCHACHERMAYER, W. A general version of the fundamental theorem of asset pricing. Mathematische Annalen, v. 300, 1994p. 463-520.
Duffie , D.; Pan , J. & Singleton , K. Transform analysis and asset pricing for affine jump-diffusions. Econometrica, v. 68, n. 6, 2000, p.1343-1376.
Eberlein , E.; Keller , U. & PRAUSE, K. New insights into smile, mispricing and value at risk. Journal of Business, v. 71, n. 3, 1998, p. 371-405.
EsscherSSCHER, F. On the probability function in the collective theory of risk. Skandinavisk Aktuarietidskrift, v. 15, p. 175-195, 1932.
FAJARDO, J.; Schuschny A. & A. Lévy Processes and The Brazilian Market. Brazilian Review of Econometrics, v.21, n. 2, 2001, p. 263-289.
FAJARDO, J. & Farias , A. R. Generalized Hyperbolic Distributions and Brazilian Data. Brazilian Review of Econometrics, v.24, n. 2, 2004, p. 1-21.
FAJARDO, J. & Farias , A. R. Multivariate affine generalized hyperbolic distributions: An empirical investigation, International Review of Financial Analysis, v. 18, n. 4, Sept. 2009, p. 174-184.
FAJARDO, J. & Farias , A. R. Derivative pricing using multivariate affine generalized hyperbolic distributions, Journal of Banking &Finance, v. 34, n. 7, July 2010, p. 1607-1617.
FAJARDO, J.; ORNELAS. J. R. H. & FARIAS, A. R. Analyzing the use of generalized hyperbolic distributions to value at risk calculations. Revista de Economia Aplicada, v. 9, 2005, p. 25-38.
French , d. w. & Martin l. j. The measurement of option mispricing. Journal of Banking and Finance, v. 12, 1988, p. 537-550.
GERBER, H.U. & Shiu , E.S.W. Option pricing by Esscher transforms. Transactions of the Society of Actuaries, v. 46, 1994, p. 99–191.
GERBER, H.U. & Shiu , E.S.W. Actuarial bridges to dynamic hedging and option pricing. Insurance: Mathematics and Economics, v. 18, n. 3, 1996, p. 183–218.
Grigelionis , B. Processes of Meixner type. Lithuanian Mathematical Journal, v. 39, n. 1, 1999, p. 33-41.
Grigelionis , B. Generalized z-distributions and related stochastic processes. Lithuanian Mathematical Journal, v. 41, n. 3, 2001, p. 303-319.
Heston , S. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, v. 6, n. 2, 1993, p. 327-343.
Jones , E. P. Option arbitrage and strategy with large price changes. Journal of Financial Economics, v.13, n. 1, 1984, p. 91-113.
MERTON, R. C. Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, v. 3, 1976, p. 125-144.
Naik , V. & Lee , M. General equilibrium pricing of options on the market portfolio with discontinuous returns. Review of Financial Studies, v. 3, n. 4, 1990, p. 493-521.
MADAN, D.; Carr P. & Chang , E. The variance gamma process and option pricing. European Finance Review, v. 2, 1998, p. 79-105.
Schoutens , W. Meixner processes in finance. Eindhoven, Eurandom Report, n. 002, 2001.
Schoutens , W. Lévy processes in finance: pricing financial derivatives. John Wiley & Sons, 2003.
Schoutens , W. & Teugels , J.L. Lévy processes, polynomials and martingales. Communication in Statistics Stochastic Models, v. 14, n. 1 e 2, 1998, p. 335–349.
STEPHENS, M. A. Asymptotic results for goodness of fi t statistics with unknown parameters. The Annals of Statistics, v. 4, n. 2, 1976, p. 357-369.
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