Borak, Szymon and Misiorek, Adam and Weron, Rafal (2010): Models for Heavytailed Asset Returns.

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Abstract
Many of the concepts in theoretical and empirical finance developed over the past decades – including the classical portfolio theory, the BlackScholesMerton option pricing model or the RiskMetrics variancecovariance approach to VaR – rest upon the assumption that asset returns follow a normal distribution. But this assumption is not justified by empirical data! Rather, the empirical observations exhibit excess kurtosis, more colloquially known as fat tails or heavy tails. This chapter is intended as a guide to heavytailed models. We first describe the historically oldest heavytailed model – the stable laws. Next, we briefly characterize their recent lightertailed generalizations, the socalled truncated and tempered stable distributions. Then we study the class of generalized hyperbolic laws, which – like tempered stable distributions – can be classified somewhere between infinite variance stable laws and the Gaussian distribution. Finally, we provide numerical examples.
Item Type:  MPRA Paper 

Original Title:  Models for Heavytailed Asset Returns 
Language:  English 
Keywords:  Heavytailed distribution; Stable distribution; Tempered stable distribution; Generalized hyperbolic distribution; Asset return; Random number generation; Parameter estimation 
Subjects:  ?? C16 ?? C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C13  Estimation: General G  Financial Economics > G3  Corporate Finance and Governance > G32  Financing Policy ; Financial Risk and Risk Management ; Capital and Ownership Structure ; Value of Firms ; Goodwill C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C15  Statistical Simulation Methods: General 
Item ID:  25494 
Depositing User:  Rafal Weron 
Date Deposited:  28. Sep 2010 20:15 
Last Modified:  30. Dec 2015 14:30 
References:  Atkinson, A. C. (1982). The simulation of generalized inverse Gaussian and hyperbolic random variables, SIAM Journal of Scientific & Statistical Computing 3: 502–515. BarndorffNielsen, O. E. (1977). Exponentially decreasing distributions for the logarithm of particle size, Proceedings of the Royal Society London A 353: 401–419. BarndorffNielsen, O. E. (1995). Normal\\Inverse Gaussian Processes and the Modelling of Stock Returns, Research Report 300, Department of Theoretical Statistics, University of Aarhus. BarndorffNielsen, O. E. and Blaesild, P. (1981). Hyperbolic distributions and ramifications: Contributions to theory and applications, in C. Taillie, G. Patil, B. Baldessari (eds.) Statistical Distributions in Scientific Work, Volume 4, Reidel, Dordrecht, pp. 19–44. BaroneAdesi, G., Giannopoulos, K., and Vosper, L., (1999). VaR without correlations for portfolios of derivative securities, Journal of Futures Markets 19(5): 583–602. Basle Committee on Banking Supervision (1995). An internal modelbased approach to market risk capital requirements, http://www.bis.org. Bianchi, M. L., Rachev, S. T., Kim, Y. S. and Fabozzi, F. J. (2010). Tempered stable distributions and processes in finance: Numerical analysis, in M. Corazza, P. Claudio, P. (Eds.) Mathematical and Statistical Methods for Actuarial Sciences and Finance, Springer. Bibby, B. M. and Sørensen, M. (2003). Hyperbolic processes in finance, in S. T. Rachev (ed.) Handbook of Heavytailed Distributions in Finance, North Holland. Blaesild, P. and Sorensen, M. (1992). HYP – a Computer Program for Analyzing Data by Means of the Hyperbolic Distribution, Research Report 248, Department of Theoretical Statistics, Aarhus University. Boyarchenko, S. I. and Levendorskii, S. Z. (2000). Option pricing for truncated Levy processes, International Journal of Theoretical and Applied Finance 3: 549–552. Brcich, R. F., Iskander, D. R. and Zoubir, A. M. (2005). The stability test for symmetric alpha stable distributions, IEEE Transactions on Signal Processing 53: 977–986. Buckle, D. J. (1995). Bayesian inference for stable distributions, Journal of the American Statistical Association 90: 605–613. Carr, P., Geman, H., Madan, D. B. and Yor, M. (2002). The fine structure of asset returns: an empirical investigation, Journal of Business 75: 305–332. Chambers, J. M., Mallows, C. L. and Stuck, B. W. (1976). A Method for Simulating Stable Random Variables, Journal of the American Statistical Association 71: 340–344. Chen, Y., Hardle, W. and Jeong, S.O. (2008). Nonparametric risk management with generalized hyperbolic distributions, Journal of the American Statistical Association 103: 910–923. Cont, R., Potters, M. and Bouchaud, J.P. (1997). Scaling in stock market data: Stable laws and beyond, in B. Dubrulle, F. Graner, D. Sornette (eds.) Scale Invariance and Beyond, Proceedings of the CNRS Workshop on Scale Invariance, Springer, Berlin. D’Agostino, R. B. and Stephens, M. A. (1986). GoodnessofFit Techniques, Marcel Dekker, New York. Dagpunar, J. S. (1989). An Easily Implemented Generalized Inverse Gaussian Generator, Communications in Statistics – Simulations 18: 703–710. Danielsson, J., Hartmann, P. and De Vries, C. G. (1998). The cost of conservatism: Extreme returns, value at risk and the Basle multiplication factor, Risk 11: 101–103. Dominicy, Y. and Veredas, D. (2010). The method of simulated quantiles, ECARES working paper, 2010–008. DuMouchel, W. H. (1971). Stable Distributions in Statistical Inference, Ph.D. Thesis, Department of Statistics, Yale University. DuMouchel, W. H. (1973). On the Asymptotic Normality of the Maximum–Likelihood Estimate when Sampling from a Stable Distribution, Annals of Statistics 1(5): 948–957. Eberlein, E. and Keller, U. (1995). Hyperbolic distributions in finance, Bernoulli 1: 281–299. Fama, E. F. and Roll, R. (1971). Parameter Estimates for Symmetric Stable Distributions, Journal of the American Statistical Association 66: 331–338. Fan, Z. (2006). Parameter estimation of stable distributions, Communications in Statistics – Theory and Methods 35(2): 245–255. Fragiadakis, K., Karlis, D. and Meintanis, S. G. (2009). Tests of fit for normal inverse Gaussian distributions, Statistical Methodology 6: 553–564. Garcia, R., Renault, E. and Veredas, D. (2010). Estimation of stable distributions by indirect inference, Journal of Econometrics, Forthcoming. Grabchak, M. (2010). Maximum likelihood estimation of parametric tempered stable distributions on the real line with applications to finance, Ph.D. thesis, Cornell University. Grabchak, M. and Samorodnitsky, G. (2010). Do financial returns have finite or infinite variance? A paradox and an explanation, Quantitative Finance, DOI: 10.1080/14697680903540381. Guillaume, D. M., Dacorogna, M. M., Dave, R. R., Muller, U. A., Olsen, R. B. and Pictet, O. V. (1997). From the birds eye to the microscope: A survey of new stylized facts of the intradaily foreign exchange markets, Finance & Stochastics 1: 95–129. Janicki, A. and Weron, A. (1994a). Can one see αstable variables and processes, Statistical Science 9: 109–126. Janicki, A. and Weron, A. (1994b). Simulation and Chaotic Behavior of αStable Stochastic Processes, Marcel Dekker. Karlis, D. (2002). An EM type algorithm for maximum likelihood estimation for the Normal Inverse Gaussian distribution, Statistics and Probability Letters 57: 43–52. Karlis, D. and Lillestol, J. (2004). Bayesian estimation of NIG models via Markov chain Monte Carlo methods, Applied Stochastic Models in Business and Industry 20(4): 323–338. Kawai, R. and Masuda, H. (2010). On simulation of tempered stable random variates, Preprint, Kyushu University. Kogon, S. M. and Williams, D. B. (1998). Characteristic function based estimation of stable parameters, in R. Adler, R. Feldman, M. Taqqu (eds.), A Practical Guide to Heavy Tails, Birkhauser, pp. 311–335. Koponen, I. (1995). Analytic approach to the problem of convergence of truncated Levy flights towards the Gaussian stochastic process, Physical Review E 52: 1197–1199. Koutrouvelis, I. A. (1980). Regression–Type Estimation of the Parameters of Stable Laws, Journal of the American Statistical Association 75: 918–928. Kuester, K., Mittnik, S., and Paolella, M.S. (2006). ValueatRisk prediction: A comparison of alternative strategies, Journal of Financial Econometrics 4(1): 53–89. Kuchler, U., Neumann, K., Sørensen, M. and Streller, A. (1999). Stock returns and hyperbolic distributions, Mathematical and Computer Modelling 29: 1–15. Lombardi, M. J. (2007). Bayesian inference for αstable distributions: A random walk MCMC approach, Computational Statistics and Data Analysis 51(5): 2688–2700. Madan, D. B. and Seneta, E. (1990). The variance gamma (V.G.) model for share market returns, Journal of Business 63: 511–524. Mandelbrot, B. B. (1963). The variation of certain speculative prices, Journal of Business 36: 394–419. Mantegna, R. N. and Stanley, H. E. (1994). Stochastic processes with ultraslow convergence to a Gaussian: The truncated Levy flight, Physical Review Letters 73: 2946–2949. Matacz, A. (2000). Financial Modeling and Option Theory with the Truncated Levy Process, International Journal of Theoretical and Applied Finance 3(1): 143–160. Matsui, M. and Takemura, A. (2006). Some improvements in numerical evaluation of symmetric stable density and its derivatives, Communications in Statistics – Theory and Methods 35(1): 149–172. Matsui, M. and Takemura, A. (2008). Goodnessoffit tests for symmetric stable distributions – empirical characteristic function approach, TEST 17(3): 546–566. McCulloch, J. H. (1986). Simple consistent estimators of stable distribution parameters, Communications in Statistics – Simulations 15: 1109–1136. McNeil, A. J., Rudiger, F. and Embrechts, P. (2005). Quantitative Risk Management, Princeton University Press, Princeton, NJ. Michael, J. R., Schucany, W. R. and Haas, R. W. (1976). Generating Random Variates Using Transformations with Multiple Roots, The American Statistician 30: 88–90. Mittnik, S., Doganoglu, T. and Chenyao, D. (1999). Computing the Probability Density Function of the Stable Paretian Distribution, Mathematical and Computer Modelling 29: 235–240. Mittnik, S. and Paolella, M. S. (1999). A simple estimator for the characteristic exponent of the stable Paretian distribution, Mathematical and Computer Modelling 29: 161–176. Mittnik, S., Rachev, S. T., Doganoglu, T. and Chenyao, D. (1999). Maximum Likelihood Estimation of Stable Paretian Models, Mathematical and Computer Modelling 29: 275–293. Nolan, J. P. (1997). Numerical Calculation of Stable Densities and Distribution Functions, Communications in Statistics – Stochastic Models 13: 759–774. Nolan, J. P. (2001). Maximum Likelihood Estimation and Diagnostics for Stable Distributions, in O. E. BarndorffNielsen, T. Mikosch, S. Resnick (eds.), Levy Processes, Brikhauser, Boston. Nolan, J. P. (2010). Stable Distributions – Models for Heavy Tailed Data, Birkhauser, Boston. In progress, Chapter 1 online at academic2.american.edu/∼jpnolan. Ojeda, D. (2001). Comparison of stable estimators, Ph.D. Thesis, Department of Mathematics and Statistics, American University. Paolella, M. S. (2001). Testing the stable Paretian assumption, Mathematical and Computer Modelling 34: 1095–1112. Paolella, M. S. (2007). Intermediate Probability: A Computational Approach, Wiley, Chichester. Peters, G. W., Sisson, S. A. and Fan, Y. (2009). Likelihoodfree Bayesian inference for αstable models, Preprint: http://arxiv.org/abs/0912.4729. Poirot, J. and Tankov, P. (2006). Monte Carlo option pricing for tempered stable (CGMY) processes, AsiaPacific Financial Markets 13(4): 327344. Prause, K. (1999). The Generalized Hyperbolic Model: Estimation, Financial Derivatives, and Risk Measures, Ph.D. Thesis, Freiburg University, http://www.freidok.unifreiburg.de/volltexte/15. Press, S. J. (1972). Estimation in Univariate and Multivariate Stable Distribution, Journal of the American Statistical Association 67: 842–846. Protassov, R. S. (2004). EMbased maximum likelihood parameter estimation for multivariate generalized hyperbolic distributions with fixed λ, Statistics and Computing 14: 67–77. Rachev, S. and Mittnik, S. (2000). Stable Paretian Models in Finance, Wiley. Rosinski, J. (2007). Tempering stable processes, Stochastic Processes and Their Applications 117(6): 677–707. Ross, S. (2002). Simulation, Academic Press, San Diego. Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non–Gaussian Random Processes, Chapman & Hall. Shuster, J. (1968). On the Inverse Gaussian Distribution Function, Journal of the American Statistical Association 63: 1514–1516. Stahl, G. (1997). Three cheers, Risk 10: 67–69. Stute, W., Manteiga, W. G. and Quindimil, M.P. (1993). Bootstrap Based GoodnessOfFitTests, Metrika 40: 243–256. Venter, J. H. and de Jongh, P. J. (2002). Risk estimation using the Normal Inverse Gaussian distribution, The Journal of Risk 4: 1–23. Weron, R. (1996). On the ChambersMallowsStuck Method for Simulating Skewed Stable Random Variables, Statistics and Probability Letters 28: 165–171. See also R. Weron (1996) Correction to: On the ChambersMallowsStuck Method for Simulating Skewed Stable Random Variables, Working Paper, Available at MPRA: http://mpra.ub.unimuenchen.de/20761/. Weron, R. (2001). Levy–Stable Distributions Revisited: Tail Index > 2 Does Not Exclude the Levy–Stable Regime, International Journal of Modern Physics C 12: 209–223. Weron, R. (2004). Computationally Intensive Value at Risk Calculations, in J. E. Gentle, W. Hardle, Y. Mori (eds.) Handbook of Computational Statistics, Springer, Berlin, 911–950. Weron, R. (2006). Modeling and Forecasting Electricity Loads and Prices: A Statistical Approach, Wiley, Chichester. Zolotarev, V. M. (1964). On representation of stable laws by integrals, Selected Translations in Mathematical Statistics and Probability 4: 84–88. Zolotarev, V. M. (1986). One–Dimensional Stable Distributions, American Mathematical Society. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/25494 