Optimal Portfolio Selection and Risk Management: A Comparison between the Stable Paretian Approach and the Gaussian One

  • Sergio Ortobelli
  • Svetlozar Rachev
  • Isabella Huber
  • Almira Biglova


This paper analyzes stable Paretian models in portfolio theory, risk management and option pricing theory. Firstly, we examine investor’s optimal choices when we assume respectively either Gaussian or stable non-Gaussian distributed index returns. Thus, we approximate discrete time optimal allocations assuming different distributional assumptions and considering several term structure scenarios. Secondly, we compare some stable approaches to compute VaR for heavy-tailed return series. These models are subject to backtesting on out-of-sample data in order to assess their forecasting power. Finally, when asset prices are log-stable distributed, we propose a numerical valuation of option prices and we describe and compare delta hedging strategies when asset prices are either log-stable distributed or log-normal distributed.


Option Price Optimal Portfolio Portfolio Selection Asset Return Efficient Frontier 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Ahlburg, D.A.(1992): A comment on error measures, International Journal of Forecasting 8,99–111.Google Scholar
  2. [2]
    Akgiray, V. (1989): Conditional heteroskedasticity in time series of stock returns: evidence and forecast, Journal of Business 62, 55–80.CrossRefGoogle Scholar
  3. [3]
    Armstrong, J.S., and F. Collopy (1992): Error measures for generalizing about forecasting methods: Empirical comparisons, International Journal of Forecasting, n.8, 69–80.CrossRefGoogle Scholar
  4. [4]
    Barndorff-Nielsen O.E. (1994): Gaussian inverse Gaussian processes and the modeling of stock returns, Technical Report, Aarhus University.Google Scholar
  5. [5]
    Bawa, V. S. (1976): Admissible portfolio for all individuals, Journal of Finance 31, 1169–1183.CrossRefGoogle Scholar
  6. [6]
    Bawa, V. S. (1978): Safety-first stochastic dominance and optimal portfolio choice, Journal of Financial and Quantitative Analysis, 255–271.Google Scholar
  7. [7]
    Blattberg, R.C. and N.J. Gonedes (1974): A comparison of the stable and student distributions as statistical models for stock prices, Journal of Business 47, 244–280.CrossRefGoogle Scholar
  8. [8]
    Boender, G.C.E.(1997): A hybrid simulation/optimization scenario model for asset/liability management, European Journal of Operation Research 99, 126–135.Google Scholar
  9. [9]
    Bollerslev, T. (1986): Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31, 307–327.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Box G.E.P. and G.M. Jenkins (1976): Time series analysis: forecasting and control, 2nd ed. San Francisco: Holden-DayMATHGoogle Scholar
  11. [11]
    Brokwell, P.J. and R.A. Davis (1991): Time series: theory and methods, 2nd ed. New York: Springer.CrossRefGoogle Scholar
  12. [12]
    Cadenillas A. and S.R. Pliska (1999): Optimal trading of a security when there are taxes and transaction costs, Finance and Stochastics 3, 137–165.MATHCrossRefGoogle Scholar
  13. [13]
    Campbell, J. (1987): Stock returns and the term structure, Journal of Financial Economics 18, 373–399.CrossRefGoogle Scholar
  14. [14]
    Chamberlein, G. (1983): A characterization of the distributions that imply mean variance utility functions, Journal of Economic Theory 29, 975–988.Google Scholar
  15. [15]
    Chambers, J., S.J. Mallows and B. Stuck (1976): A method for simulating stable random variables, Journal of the American Statistical Association 71, 340–344.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Chen, N. F. and J., Ingersoll (1983): Exact pricing in linear factor models with finitely many assets: a note, Journal of Finance 38, 985–988.CrossRefGoogle Scholar
  17. [17]
    Cheng, B. and S., Rachev (1995): Multivariate stable futures prices, Mathematical Finance 5, 133–153.MATHCrossRefGoogle Scholar
  18. [18]
    Clark, P.K. (1973): A subordinated stochastic process model with finite variance for speculative prices, Econometrica 41, 135–155.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Connor, G. (1984): A unified beta pricing theory, Journal of Economic Theory 34, 13–31.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Consiglio, A., I. Massabo and S. Ortobelli (2001): Non-Gaussian distribution for VaR calculation: an assessment for the Italian Market, Proceeding IFAC SME 2001 Symposium.Google Scholar
  21. [21]
    Cvitanic, J., and I., Karatzas (1992): Convex duality in constrained portfolio optimization, Annals of Applied Probability 2, 767–818.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    Davis, M.H.A. and A.R. Norman (1990): Portfolio selection with transaction cost, Mathematical Operation Research 15, 676–713.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Davis, R. (1996): Gauss-Newton and M-estimation for ARMA processeswith infinite variance, Stochastic Processes and Applications 63, 75–95.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Davis, R.A. and S.I Resnick (1986): Limit theory for the sample covariance and correlation functions of moving averages, Annals of Statistics 14, 533–558.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    Davis, R.A. and S.I Resnick (1996): Limit theory for bilinear processes with heavy tailed noise, Annals of Applied Probability 6, 1191–1210.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    Davis, R.A., T. Mikosch and B. Basrak (1999): Sample ACF of multivariate stochastic recurrence equations with applications to GARCH, Technical Report, University of Groningen.Google Scholar
  27. [27]
    De Haan, L., S.I. Resnick, H. Rootzen and CG. Vries (1989): Extremal behavior of solutions to a stochastic difference equation with applications to ARCH processes, Stochastic Processes and Applications 32, 213–224.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Duan, J.C. (1995): The GARCH option pricing model, Mathematical Finance 5, 13–32.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    Dutta P. K. (1994): Bankruptcy and expected utility maximization, Journal of Economic Dynamics and Control 18, 539–560.CrossRefGoogle Scholar
  30. [30]
    Dybvig, P. (1985): Acknowledgment: Kinks on the mean variance frontier, Journal of Finance 40, 245.Google Scholar
  31. [31]
    Dybvig, P. (1983): An explicit bound on individual assets’ deviations from APT pricing in a finite economy, Journal of Financial Economics 12, 483–496.CrossRefGoogle Scholar
  32. [32]
    Dybvig, P. and S. Ross (1986): Tax clienteles and asset pricing, Journal of Finance 41, 751–762.Google Scholar
  33. [33]
    Eberlein, E. and K. Keller (1995): Hyperbolic distributions in Finance, Bernoulli 1, 281–299.MATHCrossRefGoogle Scholar
  34. [34]
    Engle R.F. (1982): Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. Inflation, Econometrica 50, 987–1008.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    Fama, E. (1963): Mandelbrot and the stable paretian hypothesis Journal of Business 36, 420–429.CrossRefGoogle Scholar
  36. [36]
    Fama, E. (1965a): The behavior of stock market prices Journal of Business 38, 34–105.CrossRefGoogle Scholar
  37. [37]
    Fama, E. (1965b): Portfolio analysis in a stable paretian market Management Science 11, 404–419.MATHCrossRefGoogle Scholar
  38. [38]
    Fama, E. and K., French (1988): Permanent and temporary components of stock prices, Journal of political Economy, 96, 246–273.CrossRefGoogle Scholar
  39. [39]
    Feller, W. (1966): An introduction to probability theory and its applications II. New York, Wiley.MATHGoogle Scholar
  40. [40]
    Fildes, R. (1992): The evaluation of extrapolative forecasting methods, International Journal of Forecasting 8, 81–98.CrossRefGoogle Scholar
  41. [41]
    Föllmer, H. and D., Sondermann (1986): Hedging of non-redundant contingent claims, in Contributions to Mathematical Economics, W. Hildenbrand and A. Mas Aollell eds. 205–223.Google Scholar
  42. [42]
    Föllmer, H and M., Schweizer (1989): Microeconomic approach to diffusion models for stock prices, Mathematical Finance 3, 1–23.CrossRefGoogle Scholar
  43. [43]
    French, K., G., Schwert and R., Stambaugh (1987): Expected stock returns and volatility, Journal of Financial Economics 19, 3–29.CrossRefGoogle Scholar
  44. [44]
    Gamrowski, B. and S., Rachev (1999): A testable version of the Pareto-stable CAPM, Mathematical and computer modeling 29, 61–81.MATHCrossRefGoogle Scholar
  45. [45]
    Gamrowski, B. and S., Rachev (1994): Stable models in testable asset pricing, in Approximation, probability and related fields. New York: Plenum Press.Google Scholar
  46. [46]
    Giacometti, R. and S., Ortobelli (2003): “Risk measures for asset allocation models”, in Chapter 6 ”New Risk Measures in Investment and Regulation” Elsevier Science Ltd., 69–86.Google Scholar
  47. [47]
    Gilster, J. and W. Lee (1984): The effect of transaction costs and different borrowing and lending rates on the option pricing model: a note, Journal of Finance 39, 1215–1222.CrossRefGoogle Scholar
  48. [48]
    Götzenberger, G., S., Rachev and E., Schwartz (2001): Performance measurements: the stable paretian approach, to appear in Applied Mathematics Reviews, Vol. 1, World Scientific Publ. 2000, 329–406.Google Scholar
  49. [49]
    Grinblatt, M. and S., Titman (1983): Factor pricing in a finite economy, Journal of Financial Economics 12, 497–508.CrossRefGoogle Scholar
  50. [50]
    Hardin, Jr. (1984): Skewed stable variable and processes, Technical Report 79, Center for Stochastic Processes at the University of North Carolina, Chapel Hill.Google Scholar
  51. [51]
    Hartvig, N.V. J.L. Jensen, and J. Pedersen (2001): A class of risk neutral densities with heavy tails, Finance and Stochastics 5, 115–128.MathSciNetMATHCrossRefGoogle Scholar
  52. [52]
    He, H., and N. Pearson (1993): Consumption and portfolio policies with incomplete markets and short sale constraints: The infinite dimensional case, Journal of Economic Theory 54, 259–304.MathSciNetCrossRefGoogle Scholar
  53. [53]
    Hofmann, N., E., Platen, M. Schweizer (1982): Option pricing under incompleteness and stochastic volatility, Technical Report, Department of Mathematics, University of BonnGoogle Scholar
  54. [54]
    Huberman, G. (1982): A simple approach to arbitrage pricing theory, Journal of Economic Theory 28, 183–191.MATHCrossRefGoogle Scholar
  55. [55]
    Hurst, S.H., E. Platen and S. Rachev (1997): Subordinated market index model: a comparison, Financial Engineering and the Japanese Markets 4, 97–124.MATHCrossRefGoogle Scholar
  56. [56]
    Ingersoll, J. Jr. (1987): Theory of financial decision making, Totowa: Rowman&Littlefield.Google Scholar
  57. [57]
    Janicki, A. and A., Weron (1994): Simulation and chaotic behavior of stable stochastic processes, New York: Marcel Dekker.Google Scholar
  58. [58]
    Jobson, J.D. (1992): Applied Multivariate Data Analysis, Heidelberg: Springer-Verlag.MATHCrossRefGoogle Scholar
  59. [59]
    Kallsen, J. and M. Taqqu (1998): Option pricing in ARCH-Type models, Mathematical Finance 8, 13–26.MathSciNetMATHCrossRefGoogle Scholar
  60. [60]
    Kesten H. (1973): Random difference equations and renewal theory for products of random matrices, Acta Mathematica 131, 207–248.MathSciNetMATHCrossRefGoogle Scholar
  61. [61]
    Khindanova, I., S., Rachev and E., Schwartz (2001): Stable modeling of Value at Risk, Mathematical and Computer Modelling 34, 1223–1259.MathSciNetMATHCrossRefGoogle Scholar
  62. [62]
    Klebanov, L.B., Rachev S., Szekely G. (1999): Pre-limit theorems and their applications, Acta Applicandae Mathematicae 58, 159–174.MathSciNetMATHCrossRefGoogle Scholar
  63. [63]
    Klebanov, L.B., Rachev S., Safarian G. (2000): Local pre-limit theorems and their applications to finance, Applied Mathematics Letters 13, 70–73.MathSciNetCrossRefGoogle Scholar
  64. [64]
    Korn R. (1998): Portfolio optimization with strictly positive transaction costs and impulse control, Finance and Stochastics 2, 85–114.MathSciNetMATHCrossRefGoogle Scholar
  65. [65]
    Kraus, A. and R., Litzenberger (1976): Skewness preference and the valuation of risk assets, Journal of Finance 31, 1085–1100.Google Scholar
  66. [66]
    Lamantia, F., S., Ortobelli, and S., Rachev (2003): Value at risk with stable distributed returns, Technical Report, University of Bergamo, to appear in Annals of Operation Research.Google Scholar
  67. [67]
    Leland H. (1985): Option pricing and replication with transaction costs, Journal of Finance 40, 1283–1301.CrossRefGoogle Scholar
  68. [68]
    Levy, H. (1992): Stochastic dominance and expected utility: survey and analysis, Management Science 38, 555–593.MATHCrossRefGoogle Scholar
  69. [69]
    Li, D. and W.L., Ng (2000): Optimal dynamic portfolio selection: multiperiod meanvariance formulation, Mathematical Finance 10, 387–406.MathSciNetMATHCrossRefGoogle Scholar
  70. [70]
    Lo, A. (1991): Long term memory in stock market prices, Econometrica 59, 1279–1313.MATHCrossRefGoogle Scholar
  71. [71]
    Longestaey, J. and P. Zangari (1996). RiskMetrics — Technical Document. J.P. Morgan, Fourth edition, New York.Google Scholar
  72. [72]
    Mandelbrot, B. (1963a): New methods in statistical economics, Journal of Political Economy 71, 421–440.CrossRefGoogle Scholar
  73. [73]
    Mandelbrot, B. (1963b): The variation of certain speculative prices, Journal of Business 26, 394–419.Google Scholar
  74. [74]
    Mandelbrot, B. (1967): The variation of some other speculative prices, Journal of Business 40, 393–413.CrossRefGoogle Scholar
  75. [75]
    Mandelbrot, B. and M., Taylor (1967): On the distribution of stock price differences, Operations Research 15, 1057–1062.CrossRefGoogle Scholar
  76. [76]
    Markowitz, H. (1959): Portfolio selection; efficient diversification of investment, New York: Wiley.Google Scholar
  77. [77]
    McCulloch J.H. (1996): Financial applications of stable distributions, in Handbook of Statistics 14, G.S. Maddala and C.R. Rao eds., Elsevier Science.Google Scholar
  78. [78]
    Mittnik, S. and S., Rachev (1993): Modeling asset returns with alternative stable distributions, Econometric Reviews 12, 261–330.MathSciNetMATHCrossRefGoogle Scholar
  79. [79]
    Mittnik, S., S., Rachev and M., Paolella (1996): Integrated stable GARCH processes, Technical Report Institute of Statistics and Econometrics, Christian Albrechts University at KielGoogle Scholar
  80. [80]
    Milne F. (1988): Arbitrage and diversification a general equilibrium asset economy, Econometrica 56, 815–840.MathSciNetMATHCrossRefGoogle Scholar
  81. [81]
    Mikosh, T., T. Gadrich, C. Kluppelberg, and R.J. Adler (1995): Parameter estimation for ARMA models with infinite variance innovations, The Annals of Statistics, 23, 305–326.MathSciNetCrossRefGoogle Scholar
  82. [82]
    Mikosh, T., and C. Starica (1998): Limit Theory for the sample autocorrelations and extremes of a GARCH(1,1) process, Tecnical Report, University of Groningen.Google Scholar
  83. [83]
    Morton A., and S.R., Pliska (1995): Optimal portfolio management with fixed transaction costs, Mathematical Finance 5, 337–356.MATHCrossRefGoogle Scholar
  84. [84]
    Mossin, J. (1966): Equilibrium in a capital asset market, Econometrica 34, 768–783.CrossRefGoogle Scholar
  85. [85]
    Nolan, J. (1997): Numerical computation of stable densities and distribution functions, Communications in Statistics Stochastic Models 13, 759–774.MathSciNetMATHCrossRefGoogle Scholar
  86. [86]
    Ortobelli, S. (2001): The classification of parametric choices under uncertainty: analysis of the portfolio choice problem, Theory and Decision 51, 297–327.MathSciNetMATHCrossRefGoogle Scholar
  87. [87]
    Ortobelli, S. I., Huber, M., Höchstötter, and S., Rachev (2001): A comparison among Gaussian and non-Gaussian portfolio choice models, Proceeding IFAC SME 2001 Symposium.Google Scholar
  88. [88]
    Ortobelli, S. and S. Rachev (2001): Safety first analysis and stable paretian approach to portfolio choice theory. Mathematical and Computer Modelling 34: 1037–1072.MathSciNetMATHCrossRefGoogle Scholar
  89. [89]
    Ortobelli S, I., Huber, S. Rachev and E. Schwartz (2002): Portfolio choice theory with non-Gaussian distributed returns Chapter 14 in the Handbook of Heavy Tailed Distributions in Finance. North Holland Handbooks of Finance (Series Editor W. T. Ziemba).Google Scholar
  90. [90]
    Ortobelli, S., I., Huber, E. Schwartz (2002): Portfolio selection with stable distributed returns, Mathematical Methods of Operations Research 55, 265–300.MathSciNetMATHCrossRefGoogle Scholar
  91. [91]
    Ortobelli, S., A., Biglova, I., Huber, B., Racheva and S., Stoyanov (2003) Portfolio choice with heavy tailed distributions, Technical Report 22, University of Bergamo, to appear in Journal of Concrete and Applicable Mathematics.Google Scholar
  92. [92]
    Osborne, M. F. (1959): Brownian motion in the stock market, Operation Research 7, 145–173.MathSciNetCrossRefGoogle Scholar
  93. [93]
    Owen, J. and R., Rabinovitch (1983): On the class of elliptical distributions and their applications to the theory of portfolio choice, Journal of Finance 38, 745–752.CrossRefGoogle Scholar
  94. [94]
    Panorska, A., S. Mittnik and S., Rachev (1995): Stable GARCH models for financial time series, Applied Mathematics Letters 815, 33–37.CrossRefGoogle Scholar
  95. [95]
    Praetz, P. (1972): The distribution of share price changes, Journal of Business 45, 49–55.CrossRefGoogle Scholar
  96. [96]
    Press W.H., S.A., Teukolsky, W.T. Vetterling, B.P. Flannery (1992): Numerical recipes in C: the art of scientific computing, 2nd ed. New York: Cambridge University Press.Google Scholar
  97. [97]
    Pyle, D. and S., Turnovsky (1970): Safety first and expected utility maximization in mean standard deviation portfolio selection, Review of Economic Statistics 52, 75–81.CrossRefGoogle Scholar
  98. [98]
    Pyle, D. and S., Turnovsky (1971): Risk aversion in change-constrained portfolio selection, Management Science 18, 218–225MATHCrossRefGoogle Scholar
  99. [99]
    Rachev, S. (1991): Probability metrics and the stability of stochastic models, New York: Wiley.MATHGoogle Scholar
  100. [100]
    Rachev, S.T., S. Ortobelli, S. Schwartz, and E. Schwartz (2002): The problem of optimal asset allocation with stable disctributed returns, Technical Report, University of California at Santa Barbara, to appear in Stochastic Processes and Functional Analysis: Recent Advanced, A volume in honor of M.M. Rao, Marcel Dekker Inc.Google Scholar
  101. [101]
    Rachev, S. H., Xin (1993): Test on association of random variables in the domain of attraction of multivariate stable law, Probability and Mathematical Statistics, 14, 125–141.MathSciNetMATHGoogle Scholar
  102. [102]
    Rachev, S.T., S., Ortobelli, S., and E., Schwartz, (2002): “The problem of optimal asset allocation with stable distributed returns”, Technical Report, University of California at Santa Barbara, “Stochastic Processes and Functional Analysis” Marcel Dekker Editor.Google Scholar
  103. [103]
    Rachev, S., E., Schwartz and I., Khindanova (2000): Stable modeling of credit risk, Technical Report, Anderson School of Management, Department of Finance.Google Scholar
  104. [104]
    Ross, S. (1975): Return, risk and arbitrage, in Studies in Risk and Return; Irwin Friend and J. Bicksler, eds. Cambridge: Mass.: Ballinger Publishing Co.Google Scholar
  105. [105]
    Ross, S. (1976): The arbitrage theory of capital asset pricing, Journal of Economic Theory 13, 341–360.MathSciNetCrossRefGoogle Scholar
  106. [106]
    Ross, S. (1978): Mutual fund separation in financial theory-the separating distributions, Journal of Economic Theory 17, 254–286.MathSciNetMATHCrossRefGoogle Scholar
  107. [107]
    Ross, S. (1987): Arbitrage and martingale with taxation, Journal of Political Economy 95, 371–393.CrossRefGoogle Scholar
  108. [108]
    Rothschild, M. and J., Stiglitz (1970): Increasing risk: I. definition, Journal of Economic Theory 2, 225–243.MathSciNetCrossRefGoogle Scholar
  109. [109]
    Roy. A.D. (1952): Safety-first and the holding of assets, Econometrica 20, 431–449.MATHCrossRefGoogle Scholar
  110. [110]
    Roy, S. (1995): Theory of dynamic portfolio choice for survival under uncertainty, Mathematical Social Sciences 30, 171–194MathSciNetMATHCrossRefGoogle Scholar
  111. [111]
    Samorodnitsky, G. and M.S., Taqqu (1994): Stable non Gaussian random processes: stochastic models with infinite variance, New York: Chapman and Hall.MATHGoogle Scholar
  112. [112]
    Samuelson, P.A. (1955): Brownian motion in stock market, Unpublished Manuscript.Google Scholar
  113. [113]
    Shaked, M. and G., Shanthikumar (1994): Stochastic orders and their applications, New York: Academic Press Inc. Harcourt Brace&Company.MATHGoogle Scholar
  114. [114]
    Simaan, Y. (1993): Portfolio selection and asset pricing-Three parameter framework, Management Science 5, 568–577.CrossRefGoogle Scholar
  115. [115]
    Tesler, L.G. (1955/6): Safety first and hedging, Review of Economic Studies 23, 1–16.CrossRefGoogle Scholar
  116. [116]
    Tokat, Y, S., Rachev and E., Schwartz (2003): The stable non-Gaussian asset allocation: a comparison with the classical Gaussian approach, Journal of Economic Dynamics and Control, 27, 937–969.MATHCrossRefGoogle Scholar
  117. [117]
    Young, M.R. (1998): A minimax portfolio selection rule with linear programming solution, Management Science, 44, 673–683.MATHCrossRefGoogle Scholar
  118. [118]
    Zolatorev, V. M. (1986): One-dimensional stable distributions, Amer. Math. Soc. Transl. of Math. Monographs 65, Providence: RI. Transl. of the original 1983 Russian.Google Scholar

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© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Sergio Ortobelli
    • 1
  • Svetlozar Rachev
    • 2
    • 3
  • Isabella Huber
    • 2
  • Almira Biglova
    • 2
    • 4
  1. 1.Department MSIAUniversity of BergamoBergamoItaly
  2. 2.Institut für Statistik und Mathematische WirtschaftstheorieUniversität KarlsruheKarlsruheGermany
  3. 3.Department of Statistics and Applied ProbabilityUniversity of CaliforniaSanta BarbaraUSA
  4. 4.Ufa State Aviation Technical UniversityRussia

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