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2012 | 32 | 1-2 | 47-58
Tytuł artykułu

Normality assumption for the log-return of the stock prices

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Abstrakty
EN
The normality of the log-returns for the price of the stocks is one of the most important assumptions in mathematical finance. Usually is assumed that the price dynamics of the stocks are driven by geometric Brownian motion and, in that case, the log-return of the prices are independent and normally distributed. For instance, for the Black-Scholes model and for the Black-Scholes pricing formula [4] this is one of the main assumptions. In this paper we will investigate if this assumption is verified in the real world, that is, for a large number of company stock prices we will test the normality assumption for the log-return of their prices. We will apply the Kolmogorov-Smirnov [10, 5], the Shapiro-Wilks [17, 16] and the Anderson-Darling [1, 2] tests for normality to a wide number of company prices from companies quoted in the Nasdaq composite index.
Twórcy
  • Department of Mathematics, CMA of FCT/UNL
Bibliografia
  • [1] T.W. Anderson and D.A. Darling, Asymptotic theory of certain goodness-of-fit criteria based on stochastic processes, Ann. Math. Statist. 23 (1952) 193-212. doi: 10.1214/aoms/1177729437
  • [2] T.W. Anderson and D.A. Darling, A test of goodness of fit, Journal of the American Statistical Association 49 (268) (1954) 765-769. doi: 10.1080/01621459.1954.10501232
  • [3] Z.W. Birnbaum, Numerical tabulation of the distribution of Kolmogorov's statistic for finite sample size, Journal of the American Statistical Association 47 (259) (1952) 425-441. doi: 10.1080/01621459.1952.10501182
  • [4] T. Bjork, Arbitrage Theory in Continuous Time (Oxford University Press, 1998). doi: 10.1093/0198775180.001.0001
  • [5] D.A. Darling, The Kolmogorov-Smirnov, Cramer-von Mises tests, Ann. Math. Statist. 28 (4) (1957) 823-838. doi: 10.1214/aoms/1177706788
  • [6] D.E.A. Giles, A saddlepoint approximation to the distribution function of the Anderson-Darling test statistic, Communications in Statistics - Simulation and Computation 30 (4) (2001) 899-905. doi: 10.1081/SAC-100107787
  • [7] H.L. Harter, Expected values of normal order statistics, Biometrika 48 (1-2) (1961) 151-165. doi: 10.1093/biomet/48.1-2.151
  • [8] I. Karatzas and Shreve, Brownian Motion and Stochastic Calculus (Springer-Verlag, 2000).
  • [9] P.A. Lewis, Distribution of the Anderson-Darling statistic, Ann. Math. Statist. 32 (4) (1961) 1118-1124. doi: 10.1214/aoms/1177704850
  • [10] H.W. Lilliefors, On the Kolmogorov-Smirnov test for normality with mean and variance unknown, Journal of the American Statistical Association 62 (318) (1967) 399-402. doi: 10.1080/01621459.1967.10482916
  • [11] F.J. Massey, The Kolmogorov-Smirnov test for goodness of fit, Journal of the American Statistical Association 46 (253) (1951) 68-78. doi: 10.1080/01621459.1951.10500769
  • [12] B. Oksendall, Stochastic Differential Equations (Springer-Verlag, 1998). doi: 10.1007/978-3-662-03620-4
  • [13] J.P. Royston, An extension of Shapiro and Wilk's W test for normality to large samples, Journal of the Royal Statistical Society. Series C (Applied Statistics) 31 (1982) 115-124.
  • [14] J.P. Royston, A simple method for evaluating the Shapiro-Francia W' test of non-normality, Journal of the Royal Statistical Society. Series D (The Statistician) 32 (1983) 287-300. doi: 10.2307/2987935
  • [15] K. Sarkadi, The consistency of the Shapiro-Francia test, Biometrika 62 (2) (1975) 445-450.
  • [16] S.S. Shapiro and R.S. Francia, An approximate analysis of variance test for normality, Journal of the American Statistical Association 67 (1972) 215-216. doi: 10.1080/01621459.1972.10481232
  • [17] S.S. Shapiro and M. Wilk, An analysis of variance test for normality (Complete Samples), Biometrika 52 (1965) 591-611.
  • [18] M.A. Stephens, Use of the Kolmogorov-Smirnov, Cramer-Von Mises and related statistics without extensive tables, Journal of the Royal Statistical Society. Series B (Methodological) 32 (1) (1970) 115-122.
  • [19] M.A. Stephens, EDF statistics for goodness of fit and some comparisons, Journal of the American Statistical Association 69 (347) (1974) 730-737. doi: 10.1080/01621459.1974.10480196
  • [20] M.A. Stephens, Asymptotic results for goodness-of-fit statistics with unknown parameters, Annals of Statistics 4 (2) (1976) 357-369. doi: 10.1214/aos/1176343411
  • [21] M.A. Stephens, Goodness of Fit with Special Reference to Tests for Exponentiality, Technical Report No. 262 (Department of Statistics, Stanford University, Stanford, CA, 1977).
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1143
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