Journal of Advances in Economics and Finance

Download PDF (594.1 KB) PP. 79 - 90 Pub. Date: May 10, 2019

DOI: https://dx.doi.org/10.22606/jaef.2019.42004

• Ivivi Joseph Mwaniki^*
  School of Mathematics, Division of statistics and actuarial Science, University of Nairobi, Kenya

A search for a distribution which adequately describes the dynamics of log returns has been a subject of study for many years. Empirical evidence has resulted in stylized facts of returns. Arguably, in this study, the three components of returns, mean equation part, the changing variance part and the resulting residuals are determined and their corresponding parameters estimated within the proposed framework. Spectral density analysis is used to trace the seasonality component inherent in the standardized residuals. Empirical data sets from eight different indexes and common stock are applied to the model, and results tabulated in support of the resulting framework.

TGARCH, student t distribution, spectral density, seasonality.

[1] S. J. Press, “A compound events model for security prices,” Journal of business, pp. 317–335, 1967.

[2] P. K. Clark, “A subordinated stochastic process model with finite variance for speculative prices,” Econometrica: journal of the Econometric Society, pp. 135–155, 1973.

[3] R. C. Merton, “Option pricing when underlying stock returns are discontinuous,” Journal of financial economics, vol. 3, no. 1-2, pp. 125–144, 1976.

[4] S. Beckers, “A note on estimating the parameters of the diffusion-jump model of stock returns,” Journal of Financial and Quantitative Analysis, vol. 16, no. 01, pp. 127–140, 1981.

[5] C. A. Ball and W. N. Torous, “On jumps in common stock prices and their impact on call option pricing,” The Journal of Finance, vol. 40, no. 1, pp. 155–173, 1985.

[6] P. Jorion, “On jump processes in the foreign exchange and stock markets,” Review of Financial Studies, vol. 1, no. 4, pp. 427–445, 1988.

[7] R. F. Engle, “Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation,” Econometrica: Journal of the Econometric Society, pp. 987–1007, 1982.

[8] T. Bollerslev, “Generalized autoregressive conditional heteroskedasticity,” Journal of econometrics, vol. 31, no. 3, pp. 307–327, 1986.

[9] R. Cont, “Empirical properties of asset returns: stylized facts and statistical issues,” Quantitave finance, vol. 1, pp. 223–236, 2001.

[10] T. H. Rydberg, “Realistic statistical modelling of financial data,” International Statistical Review/Revue Internationale de Statistique, pp. 233–258, 2000.

[11] E. Eberlein and U. Keller, “Hyperbolic distributions in finance,” Bernoulli, pp. 281–299, 1995.

[12] O. E. Barndorff-Nielsen, “Normal inverse gaussian distributions and stochastic volatility modelling,” Scandinavian Journal of statistics, vol. 24, no. 1, pp. 1–13, 1997.

[13] D. A. Hsieh, “The statistical properties of daily foreign exchange rates: 1974–1983,” Journal of international economics, vol. 24, no. 1-2, pp. 129–145, 1988.

[14] P. Christoffersen and K. Jacobs, “Which garch model for option valuation?” Management science, vol. 50, no. 9, pp. 1204–1221, 2004.

[15] B. Eraker, M. Johannes, and N. Polson, “The impact of jumps in volatility and returns,” The Journal of Finance, vol. 58, no. 3, pp. 1269–1300, 2003.

[16] D. S. Bates, “Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options,” Review of financial studies, vol. 9, no. 1, pp. 69–107, 1996.

[17] J. Pan, “The jump-risk premia implicit in options: Evidence from an integrated time-series study,” Journal of financial economics, vol. 63, no. 1, pp. 3–50, 2002.

[18] D. Duffie, J. Pan, and K. Singleton, “Transform analysis and asset pricing for affine jump-diffusions,” Econometrica, vol. 68, no. 6, pp. 1343–1376, 2000.

[19] W. H. Chan and J. M. Maheu, “Conditional jump dynamics in stock market returns,” Journal of Business & Economic Statistics, vol. 20, no. 3, pp. 377–389, 2002.

[20] L. Stentoft, “American option pricing using garch models and the normal inverse gaussian distribution,” Journal of Financial Econometrics, vol. 6, no. 4, pp. 540–582, 2008.

[21] I. J. Mwaniki, “Modeling heteroscedastic skewed and leptokurtic returns in discrete time,” Journal of Applied finance and banking, vol. 9, no. 5, pp. 1–14, 2019.

[22] G. Bakshi, C. Cao, and Z. Chen, “Empirical performance of alternative option pricing models,” The Journal of finance, vol. 52, no. 5, pp. 2003–2049, 1997.

[23] B. Dumas, J. Fleming, and R. E. Whaley, “Implied volatility functions: Empirical tests,” The Journal of Finance, vol. 53, no. 6, pp. 2059–2106, 1998.

[24] S. R. Das and R. K. Sundaram, “Of smiles and smirks: A term structure perspective,” Journal of financial and quantitative analysis, vol. 34, no. 02, pp. 211–239, 1999.

[25] J. M. Maheu and T. H. McCurdy, “News arrival, jump dynamics, and volatility components for individual stock returns,” The Journal of Finance, vol. 59, no. 2, pp. 755–793, 2004.

[26] R. F. Engle and V. K. Ng, “Measuring and testing the impact of news on volatility,” The journal of finance, vol. 48, no. 5, pp. 1749–1778, 1993.

[27] L. R. Glosten, R. Jagannathan, and D. E. Runkle, “On the relation between the expected value and the volatility of the nominal excess return on stocks,” The journal of finance, vol. 48, no. 5, pp. 1779–1801, 1993.

[28] J.-M. Zakoian, “Threshold heteroskedastic models,” Journal of Economic Dynamics and control, vol. 18, no. 5, pp. 931–955, 1994.

[29] D. B. Madan and K. Wang, “Nonrandom price movements,” Finance Research Letters, 2016.