Effective Truncation of a Student’s *t*-Distribution by Truncation of the Chi Distribution in a Chi-Normal Mixture

Author(s)
Daniel T. Cassidy

ABSTRACT

A Student’s*t*-distribution is obtained from a weighted average over the standard deviation of a normal distribution, σ, when 1/σ is distributed as chi. Left truncation at q of the chi distribution in the mixing integral leads to an effectively truncated Student’s *t*-distribution with tails that decay as exp (-q^{2}t^{2}). The effect of truncation of the chi distribution in a chi-normal mixture is investigated and expressions for the pdf, the variance, and the kurtosis of the t-like distribution that arises from the mixture of a left-truncated chi and a normal distribution are given for selected degrees of freedom __<__5. This work has value in pricing financial assets, in understanding the Student’s *t*--distribution, in statistical inference, and in analysis of data.

A Student’s

Cite this paper

D. Cassidy, "Effective Truncation of a Student’s*t*-Distribution by Truncation of the Chi Distribution in a Chi-Normal Mixture," *Open Journal of Statistics*, Vol. 2 No. 5, 2012, pp. 519-525. doi: 10.4236/ojs.2012.25067.

D. Cassidy, "Effective Truncation of a Student’s

References

[1] Student, “The Probable Error of a Mean,” Biometrika, Vol. 6, No. 1, 1908, pp. 1-25.

[2] S. L. Zabell, “On Student’s 1908 Article ‘The Probable Error of a Mean’,” Journal of the American Statistical Association, Vol. 103, No. 481, 2008, pp. 1-7. doi:10.1198/016214508000000030

[3] S. Nadarajah, “Explicit Expressions for Moments of t Order Statistic,” Comptes Rendus de l’Académie des Sciences—Series I, Vol. 345, No. 9, 2007, pp. 523-526. doi:10.1016/j.crma.2007.10.027

[4] P. D. Praetz, “The Distribution of Share Price Changes,” The Journal of Business, Vol. 45, No. 1, 1972, pp. 49-55. doi:10.1086/295425

[5] R. C. Blattberg and N. Gonedes, “A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices,” The Journal of Business, Vol. 47, No. 2, 1974, pp. 244-280. doi:10.1086/295634

[6] E. Platen and R. Sidorowicz, “Empirical Evidence on Student-t Log-Returns of Diversified World Stock Indices,” Research Paper 194, Quantitative Finance Research Centre, University of Technology, Sydney, 2007. http://www.business.uts.edu.au/qfrc/research/research_papers/rp194.pdf

[7] A. Gerig, J. Vicente and M. Fuentes, “Model for Non-Gaussian Intraday Stock Returns,” Physical Review E, Vol. 80, No. 6, 2009, pp. 1-4. doi:10.1103/PhysRevE.80.065102

[8] L. Moriconi, “Delta Hedged Option Valuation with Underlying Non-Gaussian Returns,” Physica A, Vol. 380, No. 1, 2007, pp. 343-350. doi:10.1016/j.physa.2007.01.018

[9] D. T. Cassidy, M. J. Hamp and R. Ouyed, “Pricing European Options with a Log Student’s t-Distribution: A Gosset Formula,” Physica A, Vol. 389, No. 24, 2010, pp. 5736- 5748. doi:10.1016/j.physa.2010.08.037

[10] J.-P. Bouchaud and D. Sornette, “The Black-Scholes Option Pricing Problem in Mathematical Finance: Generalization and Extensions for a Large Class of Stochastic Processes,” Journal de Physique I (France), Vol. 4, No. 6, 1994, pp. 863-881. doi:10.1051/jp1:1994233

[11] J. L. McCauley, G. H. Gunaratne and K. E. Bassler, “Martingale Option Pricing,” Physica A, Vol. 380, No. 1, 2007, pp. 351-356. doi:10.1016/j.physa.2007.02.038

[12] G. C. Lim, G. M. Martin and V. L. Martin, “Pricing Currency Options in the Presence of Time-Varying Volatility and Non-Normalities,” Journal of Multinational Financial Management, Vol. 16, No. 3, 2006, pp. 291-314. doi:10.1016/j.mulfin.2005.08.004

[13] J. N. Lye and V. L. Martin, “Robust Estimation, NonNormalities and Generalized Exponential Distributions,” Journal of the American Statistical Association, Vol. 88, No. 421, 1993, pp. 261-267. doi:10.2307/2290721

[1] Student, “The Probable Error of a Mean,” Biometrika, Vol. 6, No. 1, 1908, pp. 1-25.

[2] S. L. Zabell, “On Student’s 1908 Article ‘The Probable Error of a Mean’,” Journal of the American Statistical Association, Vol. 103, No. 481, 2008, pp. 1-7. doi:10.1198/016214508000000030

[3] S. Nadarajah, “Explicit Expressions for Moments of t Order Statistic,” Comptes Rendus de l’Académie des Sciences—Series I, Vol. 345, No. 9, 2007, pp. 523-526. doi:10.1016/j.crma.2007.10.027

[4] P. D. Praetz, “The Distribution of Share Price Changes,” The Journal of Business, Vol. 45, No. 1, 1972, pp. 49-55. doi:10.1086/295425

[5] R. C. Blattberg and N. Gonedes, “A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices,” The Journal of Business, Vol. 47, No. 2, 1974, pp. 244-280. doi:10.1086/295634

[6] E. Platen and R. Sidorowicz, “Empirical Evidence on Student-t Log-Returns of Diversified World Stock Indices,” Research Paper 194, Quantitative Finance Research Centre, University of Technology, Sydney, 2007. http://www.business.uts.edu.au/qfrc/research/research_papers/rp194.pdf

[7] A. Gerig, J. Vicente and M. Fuentes, “Model for Non-Gaussian Intraday Stock Returns,” Physical Review E, Vol. 80, No. 6, 2009, pp. 1-4. doi:10.1103/PhysRevE.80.065102

[8] L. Moriconi, “Delta Hedged Option Valuation with Underlying Non-Gaussian Returns,” Physica A, Vol. 380, No. 1, 2007, pp. 343-350. doi:10.1016/j.physa.2007.01.018

[9] D. T. Cassidy, M. J. Hamp and R. Ouyed, “Pricing European Options with a Log Student’s t-Distribution: A Gosset Formula,” Physica A, Vol. 389, No. 24, 2010, pp. 5736- 5748. doi:10.1016/j.physa.2010.08.037

[10] J.-P. Bouchaud and D. Sornette, “The Black-Scholes Option Pricing Problem in Mathematical Finance: Generalization and Extensions for a Large Class of Stochastic Processes,” Journal de Physique I (France), Vol. 4, No. 6, 1994, pp. 863-881. doi:10.1051/jp1:1994233

[11] J. L. McCauley, G. H. Gunaratne and K. E. Bassler, “Martingale Option Pricing,” Physica A, Vol. 380, No. 1, 2007, pp. 351-356. doi:10.1016/j.physa.2007.02.038

[12] G. C. Lim, G. M. Martin and V. L. Martin, “Pricing Currency Options in the Presence of Time-Varying Volatility and Non-Normalities,” Journal of Multinational Financial Management, Vol. 16, No. 3, 2006, pp. 291-314. doi:10.1016/j.mulfin.2005.08.004

[13] J. N. Lye and V. L. Martin, “Robust Estimation, NonNormalities and Generalized Exponential Distributions,” Journal of the American Statistical Association, Vol. 88, No. 421, 1993, pp. 261-267. doi:10.2307/2290721