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Describing n-day returns with Student’s t-distributions

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  • Cassidy, Daniel T.

Abstract

Prices for European call options can be calculated for returns that follow a Student’s t-distribution if the t-distribution is truncated or if the value of the asset is capped. The distributions for n-fold convolution of a Student’s t-distribution and a truncated Student’s t-distribution, both with ν=3, are considered in this work. It is shown that a truncated Student’s t-distribution under n-fold self-convolution becomes normal-like whereas a Student’s t-distribution retains the fat tails of the original distribution under n-fold self-convolution. These results can be used to explain the development of the distribution of n-day returns from a truncated Student’s t-distribution for the daily returns to normal as n increases from 1 to 10 or 100. A truncated Student’s t-distribution with 3±0.5 degrees of freedom fits the daily returns of the DJIA and S&P 500 indices.

Suggested Citation

  • Cassidy, Daniel T., 2011. "Describing n-day returns with Student’s t-distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(15), pages 2794-2802.
    • Handle: RePEc:eee:phsmap:v:390:y:2011:i:15:p:2794-2802
    DOI: 10.1016/j.physa.2011.03.019
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    Cited by:

    1. Wang, Xiao-Tian & Li, Zhe & Zhuang, Le, 2017. "European option pricing under the Student’s t noise with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 469(C), pages 848-858.
    2. Till Massing, 2019. "What is the best Lévy model for stock indices? A comparative study with a view to time consistency," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 33(3), pages 277-344, September.
    3. Till Massing, 2018. "Simulation of Student–Lévy processes using series representations," Computational Statistics, Springer, vol. 33(4), pages 1649-1685, December.
    4. Lasko Basnarkov & Viktor Stojkoski & Zoran Utkovski & Ljupco Kocarev, 2019. "Option Pricing With Heavy-Tailed Distributions Of Logarithmic Returns," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(07), pages 1-35, November.
    5. Daniel T. Cassidy & Michael J. Hamp & Rachid Ouyed, 2013. "Log Student’s t -distribution-based option sensitivities: Greeks for the Gosset formulae," Quantitative Finance, Taylor & Francis Journals, vol. 13(8), pages 1289-1302, July.
    6. Massing, Till & Ramos, Arturo, 2021. "Student’s t mixture models for stock indices. A comparative study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 580(C).

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