Models for stock returns

Abstract

Historically, the normal variance model has been used to describe stock return distributions. This model is based on taking the conditional stock return distribution to be normal with its variance itself being a random variable. The form of the actual stock return distribution will depend on the distribution for the variance. In practice, the distributions chosen for the variance appear to be very limited. In this note, we derive a comprehensive collection of formulas for the actual stock return distribution, covering some sixteen flexible families. The corresponding estimation procedures are derived by the method of moments and the method of maximum likelihood. We feel that this work could serve as a useful reference and lead to improved modelling with respect to stock market returns.

Keywords:

• Beta function
• Gamma function
• Generalized hypergeometric function
• Incomplete gamma function
• Kummer function
• Modified Bessel function
• Normal distribution
• Stock market returns

Acknowledgements

The author would like to thank the Editors-in-Chief, the Managing Editor and the three referees for carefully reading the paper and for their comments which greatly improved the paper.