Ask HN: Probability Distribution of Stock Returns

1 points by billziss 2 years ago | 2 comments

Is there a modern theory for the probability distribution of stock returns?

It is relatively easy to deduce that under idealized conditions stock returns follow a log normal distribution. One arrives at this by considering the product of ratios of prices ("stock returns"), applying a natural logarithm to convert the product into a sum and then applying the Central Limit Theorem under the condition that the ratios are iid (independent and identically distributed) and have finite mean and variance.

The problem is of course that we cannot just assume that returns are iid or that they have finite variance. So I am seeking alternative theories that try to address these shortcomings.

I am aware of Mandelbrot's Multifractal Model of Asset Returns. Is this considered SOA in the field? Is there something else that is considered a better model or easier to work with?

nabla9 2 years ago |

For synthetic distributions Mandelbrot's Multifractal is good enough.

You may consider using real historical distributions derived from real data. You can record different distributions from different time periods and different economic conditions.

billziss 2 years ago | |

Thanks for your answer.

I am using real data of course, but I am also seeking a deeper understanding and ideally one based on first principles. This is why I asked this question.

It seems to me that if we could correctly characterize the dependence of the returns we could perhaps arrive at the correct theoretical answer that also coincides with practical observation. Mandelbrot almost gets there, but his use of multifractal trading time seems brilliant but also somewhat arbitrary (at least in the papers of his I read).