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

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.