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.A Student’s t-distribution arises as a mixture of a normal pdf with a variance that is distributed as an inverse chi squared distribution. Values of infinity and of zero are not observed for the volatility of returns. Thus the distribution of the variance (i.e., the square of the volatility) should be truncated to agree with observations. It is shown that the mixture of a normal pdf with a variance that is distributed as a truncated inverse chi squared distribution yields an effectively truncated Student’s t-distribution, for which the tails develop as exp(−t2) for large t. This exponential development of the tails effectively truncates the t-distribution.