A Note on the Cumulative Distribution Function of the Pearson Type IV Distribution for Financial Applications

Abstract

The cumulative distribution function of the Pearson type IV distribution is of complex form and includes a complex hypergeometric function. Although the mathematical form is complex, the resulting imaginary part is actually of the order which is attributed to the series summation of the hypergeometric function, where each term is complex, but summing up terms the complex part converges slowly to zero. This requires the use of software that can support complex hypergeometric functions, and several terms have to be summed up to achieve the negligible imaginary part. In this note we examine the transformation of the complex cumulative distribution to other hypergeometric functions that have a real contribution in each summing term, and the recurrence relation for the calculation of the expression is provided.