Bona studied the distribution of statistics on Stirling permutations. He showed that the generating polynomial of descents has only real roots. We generalize this result by applying the recent theory of stable polynomials developed by Borcea and Branden. We introduce a suitable multivariate generating polynomial $f(z_1, \dots, z_m)$, and prove that it is stable, in the sense that whenever the $z_i$'s lie on the upper half-plane $f$ does not vanish. Connections to multivariate Eulerian polynomials (Branden), Polya urns (Janson) and the recent proof of MCP conjecture (Branden, Haglund, Visontai, Wagner) will be presented as well.