Positive volatility simulation in the Heston model

In the Heston stochastic volatility model, the transition probability of the variance process can be represented by a non-central chi-square density. We focus on the case when the number of degrees of freedom is small and the zero boundary is attracting and attainable, typical in foreign exchange markets. We prove a new representation for this density based on sums of powers of generalized Gaussian random variables. Further we prove Marsaglia's polar method extends to this distribution, providing an exact method for generalized Gaussian sampling. The advantages are that for the mean-reverting square-root process in the Heston model and Cox-Ingersoll-Ross model, we can generate samples from the true transition density simply, efficiently and robustly.