Stochastic Calculus of Heston’s Stochastic-Volatility Model

The Heston stochastic-volatility model is a square-root diffusion model for the stochasticvariance. It gives rise to a singular diffusion for the distribution as noted by Feller (1951). Hence, there is an order constraint on the relationship between the limit that the variance goes to zero and the limit that time-step goes to zero, so that any non-trivial transformation of the Heston model leads to a transformed diffusion in the Itô Calculus. Several transformations are introduced that lead to proper diffusions and preservation of the nonnegativity of the variance in a perfect-square form. An exact, nonsingular solution is found for a special combination of the Heston stochastic volatility parameters. A computationally simple and practical simulation recipe of solutions of the Heston model is introduced that is consistent with the proper diffusion scaling for the time-step and the variance when both are small.