On the RND under Hestons stochastic volatility model

We consider Hestons (1993) stochastic volatility model for valuation of European options to which (semi) closed form solutions are available and are given in terms of characteristic functions. We prove that the class of scale-parameter distributions with mean being the forward spot price satisfies Hestons solution. Thus, we show that any member of this class could be used for the direct risk-neutral valuation of the option price under Hestons SV model. In fact, we also show that any RND with mean being the forward spot price that satisfies Hestons option valuation solution, must be a member of a scale-family of distributions in that mean. As particular examples, we show that one-paramet