We combine singular perturbation techniques with an effective media argument to analyze the general Heston model:
We first show that the marginal probability density Q(T, F) satisfies an effective 1‐d forward equation through O(ε2). We analyze this 1‐d forward equation using an effective media approach. For any given expiry date Tex, this analysis yields effective SABR parameters αeff, ρeff, νeff and shows that the marginal density of the SABR model at Tex matches the marginal density of the Heston model at Tex, again to within O(ε2). Thus, the implied volatility smile σN(Tex, K) for the Heston model is given by the original SABR implied vol formula with these parameters to within O(ε2)