From Volatility Smile to Risk Neutral Probability and Closed Form Solution of Local Volatility Function

The risk neutral measure is identified as a symmetric location-scale family of distribution in the local regime of the λ model. A partial differential equation is derived as the transformation between the implied volatility surface and such risk neutral probability. Given a well-interpolated volatility surface from market data, the risk neutral probability and the implied rate of growth can be extracted in a model-free manner. On the other hand, assuming a priori knowledge on the rate of growth and the risk neutral probability being an symmetric λ distribution, the closed form solution of the local volatility function be derived from Fokker-Planck equation. Based on such solution, I discuss possible forms of diffusion process, implement a leptokurtic extension of Weiner process, and discover a mean-reverting process that bridges between the Ornstein-Uhlenbeck process and Bessel process.