Option Pricing when Underlying Process is Unknown

This paper proves that a simultaneously delta-neutral and gamma-neutral position can be established with the underlying asset and two options on the same asset, without any assumption on the underlying process of the asset price. This hedging strategy leads to the same fundamental partial differential equation as derived by Black and Scholes (1973), except that the variance function is the market price of convexity, which is not necessarily the true variance function of the underlying process. This result explains why the implied volatility can be persistently different from the historical volatility of the underlying asset price. In addition, the delta-gamma-neutral strategy indicates that the observed option prices that exhibit volatility smile, smirk, or term structure, are all mispriced.