Revisiting the Implied Remaining Variance framework of Carr and Sun (2014): Locally consistent dynamics and sandwiched martingales

Implied volatility is at the very core of modern finance, notwithstanding standard option pricing models continue to derive option prices starting from the joint dynamics of the underlying asset price and the spot volatility. These models often cause difficulties: no closed formulas for prices, demanding calibration techniques, unclear maps between spot and implied volatility. Inspired by the practice of using implied volatility as quoting system for option prices, models for the joint dynamics of the underlying asset price and the implied volatility have been proposed to replace standard option pricing models. Starting from Carr and Sun (2014), we develop a framework based on the Implied Remaining Variance where minimal conditions for absence of arbitrage are identified, and smile bubbles are dealt with. The key concepts arising from the new IRV framework are those of locally consistent dynamics and sandwiched martingale. Within the new IRV framework, the results of Schweizer and Wissel (2008b) are reformulated, while those of El Amrani, Jacquier and Martini (2021) are independently derived.