  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.
