Previous literature has identified an effect, dubbed the Zumbach effect, that is nonzero empirically but conjectured to be zero in any conventional stochastic volatility model. Essentially this effect corresponds to the property that past squared returns forecast future volatilities better than...

We use the Itô Decomposition Formula (see Alòs (2012)) to express certain conditional expectations as exponentials of iterated integrals. As one application, we compute an exact formal expression for the leverage swap for any stochastic volatility model expressed in forward variance form. As...

This work is concerned with forest and cumulant type expansions of general random variables on a filtered probability spaces. We establish a “broken exponential martingale” expansion that generalizes and unifies the exponentiation result of Alòs, Gatheral, and Radoičić ́ (SSRN'17; [AGR20])...

In this paper it is proved that the Black-Scholes implied volatility satisfies a second order non-linear partial differential equation. The obtained PDE is then used to construct an algorithm for fast and accurate polynomial approximation for Black-Scholes implied volatility that improves on the...

We present a simple rational approximation to the solution of the rough Heston Riccati equation valid in a region of its domain relevant to option valuation. Pricing under rough Heston using this approximation is both fast and very accurate

We study an expansion of the cumulative distribution function of the standard normal random variable that results in a family of closed form approximations that converge at 0. One member of the family that has only five explicit constants offers the absolute error of 5.79 10^{-6} across the...

We show that an explicit approximate implied volatility formula can be obtained from a Black–Scholes formula approximation that is 2% accurate. The relative error of the approximate implied volatility is uniformly bounded for options with any moneyness and with arbitrary large or small option...

We introduce a closed form approximation for the implied volatility of ATM-forward options. The relative error of this approximation is uniformly bounded for all option maturities and implied volatilities. The approximation is extremely precise, having relative error less than 10−6 for all...

We establish bounds on Black-Scholes implied volatility that improve on the uniform bounds previously derived by Tehranchi. Our upper bound is uniform, while the lower bound holds for most options likely to be encountered in practical applications. We further demonstrate the practical...