A small-time Edgeworth expansion of the density of an asset price is given under a general stochastic volatility model, from which asymptotic expansions of put option prices and at-the-money implied volatilities follow. A limit theorem for at-the-money implied volatility skew and curvature is...

Rough volatility models are known to fit the volatility surface remarkably well with very few parameters. On the other hand, the classical Heston model is highly tractable allowing for fast calibration. We present here the rough Heston model which offers the best of both worlds. Even better, we...

We consider an exchange who wishes to set suitable make-take fees to attract liquidity on its platform. Using a principal-agent approach, we are able to describe in quasi-explicit form the optimal contract to propose to a market maker. This contract depends essentially on the market maker...

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...

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 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])...