We consider an auction market in which market makers fill the order book during a given time period while some other investors send market orders. We define the clearing price of the auction as the price maximizing the exchanged volume at the clearing time according to the supply and demand of...

Market impact is the link between the volume of a (large) order and the price move during and after the execution of this order. We show that under no-arbitrage assumption, the market impact function can only be of power-law type. Furthermore, we prove that this implies that the macroscopic...

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...

Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. This leads us...

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...