It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to reproduce the behavior of both historical and implied volatilities. However, due to the non-Markovian nature of the fractional Brownian motion, they raise new issues when it comes to derivatives pricing. Using an original link between nearly unstable Hawkes processes and fractional volatility models, we compute the characteristic function of the log-price in rough Heston models. In the classical Heston model, the characteristic function is expressed in terms of the solution of a Riccati equation.Here, we show that rough Heston models exhibit quite a similar structure, the Riccati equation being replaced by a fractional Riccati equation. K E Y W O R D Sfractional Brownian motion, fractional Riccati equation, Hawkes processes, limit theorems, rough Heston models, rough volatility models Here the parameters , , 0 , and are positive, and and are two Brownian motions with correlation coefficient , that is, ⟨dW , dB ⟩ = dt.The popularity of this model is probably due to three main reasons:Mathematical Finance. 2019;29:3-38. wileyonlinelibrary.com/journal/mafi

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 to adopt the fractional stochastic volatility (FSV) model of Comte and Renault [16]. We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, H < 1/2. We demonstrate that our RFSV model is remarkably consistent with financial time series data; one application is that it enables us to obtain improved forecasts of realized volatility. Furthermore, we find that although volatility is not long memory in the RFSV model, classical statistical procedures aiming at detecting volatility persistence tend to conclude the presence of long memory in data generated from it. This sheds light on why long memory of volatility has been widely accepted as a stylized fact. Finally, we provide a quantitative market microstructurebased foundation for our findings, relating the roughness of volatility to high frequency trading and order splitting.

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We consider the model {eqnarray*}y=X\theta^*+\xi, Z=X+\Xi,{eqnarray*} where the random vector $y\in\mathbb{R}^n$ and the random $n\times p$ matrix $Z$ are observed, the $n\times p$ matrix $X$ is unknown, $\Xi$ is an $n\times p$ random noise matrix, $\xi\in\mathbb{R}^n$ is a noise independent of $\Xi$, and $\theta^*$ is a vector of unknown parameters to be estimated. The matrix uncertainty is in the fact that $X$ is observed with additive error. For dimensions $p$ that can be much larger than the sample size $n$, we consider the estimation of sparse vectors $\theta^*$. Under matrix uncertainty, the Lasso and Dantzig selector turn out to be extremely unstable in recovering the sparsity pattern (i.e., of the set of nonzero components of $\theta^*$), even if the noise level is very small. We suggest new estimators called matrix uncertainty selectors (or, shortly, the MU-selectors) which are close to $\theta^*$ in different norms and in the prediction risk if the restricted eigenvalue assumption on $X$ is satisfied. We also show that under somewhat stronger assumptions, these estimators recover correctly the sparsity pattern.Comment: Published in at http://dx.doi.org/10.1214/10-AOS793 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Through the analysis of a dataset of ultra high frequency order book updates, we introduce a model which accommodates the empirical properties of the full order book together with the stylized facts of lower frequency financial data. To do so, we split the time interval of interest into periods in which a well chosen reference price, typically the midprice, remains constant. Within these periods, we view the limit order book as a Markov queuing system. Indeed, we assume that the intensities of the order flows only depend on the current state of the order book. We establish the limiting behavior of this model and estimate its parameters from market data. Then, in order to design a relevant model for the whole period of interest, we use a stochastic mechanism that allows to switch from one period of constant reference price to another. Beyond enabling to reproduce accurately the behavior of market data, we show that our framework can be very useful for practitioners, notably as a market simulator or as a tool for the transaction cost analysis of complex trading algorithms.