Empirical properties and modeling of high frequency assets.

Summary This thesis explores theoretically and empirically some aspects of the formation and evolution of financial asset prices observed in high frequency. We begin by studying the joint dynamics of the option and its underlying. Since high-frequency data make the realized volatility process of the underlying observable, we investigate whether this information is used to price options. We find that the market does not exploit it. Stochastic volatility models are therefore to be considered as reduced-form models. Nevertheless, this study allows us to test the relevance of an empirical hedging measure that we call effective delta. It is the slope of the regression of the option price returns on those of the underlying. It provides a fairly satisfactory indicator of hedging that is independent of any modeling. For price dynamics, we turn in the following chapters to more explicit models of the market microstructure. One of the characteristics of market activity is its clustering. Hawkes processes, which are point processes with this characteristic, therefore provide an adequate mathematical framework for the study of this activity. The Markovian representation of these processes, as well as their affine character when the kernel is exponential, allow us to use the powerful analytical tools of the infinitesimal generator and Dynkin's formula to compute various quantities related to them, such as the moments or autocovariances of the number of events on a given interval. We start with a one-dimensional framework, simple enough to illuminate the approach, but rich enough to allow applications such as grouping order arrival times, predicting future market activity knowing past activity, or characterizing unusual, but nevertheless observed, forms of signature plot where the measured volatility decreases as the sampling frequency increases. Our calculations also allow us to make the calibration of Hawkes processes instantaneous by using the method of moments. The generalization to the multidimensional case then allows us to capture, with clustering, the mean reversion phenomenon that also characterizes the market activity observed at high frequency. General formulas for the signature plot are then obtained and allow us to link its shape to the relative importance of clustering or mean reversion. Our calculations also allow us to obtain the explicit form of the volatility associated with the diffusive limit, connecting the microscopic level dynamics to the volatility observed macroscopically, for example on a daily scale. Moreover, the modeling of buying and selling activities by Hawkes processes allows to compute the impact of a meta order on the asset price. We then find and explain the concave shape of this impact as well as its temporal relaxation. The analytical results obtained in the multidimensional case then provide the appropriate framework for the study of correlation. We then present general results on the Epps effect, as well as on the formation of the correlation and the lead lag.