Dealing with the Inventory Risk. A solution to the market making problem

Market makers continuously set bid and ask quotes for the stocks they have under consideration. Hence they face a complex optimization problem in which their return, based on the bid-ask spread they quote and the frequency at which they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, we consider a stochastic control problem similar to the one introduced by Ho and Stoll and formalized mathematically by Avellaneda and Stoikov. The market is modeled using a reference price $S_t$ following a Brownian motion with standard deviation $\sigma$, arrival rates of buy or sell liquidity-consuming orders depend on the distance to the reference price $S_t$ and a market maker maximizes the expected utility of its P&L over a finite time horizon. We show that the Hamilton-Jacobi-Bellman equations associated to the stochastic optimal control problem can be transformed into a system of linear ordinary differential equations and we solve the market making problem under inventory constraints. We also shed light on the asymptotic behavior of the optimal quotes and propose closed-form approximations based on a spectral characterization of the optimal quotes

Understanding the stakes of high frequency trading

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Widening of the genetic and clinical spectrum of Lamb-Shaffer syndrome, a neurodevelopmental disorder due to SOX5 haploinsufficiency

Purpose Lamb-Shaffer syndrome (LAMSHF) is a neurodevelopmental disorder described in just over two dozen patients with heterozygous genetic alterations involving SOX5, a gene encoding a transcription factor regulating cell fate and differentiation in neurogenesis and other discrete developmental processes. The genetic alterations described so far are mainly microdeletions. The present study was aimed at increasing our understanding of LAMSHF, its clinical and genetic spectrum, and the pathophysiological mechanisms involved. Methods Clinical and genetic data were collected through GeneMatcher and clinical or genetic networks for 41 novel patients harboring various types ofSOX5 alterations. Functional consequences of selected substitutions were investigated. Results Microdeletions and truncating variants occurred throughout SOX5. In contrast, most missense variants clustered in the pivotal SOX-specific high-mobility-group domain. The latter variants prevented SOX5 from binding DNA and promoting transactivation in vitro, whereas missense variants located outside the high-mobility-group domain did not. Clinical manifestations and severity varied among patients. No clear genotype-phenotype correlations were found, except that missense variants outside the high-mobility-group domain were generally better tolerated. Conclusions This study extends the clinical and genetic spectrum associated with LAMSHF and consolidates evidence that SOX5 haploinsufficiency leads to variable degrees of intellectual disability, language delay, and other clinical features.Genetics of disease, diagnosis and treatmen