Optimal Trade Execution Under Endogenous Pressure to Liquidate: Theory and Numerical Solutions

We study optimal liquidation of a trading position (so-called block order or meta-order) in a market with a linear temporary price impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the square-root law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova and Rakhlin, 2013; Farmer et al., 2013; Donier et al., 2015; T´oth et al., 2016). Mathematically, the Hamilton-Jacobi-Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task

Selective methoxylation of α-pinene to α-terpinyl methyl ether over Al3+ ion-exchanged clays

In this study, we report the use of clay-based catalysts in the methoxylation of α-pinene, for the selective synthesis of α-terpinyl methyl ether, TME. The main reaction products and intermediates were identified by GC-MS. The reaction conditions (stirring rate and catalyst load) that afford a kinetic regime were established. SAz-1 (Cheto, Arizona, USA) source clay and a montmorillonite (SD) from Porto Santo, Madeira Archipelago, Portugal, were modified by ion-exchange with Al3+ to produce catalysts with markedly different acidities and textural properties. The catalysts based on the high layer-charge SAz-1 montmorillonite proved to be the most active. Ion-exchange with Al3+, followed by thermal activation at 150°C, afforded the highest number of Brønsted acid sites - a significant proportion of which were located in the clay gallery - and this coincided with the maximum catalytic activity. The influence of various reaction conditions, to maximize α-pinene conversion and selectivity, was studied over AlSAz-1. When the reaction was performed for 1 h at 60°C, the conversion reached 65% with 65% selectivity towards the mono-ether, TME. Similar conversions and selectivities required up to 50 h over zeolites and other solid acid catalysts. The kinetic dependencies of this reaction on temperature and reagent concentration, over the selected clays were also investigated. It was established that, in the temperature and reagent concentration regime studied, the reaction was first order with respect to α-pinene. The apparent activation energies over the two catalysts, calculated from Arrhenius plots, were almost identical at 72 kJ mol-

Correlation of catalytic activity with infrared, Si-29 MAS NMR and acidity data for HCL-treated fine fractions of montmorillonites

The <2 mu m fractions of SAz-1 (Cheto, Arizona, USA) and JP (Jelsovy Potok, Slovakia) montmorillonites were treated with 6 M HCl for 30, 300 and 900 minutes at 95 degrees C. The materials obtained were investigated by X-ray fluorescence (XRF), X-ray diffraction (XRD), thermogravimetry (TG), infrared (IR) spectroscopy, and Si-29 MAS NMR spectroscopy. The number of acid sites was determined from the thermal desorption of cyclohexylamine and the catalytic activity was evaluated by reacting 2,3-dihydropyran with methanol to yield the tetrahydropyranyl ether. All the investigative methods utilised, which each provided discrete evidence for the depopulation of the octahedral sheet, confirmed that treatment of SAz-1 for 300 and 900 minutes caused complete destruction of the original structure. In contrast JP was more resistant to acid attack and the treatments resulted in materials with different levels of structural decomposition. Samples of JP and SAz-1 treated for 30 minutes exhibited Bronsted acidities commensurate with the exchange capacities of the parent materials and this was reflected in their catalytic activity. A reduction in both acidity and catalytic activity was observed after longer treatment times and this was attributed to the presence of fewer exchange sites due to the depopulation of the octahedral sheet. The Ca back-exchanged samples did not catalyse the ether forming reaction

Characterization of moderately acid-treated, size-fractionated montmorillonites using IR and MAS NMR-spectroscopy and thermal-analysis

The <2 mu m fractions of Cheto (Arizona, USA) and Jelsovy Potok (Slovakia) montmorillonites have been treated with 1 mol dm(-3) H2SO(4) for periods up to 6h at 95 degrees C. The materials obtained were investigated by X-ray fluorescence (XRF), X-ray diffraction (XRD), infrared (IR) spectroscopy, and Si-29 and Al-27 magic-angle spinning (MAS) NMR spectroscopy. Both clays were partially decomposed under the conditions utilized, but the Cheto montmorillonite, which contained a higher proportion of octahedral magnesium, was more susceptible to acid attack. XRD was insensitive to this level of acid attack but XRF, IR and Si-29 MAS NMR revealed the depopulation of the octahedral sheet. The number of acid sites was determined from the thermal desorption of cyclohexylamine and the catalytic activity was evaluated by reacting 2,3-dihydropyran with methanol to yield the tetrahydropyranyl ether. The test reaction was a more sensitive indicator of changes in clay acidity than the values determined by desorption of cyclohexylamine from acid sites

Correlation of catalytic activity with infrared, Si-29 MAS NMR and acidity data for HCL-treated fine fractions of montmorillonites

The <2 mu m fractions of SAz-1 (Cheto, Arizona, USA) and JP (Jelsovy Potok, Slovakia) montmorillonites were treated with 6 M HCl for 30, 300 and 900 minutes at 95 degrees C. The materials obtained were investigated by X-ray fluorescence (XRF), X-ray diffraction (XRD), thermogravimetry (TG), infrared (IR) spectroscopy, and Si-29 MAS NMR spectroscopy. The number of acid sites was determined from the thermal desorption of cyclohexylamine and the catalytic activity was evaluated by reacting 2,3-dihydropyran with methanol to yield the tetrahydropyranyl ether. All the investigative methods utilised, which each provided discrete evidence for the depopulation of the octahedral sheet, confirmed that treatment of SAz-1 for 300 and 900 minutes caused complete destruction of the original structure. In contrast JP was more resistant to acid attack and the treatments resulted in materials with different levels of structural decomposition. Samples of JP and SAz-1 treated for 30 minutes exhibited Bronsted acidities commensurate with the exchange capacities of the parent materials and this was reflected in their catalytic activity. A reduction in both acidity and catalytic activity was observed after longer treatment times and this was attributed to the presence of fewer exchange sites due to the depopulation of the octahedral sheet. The Ca back-exchanged samples did not catalyse the ether forming reaction