Numerical Studies on Asymptotics of European Option Under Multiscale Stochastic Volatility

Multiscale stochastic volatilities models relax the constant volatility assumption from Black-Scholes option pricing model. Such models can capture the smile and skew of volatilities and therefore describe more accurately the movements of the trading prices. Christoffersen et al. Manag Sci 55(2):1914–1932 (2009) presented a model where the underlying price is governed by two volatility components, one changing fast and another changing slowly. Chiarella and Ziveyi Appl Math Comput 224:283–310 (2013) transformed Christoffersen’s model and computed an approximate formula for pricing American options. They used Duhamel’s principle to derive an integral form solution of the boundary value problem associated to the option price. Using method of characteristics, Fourier and Laplace transforms, they obtained with good accuracy the American option prices. In a previous research of the authors (Canhanga et al. 2014), a particular case of Chiarella and Ziveyi Appl Math Comput 224:283–310 (2013) model is used for pricing of European options. The novelty of this earlier work is to present an asymptotic expansion for the option price. The present paper provides experimental and numerical studies on investigating the accuracy of the approximation formulae given by this asymptotic expansion. We present also a procedure for calibrating the parameters produced by our first-order asymptotic approximation formulae. Our approximated option prices will be compared to the approximation obtained by Chiarella and Ziveyi Appl Math Comput 224:283–310 (2013).     

L p-Perturbations of Invariant Subbundles for Linear Systems

We use Riccati's equations and the ordinary and exponential dichotomies to get simple recurrent formulae for the asymptotic integration of linear systems subjected to L _p -perturbations with arbitrary p 1. Moreover, we establish conditions which are necessary and sufficient for the persistence of one-dimensional invariant subbundles for the linear system under L _p -perturbations. In this way, we prove the sharp nature of the well known Levinson and Hartman–Wintner asymptotic theorems.     

A comparison of neighbor search algorithms for large rigid molecules

Fast determination of neighboring atoms is an essential step in molecular dynamics simulations or Monte Carlo computations, and there exists a variety of algorithms to efficiently compute neighbor lists. However, most of these algorithms are general, and not specifically designed for a given type of application. As a result, although their average performance is satisfactory, they might be inappropriate in some specific application domains. In this article, we study the case of detecting neighbors between large rigid molecules, which has applications in, e.g., rigid body molecular docking, Monte Carlo simulations of molecular self-assembly or diffusion, and rigid body molecular dynamics simulations. More precisely, we compare the traditional grid-based algorithm to a series of hierarchy-based algorithms that use bounding volumes to rapidly eliminate large groups of irrelevant pairs of atoms during the neighbor search. We compare the performance of these algorithms based on several parameters: the size of the molecules, the average distance between them, the cutoff distance, as well as the type of bounding volume used in the culling hierarchy (AABB, OBB, wrapped, or layered spheres). We demonstrate that for relatively large systems (> 100,000 atoms) the algorithm based on the hierarchy of wrapped spheres shows the best results and the traditional grid-based algorithm gives the worst timings. For small systems, however, the grid-based algorithm and the one based on the wrapped sphere hierarchy are beneficial.     

Selective dispersion of single-walled carbon nanotubes with specific chiral indices by poly(N-decyl-2,7-carbazole)

Physico-chemical methods to sort single-walled carbon nanotubes (SWNTs) by chiral index are presently lacking but are required for in-depth experimental analysis and also for potential future applications of specific species. Here we report the unexpected selectivity of poly(N-decyl-2,7-carbazole) to almost exclusively disperse semiconducting SWNTs with differences of their chiral indices (n - m) ≥ 2 in toluene. The observed selectivity complements perfectly the dispersing features of the fluorene analogue poly(9,9-dialkyl-2,7-fluorene), which disperses semiconducting SWNTs with (n - m) ≤ 2 in toluene. The dispersed samples are further purified by density gradient centrifugation and analyzed by photoluminescence excitation spectroscopy. All-atom molecular modeling with decamer model compounds of the polymers and (10,2) and (7,6) SWNTs suggests differences in the π-π stacking interaction as origin of the selectivity. We observe energetically favored complexes between the (10,2) SWNT and the carbazole decamer and between the (7,6) SWNT and the fluorene decamer, respectively. These findings demonstrate that subtle structural changes of polymers lead to selective solvation of different families of carbon nanotubes. Furthermore, chemical screening of closely related polymers may pave the way toward simple, low-cost, and index-specific isolation of SWNTs.     

Strain effects in electron spin resonance spectroscopy of quintet 2,6-bis(4'-nitrenophenyl)-4-phenylpyridine

Photolysis of 2,6-bis(4'-azidophenyl)-4-phenylpyridine in 2-methyltetrahydrofuran (2MTHF) glass at 7 K leads to quintet 2,6-bis(4'-nitrenophenyl)-4-phenylpyridine as a mixture of rotational isomers. The electron spin resonance (ESR) spectrum of this mixture of rotamers shows a considerable broadening of many transitions in the range of 0-5000 G and cannot be reproduced by computer simulations solely based on the tuning of the spin Hamiltonian parameters g, D(Q), and E(Q) alone or on predictions of DFT calculations. The best modeling of the experimental ESR spectrum is obtained only when the large line-broadening parameter of Γ(E(Q)) = 1200 MHz along with the spin Hamiltonian g = 2.003, D(Q) = 0.154 cm(-1), and E(Q) = 0.050 cm(-1) is used in the spectral simulations. The most accurate theoretical estimations of the magnetic parameters of the dinitrene in a 2MTHF glass are obtained from the B3LYP/6-311+G(d,p)+PBE/DZ/COSMO calculations of the spin-spin coupling parameters D(SS) and E(SS). Such calculations overestimate the E(Q) and D(Q) values of the dinitrene just by 1% and 10%, respectively, demonstrating that contributions of the spin-orbit coupling parameters D(SOC) and E(SOC) to the total D(Q) and E(Q) values are negligibly small. The research shows that ESR studies of polynuclear high-spin nitrenes, obtained by photolysis of rotational isomers of the starting azides, can only be successful if large E(Q) strain effects are taken into account in the spectral simulations.     

Density functional analysis of geometries and electronic structures of gold-phosphine clusters. The case of Au4(PR3)42+ and A(u4μ2-I)2(PR3)4

Geometries, ligand binding energies, electronic structure, and excitation spectra are determined for Au(4)(PR(3))(4)(2+) and Au(4)(μ(2)-I)(2)(PR(3))(4) clusters (R = PH(3), PMe(3), and PPh(3)). Density functionals including SVWN5, Xα, OPBE, LC-ωPBE, TPSS, PBE0, CAM-B3LYP, and SAOP are employed with basis sets ranging from LANL2DZ to SDD to TZVP. Metal--metal and metal--ligand bond distances are calculated and compared with experiment. The effect of changing the phosphine ligands is assessed for geometries and excitation spectra. Standard DFT and hybrid ONIOM calculations are employed for geometry optimizations with PPh(3) groups. The electronic structure of the gold--phosphine clusters examined in this work is analyzed in terms of cluster ("superatom") orbitals and d-band orbitals. Transitions out of the d band are significant in the excitation spectra. The use of different basis sets and DFT functionals leads to noticeable variations in the relative intensities of strong transitions, although the overall spectral profile remains qualitatively unchanged. The replacement of PMe(3) with PPh(3) changes the nature of the electronic transitions in the cluster due to low-lying π*-orbitals. To reproduce the experimental geometries of clusters with PPh(3) ligands, computationally less expensive PH(3) or PMe(3) ligands are sufficient for geometry optimizations. However, to predict cluster excitation spectra, the full PPh(3) ligand must be considered.