On the Approximation of the SABR Model: A Probabilistic Approach

Abstract

In this article, we derive a probabilistic approximation for three different versions of the SABR model: Normal, Log-Normal and a displaced diffusion version for the general case. Specifically, we focus on capturing the terminal distribution of the underlying process (conditional on the terminal volatility) to arrive at the implied volatilities of the corresponding European options for all strikes and maturities. Our resulting method allows us to work with a variety of parameters that cover the long-dated options and highly stress market condition. This is a different feature from other current approaches that rely on the assumption of very small total volatility and usually fail for longer than 10 years maturity or large volatility of volatility (Volvol).     

The Crystal Structure of Non‐Modified and Bipyridine‐Modified PNA Duplexes

Peptide nucleic acid (PNA) is a synthetic analogue of DNA that commonly has an N‐aminoethyl glycine backbone. The crystal structures of two PNA duplexes, one containing eight standard nucleobase pairs (GGCATGCC)₂, and the other containing the same nucleobase pairs and a central pair of bipyridine ligands, have been solved with a resolution of 1.22 and 1.10 Å, respectively. The non‐modified PNA duplex adopts a P‐type helical structure similar to that of previously characterized PNAs. The atomic‐level resolution of the structures allowed us to observe for the first time specific modes of interaction between the terminal lysines of the PNA and the backbone and the nucleobases situated in the vicinity of the lysines, which are considered an important factor in the induction of a preferred handedness in PNA duplexes. Our results support the notion that whereas PNA typically adopts a P‐type helical structure, its flexibility is relatively high. For example, the base‐pair rise in the bipyridine‐containing PNA is the largest measured to date in a PNA homoduplex. The two bipyridines bulge out of the duplex and are aligned parallel to the major groove of the PNA. In addition, two bipyridines from adjacent PNA duplexes form a π‐stacked pair that relates the duplexes within the crystal. The bulging out of the bipyridines causes bending of the PNA duplex, which is in contrast to the structure previously reported for biphenyl‐modified DNA duplexes in solution, where the biphenyls are π stacked with adjacent nucleobase pairs and adopt an intrahelical geometry. This difference shows that relatively small perturbations can significantly impact the relative position of nucleobase analogues in nucleic acid duplexes.     

Synthesis of N-phenylpyrrole and pyrazole amidines and related diamines

N-Phenylpyrrole and pyrazole nitriles 3,4,10 were prepared in one step from the corresponding aldehydes. The nitriles were converted into novel amidines 5,6,14 and related diamines 7,8,12 were also prepared from the aldehydes. The orientation of the phenyl ring to the basic function was controlled by modifying the torsional angle between the rings by methyl group substitution on the heterocycle.     

Effect of brownian diffusion on chemical kinetics

A new method of theoretical prediction of the kinetic rate constants of fast chemical reactions in solutions is presented. It takes into account the effect of finite diffusive displacements of the reacting molecules. The approach is based on the solution of the steady-state Fokker–Planck equation by the moments method of Grad developed in the theory of coagulation of aerosol particles. A comparison of the predicted rate constants with the experimental data provided by Schuh and Fischer for the self-reaction of tert-butyl radicals in n-alkanes shows a good correspondence.     

General relationships between the mobility of a chain fluid and various computed scalar metrics

We performed molecular dynamics simulations of chain systems to investigate general relationships between the system mobility and computed scalar quantities. Three quantities were found that had a simple one-to-one relationship with mobility: packing fraction, potential energy density, and the value of the static structure factor at the first peak. The chain center-of-mass mobility as a function of these three quantities could be described equally well by either a Vogel-Fulcher type or a power law equation.     

Listener sensitivity to individual talker differences in voice-onset-time

Recent findings in the domains of word and talker recognition reveal that listeners use previous experience with an individual talker's voice to facilitate subsequent perceptual processing of that talker's speech. These findings raise the possibility that listeners are sensitive to talker-specific acoustic-phonetic properties. The present study tested this possibility directly by examining listeners' sensitivity to talker differences in the voice-onset-time (VOT) associated with a word-initial voiceless stop consonant. Listeners were trained on the speech of two talkers. Speech synthesis was used to manipulate the VOTs of these talkers so that one had short VOTs and the other had long VOTs (counterbalanced across listeners). The results of two experiments using a paired-comparison task revealed that, when presented with a short- versus long-VOT variant of a given talker's speech, listeners could select the variant consistent with their experience of that talker's speech during training. This was true when listeners were tested on the same word heard during training and when they were tested on a different word spoken by the same talker, indicating that listeners generalized talker-specific VOT information to a novel word. Such sensitivity to talker-specific acoustic-phonetic properties may subserve at least in part listeners' capacity to benefit from talker-specific experience.     

Exploiting predisposition in the stereoselective synthesis of mono-, bi- and tetracyclic oxygen heterocycles: equilibration between,and trapping of,alternative di- and tetraacetals

The desymmetrisation of 1,4-difuran-2-ylbutane-1,4-diol by Sharpless asymmetric oxidation gave the corresponding desymmetrised product in > 96% ee. However, the product existed as a mixture of two interconverting isomers, both of which were mixtures of anomers. The product could be trapped in high yield with a range of reagents to give stable adducts with embedded pyran-3-one, 1,6-dioxaspiro[4.5]decane or pyrano[3,2-b]pyran ring systems. The strategy was also applied in the interconversion between alternative tetracycles and, under acidic conditions, this process was thermodynamically controlled. The selectivity of the process was rationalised by molecular modelling using the HF/6-31G* parameter set.     

Dimerization of the transmembrane domain of amyloid precursor proteins and familial Alzheimer's disease mutants

Background  

Amyloid precursor protein (APP) is enzymatically cleaved by γ-secretase to form two peptide products, either Aβ40 or the more neurotoxic Aβ42. The Aβ42/40 ratio is increased in many cases of familial Alzheimer's disease (FAD). The transmembrane domain (TM) of APP contains the known dimerization motif GXXXA. We have investigated the dimerization of both wild type and FAD mutant APP transmembrane domains.