Correcting the Bias in the Practitioner Black-Scholes Method

We address a number of technical problems with the popular Practitioner Black-Scholes (PBS) method for valuing options. The method amounts to a two-stage procedure in which fitted values of implied volatilities (IV) from a linear regression are plugged into the Black-Scholes formula to obtain predicted option prices. Firstly we ensure that the prediction from stage one is positive by using log-linear regression. Secondly, we correct the bias (see Christoffersen and Jacobs, 2004, p.298) that results from the transformation applied to the fitted values (i.e. the Black-Scholes formula) being a highly non-linear function of implied volatility. We apply the smearing technique (Duan, 1983) in order to correct this bias. An alternative means of implementing the PBS approach is to use the market option price as the dependent variable and estimate the parameters of the IV equation by the method of non-linear least squares (NLLS). A problem we identify with this method is one of model incoherency: the IV equation that is estimated does not correspond to the set of option prices used to estimate it. We use the Monte Carlo method to verify that (1) standard PBS gives biased option values, both in-sample and out-of-sample; (2) using standard (log-linear) PBS with smearing almost completely eliminates the bias; (3) NLLS gives biased option values, but the bias is less severe than with standard PBS. We are led to conclude that, of the range of possible approaches to implementing PBS, log-linear PBS with smearing is preferred on the basis that it is the only approach that results in valuations with negligible bias

A closer look at interacting dark energy with statefinder hierarchy and growth rate of structure

We investigate the interacting dark energy models by using the diagnostics of statefinder hierarchy and growth rate of structure. We wish to explore the deviations from $\Lambda$CDM and to differentiate possible degeneracies in the interacting dark energy models with the geometrical and structure growth diagnostics. We consider two interacting forms for the models, i.e., $Q_1=\beta H\rho_c$ and $Q_2=\beta H\rho_{de}$, with $\beta$ being the dimensionless coupling parameter. Our focus is the I$\Lambda$CDM model that is a one-parameter extension to $\Lambda$CDM by considering a direct coupling between the vacuum energy ($\Lambda$) and cold dark matter (CDM), with the only additional parameter $\beta$. But we begin with a more general case by considering the I$w$CDM model in which dark energy has a constant $w$ (equation-of-state parameter). For calculating the growth rate of structure, we employ the "parametrized post-Friedmann" theoretical framework for interacting dark energy to numerically obtain the $\epsilon(z)$ values for the models. We show that in both geometrical and structural diagnostics the impact of $w$ is much stronger than that of $\beta$ in the I$w$CDM model. We thus wish to have a closer look at the I$\Lambda$CDM model by combining the geometrical and structural diagnostics. We find that the evolutionary trajectories in the $S^{(1)}_3$--$\epsilon$ plane exhibit distinctive features and the departures from $\Lambda$CDM could be well evaluated, theoretically, indicating that the composite null diagnostic $\{S^{(1)}_3, \epsilon\}$ is a promising tool for investigating the interacting dark energy models.Comment: 17 pages, 4 figures; accepted for publication in JCA

A Good Sugar, D-Mannose, Suppresses Autoimmune Diabetes

It is well known that too much sugar uptake causes many health problems, including diabetes and obesity (Lustig et al. in Nature 482:27-29, 2012). However, a team of researchers led by Dr. Wanjun Chen of the National Institute of Dental and Craniofacial Research (NIDCR), National Institutes of Health (NIH), USA, have recently shown that d-mannose, a naturally occurring C-2 epimer of glucose is likely beneficial to human health. Their studies have revealed that supraphysiological levels of d-mannose that are safely achievable via drinking-water supplementation can be preventive and therapeutic to experimental autoimmune diabetes and asthmatic lung inflammation (Zhang et al. in Nat Med 23:1036-1045, 2017)

2,2′-(p-Phenyl­enedithio)diacetic acid

The complete molecule of the title compound, C10H10O4S2, is generated by a crystallographic inversion centre. In the crystal, mol­ecules are linked into a one-dimensional chain by inter­molecular O—H⋯O hydrogen bonds