On Derivations of Black-Scholes Greek Letters

Each Greek letter of an option measures the sensitivity of an option price with respect to the change in the value of a given underlying parameter such as underlying asset’s price. This article provides simple derivations of often-used five Greek letters for European call and put options within the Balck-scholes model framework. Each proof of these Greek letters bypasses complicated mathematical calculations, it is relatively simple and easy to follow. Furthermore, some calculation examples for Greek letters have been given. Keywords: Black-Scholes option pricing model, Call option, Put option, Greek letter

Canonical Least-Squares Monte Carlo Valuation of American Options: Convergence and Empirical Pricing Analysis

The paper by Liu (2010) introduces a method termed the canonical least-squares Monte Carlo (CLM) which combines a martingale-constrained entropy model and a least-squares Monte Carlo algorithm to price American options. In this paper, we first provide the convergence results of CLM and numerically examine the convergence properties. Then, the comparative analysis is empirically conducted using a large sample of the S&P 100 Index (OEX) puts and IBM puts. The results on the convergence show that choosing the shifted Legendre polynomials with four regressors is more appropriate considering the pricing accuracy and the computational cost. With this choice, CLM method is empirically demonstrated to be superior to the benchmark methods of binominal tree and finite difference with historical volatilities

Pricing American Options Using a Nonparametric Entropy Approach

This paper studies the pricing problem of American options using a nonparametric entropy approach. First, we derive a general expression for recovering the risk-neutral moments of underlying asset return and then incorporate them into the maximum entropy framework as constraints. Second, by solving this constrained entropy problem, we obtain a discrete risk-neutral (martingale) distribution as the unique pricing measure. Third, the optimal exercise strategies are achieved via the least-squares Monte Carlo algorithm and consequently the pricing algorithm of American options is obtained. Finally, we conduct the comparative analysis based on simulations and IBM option contracts. The results demonstrate that this nonparametric entropy approach yields reasonably accurate prices for American options and produces smaller pricing errors compared to other competing methods

The changes of the interspace angle after anterior correction and instrumentation in adolescent idiopathic scoliosis patients

<p>Abstract</p> <p>Background</p> <p>In idiopathic scoliosis patients, after anterior spinal fusion and instrumentation, the discs (interspace angle) between the lowest instrumented vertebra (LIV) and the next caudal vertebra became more wedged. We reviewed these patients and analyzed the changes of the angle.</p> <p>Methods</p> <p>By reviewing the medical records and roentgenograms of adolescent idiopathic scoliosis patients underwent anterior spinal fusion and instrumentation, Cobb angle of the curve, correction rate, coronal balance, LIV rotation, interspace angle were measured and analyzed.</p> <p>Results</p> <p>There were total 30 patients included. The mean coronal Cobb angle of the main curve (thoracolumbar/lumbar curve) before and after surgery were 48.9° and 11.7°, respectively, with an average correction rate of 76.1%. The average rotation of LIV before surgery was 2.1 degree, and was improved to 1.2 degree after surgery. The interspace angle before surgery, on convex side-bending films, after surgery, at final follow up were 3.2°, -2.3°, 1.8° and 4.9°, respectively. The difference between the interspace angle after surgery and that preoperatively was not significant (P = 0.261), while the interspace angle at final follow-up became larger than that after surgery, and the difference was significant(P = 0.012). The interspace angle after surgery was correlated with that on convex side-bending films (r = 0.418, P = 0.022), and the interspace angle at final follow-up was correlated with that after surgery (r = 0.625, P = 0.000). There was significant correlation between the loss of the interspace angle and the loss of coronal Cobb angle of the main curve during follow-up(r = 0.483, P = 0.007).</p> <p>Conclusion</p> <p>The interspace angle could be improved after anterior correction and instrumentation surgery, but it became larger during follow-up. The loss of the interspace angle was correlated with the loss of coronal Cobb angle of the main curve during follow-up.</p

Suppressed N fixation and diazotrophs after four decades of fertilization

Background: N fixation is one of the most important microbially driven ecosystem processes on Earth, allowing N to enter the soil from the atmosphere, and regulating plant productivity. A question that remains to be answered is whether such a fundamental process would still be that important in an over-fertilized world, as the long-term effects of fertilization on N fixation and associated diazotrophic communities remain to be tested. Here, we used a 35-year fertilization experiment, and investigated the changes in N fixation rates and the diazotrophic community in response to long-term inorganic and organic fertilization. Results: It was found that N fixation was drastically reduced (dropped by 50%) after almost four decades of fertilization. Our results further indicated that functionality losses were associated with reductions in the relative abundance of keystone and phylogenetically clustered N fixers such as Geobacter spp. Conclusions: Our work suggests that long-term fertilization might have selected against N fixation and specific groups of N fixers. Our study provides solid evidence that N fixation and certain groups of diazotrophic taxa will be largely suppressed in a more and more fertilized world, with implications for soil biodiversity and ecosystem functions

Efficacy and safety of tigecycline monotherapy vs. imipenem/cilastatin in Chinese patients with complicated intra-abdominal infections: a randomized controlled trial

<p>Abstract</p> <p>Background</p> <p>Tigecycline, a first-in-class broad-spectrum glycylcycline antibiotic, has broad-spectrum in vitro activity against bacteria commonly encountered in complicated intra-abdominal infections (cIAIs), including aerobic and facultative Gram-positive and Gram-negative bacteria and anaerobic bacteria. In the current trial, tigecycline was evaluated for safety and efficacy vs. imipenem/cilastatin in hospitalized Chinese patients with cIAIs.</p> <p>Methods</p> <p>In this phase 3, multicenter, open-label study, patients were randomly assigned to receive IV tigecycline or imipenem/cilastatin for ≤2 weeks. The primary efficacy endpoints were clinical response at the test-of-cure visit (12-37 days after therapy) for the microbiologic modified intent-to-treat and microbiologically evaluable populations. Because the study was not powered to demonstrate non-inferiority between tigecycline and imipenem/cilastatin, no formal statistical analysis was performed. Two-sided 95% confidence intervals (CIs) were calculated for the response rates in each treatment group and for differences between treatment groups for descriptive purposes.</p> <p>Results</p> <p>One hundred ninety-nine patients received ≥1 dose of study drug and comprised the modified intent-to-treat population. In the microbiologically evaluable population, 86.5% (45 of 52) of tigecycline- and 97.9% (47 of 48) of imipenem/cilastatin-treated patients were cured at the test-of-cure assessment (12-37 days after therapy); in the microbiologic modified intent-to-treat population, cure rates were 81.7% (49 of 60) and 90.9% (50 of 55), respectively. The overall incidence of treatment-emergent adverse events was 80.4% for tigecycline vs. 53.9% after imipenem/cilastatin therapy (<it>P </it>< 0.001), primarily due to gastrointestinal-related events, especially nausea (21.6% vs. 3.9%; <it>P </it>< 0.001) and vomiting (12.4% vs. 2.0%; <it>P </it>= 0.005).</p> <p>Conclusions</p> <p>Clinical cure rates for tigecycline were consistent with those found in global cIAI studies. The overall safety profile was also consistent with that observed in global studies of tigecycline for treatment of cIAI, as well as that observed in analyses of Chinese patients in those studies; no novel trends were observed.</p> <p>Trial Registration</p> <p>ClinicalTrials.gov NCT00136201</p

Risk-Neutrality of RND and Option Pricing within an Entropy Framework

This article constructs an entropy pricing framework by incorporating a set of informative risk-neutral moments (RNMs) extracted from the market-available options as constraints. Within the RNM-constrained entropic framework, a unique distribution close enough to the correct one is obtained, and its risk-neutrality is deeply verified based on simulations. Using this resultant risk-neutral distribution (RND), a sample of risk-neutral paths of the underlying price is generated and ultimately the European option&rsquo;s prices are computed. The pricing performance and analysis in simulations demonstrate that this proposed valuation is comparable to the benchmarks and can produce fairly accurate prices for options