Adiabaticity Conditions for Volatility Smile in Black-Scholes Pricing Model

Our derivation of the distribution function for future returns is based on the risk neutral approach which gives a functional dependence for the European call (put) option price, C(K), given the strike price, K, and the distribution function of the returns. We derive this distribution function using for C(K) a Black-Scholes (BS) expression with volatility in the form of a volatility smile. We show that this approach based on a volatility smile leads to relative minima for the distribution function ("bad" probabilities) never observed in real data and, in the worst cases, negative probabilities. We show that these undesirable effects can be eliminated by requiring "adiabatic" conditions on the volatility smile

T1rho and T2 relaxation times of the normal adult knee meniscus at 3T: analysis of zonal differences

Abstract Background Prior studies describe histological and immunohistochemical differences in collagen and proteoglycan content in different meniscal zones. The aim of this study is to evaluate horizontal and vertical zonal differentiation of T1rho and T2 relaxation times of the entire meniscus from volunteers without symptom and imaging abnormality. Methods Twenty volunteers age between 19 and 38 who have no knee-related clinical symptoms, and no history of prior knee surgeries were enrolled in this study. Two T1rho mapping (b-FFE T1rho and SPGR T1rho) and T2 mapping images were acquired with a 3.0-T MR scanner. Each meniscus was divided manually into superficial and deep zones for horizontal zonal analysis. The anterior and posterior horns of each meniscus were divided manually into white, red-white and red zones for vertical zonal analysis. Zonal differences of average relaxation times among each zone, and both inter- and intra-observer reproducibility were statistically analyzed. Results In horizontal zonal analysis, T1rho relaxation times of the superficial zone tended to be higher than those of the deep zone, and this difference was statistically significant in the medial meniscal segments (84.3 ms vs 76.0 ms on b-FFE, p 0.74) or good (0.60–0.74) in all meniscal segments on both horizontal and vertical zonal analysis, except for inter-class correlation coefficients of the lateral meniscus on SPGR. Compared with SPGR T1rho images, b-FFE T1rho images demonstrated more significant zonal differentiation with higher inter- and intra-observer reproducibility. Conclusions There are zonal differences in T1rho and T2 relaxation times of the normal meniscus