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The establishment of the fractional Black–Scholes option pricing model is under a major condition with the normal distribution for the state price density (SPD) function. However, the fractional Brownian motion is deemed to not be martingale with a long memory effect of the underlying asset, so that the estimation of the state price density (SPD) function is far from simple. This paper proposes a convenient approach to get the fractional option pricing model by changing variables. Further, the option price is transformed as the integral function of the cumulative density function (CDF), so it is not necessary to estimate the distribution function individually by complex approaches. Finally, it encourages to estimate the fractional option pricing model by the way of nonparametric regression and makes empirical analysis with the traded 50 ETF option data in Shanghai Stock Exchange (SSE).

Background: Depression is a typical psychosomatic disease. Shuganheweitang (SGHWT) is a clinical formula that effectively treats depression. However, the potential mechanism used by SGHWT to ameliorate depression-like behaviors is still unclear. This study investigated the effects of SGHWT on metabolic change in the liver and hypothalamus with signaling pathways involved in chronic unpredictable mild stress (CUMS)-induced depression in rats to explore the mechanism of the anti-depressive effect. Methods: A total of 52 rats were used to create a model of depression by CUMS combined with solitary rearing for 6 weeks. Open field test (OFT), sucrose preference test (SPT), forced swim test (FST), and body weight (BW) were performed to analyze the pharmacodynamic effects of SGHWT. H&E staining, Nissl staining, immunofluorescence, immunohistochemistry, and western blot were used to evaluate the mechanism of action. Untargeted metabolomics techniques by ultra-performance liquid chromatography-quantitative time-of-flight tandem mass spectrometry (UPLC-Q-TOF-MS/MS) were used to analyze all the metabolic differences in the liver and hypothalamus. Results: SGHWT improved CUMS-induced depression-like behaviors in vivo. SGHWT reduced hepatic

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