We propose a simple and robust numerical algorithm to estimate a time-dependent volatility function from a set of market observations, using the Black–Scholes (BS) model. We employ a fully implicit finite difference method to solve the BS equation numerically. To define the time-dependent volatility function, we define a cost function that is the sum of the squared errors between the market values and the theoretical values obtained by the BS model using the time-dependent volatility function. To minimize the cost function, we employ the steepest descent method. However, in general, volatility functions for minimizing the cost function are nonunique. To resolve this problem, we propose a predictor-corrector technique. As the first step, we construct the volatility function as a constant. Then, in the next step, our algorithm follows the prediction step and correction step at half-backward time level. The constructed volatility function is continuous and piecewise linear with respect to the time variable. We demonstrate the ability of the proposed algorithm to reconstruct time-dependent volatility functions using manufactured volatility functions. We also present some numerical results for real market data using the proposed volatility function reconstruction algorithm.
Under the background of the rapid development of the Internet, the development information and social status that affect economic regeneration exist in the form of literal data in massive data information. The use of Viterbi algorithm can establish a digital model of information extraction, so as to quickly and accurately find the important content information in the Internet that influences the development of reform and economic rejuvenation. Therefore, based on the Viterbi algorithm, the use of mathematical models to help Jilin economic mechanism research from qualitative to quantitative, and continuously improve the scientific level of economic research. After deep analysis of the implementation principle and algorithm flow of Viterbi algorithm, and from the need of text information extraction, an optimization and update scheme is proposed to improve the effectiveness of the algorithm and the accuracy is calculated effectively based on the word feature detection. After simulation experiments show that Viterbi algorithm has a good application value in the field of Jilin economic reform research.
This survey is the second part of the history of science of Song and Yuan dynasties and will covers the period from Jin to Yuan. Following the first part, we look at the calendrical astronomy, mathematics and medicine. In this survey we again follow Yabuuchi's work on the history of science of Song and Yuan period and Du Shiran's work on the history of science of China. We start from the sciences and mathematics of Jin which inherited those of Northern Song and see how they influenced the whole China including Yuan and Southern Song. As a conclusion the tendency to practical usages in the Southern Song as well as the suppression of Han people in Yuan prevented developments of theoretical sciences in Yuan and Ming later.
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