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In this study, we compare two methods for using neural networks to efficiently learn the price of derivatives. The first method, proposed by Funahashi (Quant Financ 21(4):575–592, 2021a), involves training the neural networks to learn the difference between the derivative price and its asymptotic expansion, rather than learning the derivative price directly. The target derivative price is then obtained by adding the approximate solution with the predicted value of neural networks. This method reduces the required amount of learning data, often by a factor of one hundred to one thousand, compared to the case where the derivative price is directly learned through neural networks. In the second method, established in this paper, the neural networks learn the difference between the derivative price written on the underlying asset price that follows the target complex stochastic process and the derivative price written on the underlying asset price that has a relatively simple stochastic process that has a closed-form solution for the target derivative prices. This method provides an alternative valuation method when no efficient approximate solution for the derivative value is observed and if one can arbitrarily determine the model parameters of the quasi-process that approximates the original process. We also propose a unified method to determine the model parameters of quasi-processes from underlying asset processes. These methods prove valuable in cases where general analytic solutions are absent, as seen in widely used financial models such as the stochastic volatility models. These cases involve time-consuming numerical calculations to generate learning data, highlighting the value of the two methods, which significantly compress calculation times. [ABSTRACT FROM AUTHOR]

One of the most important composite indicators (CIs) to assess the development of countries or regions is the human development index (HDI) which is used by the United Nations (UN) to rank countries. HDI has three dimensions including healthy life, population education, and standard of living. A total of four different sub-indicators are defined for these three dimensions. The UN evaluates and ranks all countries using a simple arithmetic or geometric average of the sub-indicators and then categorizes the countries into four different groups based on their HDI scores. To measure the HDI, the benefit-of-the-doubt (BOD) model is used by researchers instead of the geometric mean. The conventional BOD model has some main drawbacks. The first is not accounting for data uncertainty, and the second is evaluating countries using different weights for the same sub-indicators. Furthermore, BOD model is not capable of predicting countries' future HDI scores. To overcome these deficiencies, this paper proposes a common weight fuzzy BOD (CWFBOD) model to measure the HDI scores. First, to take into account the uncertainty, data are considered fuzzy numbers, and a fuzzy BOD model (FBOD) is introduced. Then, to find a set of common weights for the three dimensions of HDI, the proposed FBOD model is transformed into a multiple-objective CWFBOD model. To convert and solve the multiple objective CWFBOD model to a single objective model, a fuzzy theory approach is used. In addition, of predicting the future HDI scores of countries, an artificial neural network (ANN) is designed and trained, where the original data on sub-indicators health, education, and income are considered as the features, and the HDI scores generated by CWBOD are assumed as the target of ANN. Finally, this study applies the fuzzy C-Means clustering technique to cluster all countries into four different clusters based on the HDI scores generated by FBOD and CWFBOD models. To illustrate the ability of the proposed methodology, the HDI scores of 190 countries during the period of 2015–2021 have been estimated and predicted. The results show that the proposed integrated methodology can be effectively used to estimate and predict the HDI scores as well as to cluster countries. [ABSTRACT FROM AUTHOR]