Application of Elzaki Transform Method to Market Volatility Using the Black-Scholes Model

Black-Scholes Model (B-SM) simulates the dynamics of financial market and contains instruments such as options and puts which are major indices requiring solution. B-SM is known to estimate the correct prices of European Stock options and establish the theoretical foundation for Option pricing. Therefore, this paper evaluates the Black-Schole model in simulating the European call in a cash flow in the dependent drift and focuses on obtaining analytic and then approximate solution for the model. The work also examines Fokker Planck Equation (FPE) and extracts the link between FPE and B-SM for non equilibrium systems. The B-SM is then solved via the Elzaki transform method (ETM). The computational procedures were obtained using MAPLE 18 with the solution provided in the form of convergent series.

On the Application of Mixed Models of Probability and Convex Set for Time-Variant Reliability Analysis

In time-variant reliability problems,there are a lot of uncertain variables from different sources.Therefore,it is important to consider these uncertainties in engineering.In addition,time-variant reliability problems typically involve a complexmultilevel nested optimization problem,which can result in an enormous amount of computation.To this end,this paper studies the time-variant reliability evaluation of structures with stochastic and bounded uncertainties using a mixed probability and convex set model.In this method,the stochastic process of a limit-state function with mixed uncertain parameters is first discretized and then converted into a timeindependent reliability problem.Further,to solve the double nested optimization problem in hybrid reliability calculation,an efficient iterative scheme is designed in standard uncertainty space to determine the most probable point(MPP).The limit state function is linearized at these points,and an innovative random variable is defined to solve the equivalent static reliability analysis model.The effectiveness of the proposed method is verified by two benchmark numerical examples and a practical engineering problem.

Optimal Quota-Share and Excess-of-Loss Reinsurance and Investment with Heston’s Stochastic Volatility Model

An optimal quota-share and excess-of-loss reinsurance and investment problem is studied for an insurer who is allowed to invest in a risk-free asset and a risky asset.Especially the price process of the risky asset is governed by Heston's stochastic volatility(SV)model.With the objective of maximizing the expected index utility of the terminal wealth of the insurance company,by using the classical tools of stochastic optimal control,the explicit expressions for optimal strategies and optimal value functions are derived.An interesting conclusion is found that it is better to buy one reinsurance than two under the assumption of this paper.Moreover,some numerical simulations and sensitivity analysis are provided.

Analytical Modeling and Mechanism Analysis of Time-Varying Excitation for Surface Defects in Rolling Element Bearings

Surface defects,including dents,spalls,and cracks,for rolling element bearings are the most common faults in rotating machinery.The accurate model for the time-varying excitation is the basis for the vibration mechanism analysis and fault feature extraction.However,in conventional investigations,this issue is not well and fully addressed from the perspective of theoretical analysis and physical derivation.In this study,an improved analytical model for time-varying displacement excitations(TVDEs)caused by surface defects is theoretically formulated.First and foremost,the physical mechanism for the effect of defect sizes on the physical process of rolling element-defect interaction is revealed.According to the physical interaction mechanism between the rolling element and different types of defects,the relationship between time-varying displacement pulse and defect sizes is further analytically derived.With the obtained time-varying displacement pulse,the dynamic model for the deep groove bearings considering the internal excitation caused by the surface defect is established.The nonlinear vibration responses and fault features induced by surface defects are analyzed using the proposed TVDE model.The results suggest that the presence of surface defects may result in the occurrence of the dual-impulse phenomenon,which can serve as indexes for surface-defect fault diagnosis.

Time-varying gravity field model of Sichuan-Yunnan region based on the equivalent mass source model

High-precision time-varying gravity field is an effective way to study the internal mass movement and understanding the spatio-temporal evolution process of the geodynamic system.Compared to the satellite gravity measurement,the repeated terrestrial gravity observation can provide a more high-order signal related to the shallow crust and subsurface.However,the suitable and unified method for gravity model estimation is a key problem for further applications.In this study,we introduce the spherical hexahedron element to simulate the field source mass and forward model the change of gravity field located at the Sichuan-Yunnan region(99—104°E,23—29°N)in the four epochs from 2015 to 2017.Compared to the experimental results based on Slepian or spherical harmonics frequency domain method,this alternative approach is suitable for constructing the equivalent mass source model of regional-scale gravity data,by introducing the first-order smooth prior condition of gravity time-varying signal to suppress the high-frequency component of the signal.The results can provide a higher spatial resolution reference for regional gravity field modeling in the Sichuan-Yunnan region.

Forecasting Stock Volatility Using Wavelet-based Exponential Generalized Autoregressive Conditional Heteroscedasticity Methods

In this study,we proposed a new model to improve the accuracy of fore-casting the stock market volatility pattern.The hypothesized model was validated empirically using a data set collected from the Saudi Arabia stock Exchange(Tada-wul).The data is the daily closed price index data from August 2011 to December 2019 with 2027 observations.The proposed forecasting model combines the best maximum overlapping discrete wavelet transform(MODWT)function(Bl14)and exponential generalized autoregressive conditional heteroscedasticity(EGARCH)model.The results show the model's ability to analyze stock market data,highlight important events that contain the most volatile data,and improve forecast accuracy.The results were compared from a number of mathematical mod-els,which are the non-linear spectral model,autoregressive integrated moving aver-age(ARIMA)model and EGARCH model.The performance of the forecasting model will be evaluated based on some of error functions such as Mean absolute percentage error(MAPE),Mean absolute scaled error(MASE)and Root means squared error(RMSE).

Did weekly economic index and volatility index impact US food sales during the first year of the pandemic?

We explore the impacts of economic and financial dislocations caused by COVID-19 pandemic shocks on food sales in the United States from January 2020 to January 2021.We use the US weekly economic index(WEI)to measure economic dislocations and the Chicago Board Options Exchange volatility index(VIX)to capture the broader stock market dislocations.We validate the NARDL model by testing a battery of models using the autoregressive distributed lags(ARDL)methodology(ARDL,NARDL,and QARDL specifications).Our study postulates that an increase in WEI has a significant negative long-term effect on food sales,whereas a decrease in WEI has no statistically significant(long-run)effect.Thus,policy responses that ignore asymmetric effects and hidden cointegration may fail to promote food security during pandemics.

A COMPARISON OF FORECASTING MODELS OF THE VOLATILITY IN SHENZHEN STOCK MARKET

Based on the weekly closing price of Shenzhen Integrated Index, this article studies the volatility of Shenzhen Stock Market using three different models： Logistic, AR（1） and AR（2）. The time-variable parameters of Logistic regression model is estimated by using both the index smoothing method and the time-variable parameter estimation method. And both the AR（1） model and the AR（2） model of zero-mean series of the weekly dosing price and its zero-mean series of volatility rate are established based on the analysis results of zero-mean series of the weekly closing price, Six common statistical methods for error prediction are used to test the predicting results. These methods are： mean error （ME）, mean absolute error （MAE）, root mean squared error （RMSE）, mean absolute percentage error （MAPE）, Akaike＇s information criterion （AIC）, and Bayesian information criterion （BIC）. The investigation shows that AR（1） model exhibits the best predicting result, whereas AR（2） model exhibits predicting results that is intermediate between AR（1） model and the Logistic regression model.

Do U.S.economic conditions at the state level predict the realized volatility of oil‑price returns?A quantile machine‑learning approach

Because the U.S.is a major player in the international oil market,it is interesting to study whether aggregate and state-level economic conditions can predict the subse-quent realized volatility of oil price returns.To address this research question,we frame our analysis in terms of variants of the popular heterogeneous autoregressive realized volatility(HAR-RV)model.To estimate the models,we use quantile-regression and quantile machine learning(Lasso)estimators.Our estimation results highlights the dif-ferential effects of economic conditions on the quantiles of the conditional distribution of realized volatility.Using weekly data for the period April 1987 to December 2021,we document evidence of predictability at a biweekly and monthly horizon.

A Gibbs Sampling Algorithm to Estimate the Parameters of a Volatility Model: An Application to Ozone Data

In this work we consider a stochastic volatility model, commonly used in financial time series studies, to analyse ozone data. The model considered depends on some parameters and in order to estimate them a Markov chain Monte Carlo algorithm is proposed. The algorithm considered here is the so-called Gibbs sampling algorithm which is programmed using the language R. Its code is also given. The model and the algorithm are applied to the weekly ozone averaged measurements obtained from the monitoring network of Mexico City.

Modeling Stock Market Volatility Using GARCH Models: A Case Study of Nairobi Securities Exchange (NSE)

The aim of this paper is to use the General Autoregressive Conditional Heteroscedastic (GARCH) type models for the estimation of volatility of the daily returns of the Kenyan stock market: that is Nairobi Securities Exchange (NSE). The conditional variance is estimated using the data from March 2013 to February 2016. We use both symmetric and asymmetric models to capture the most common features of the stock markets like leverage effect and volatility clustering. The results show that the volatility process is highly persistent, thus, giving evidence of the existence of risk premium for the NSE index return series. This in turn supports the positive correlation hypothesis: that is between volatility and expected stock returns. Another fact revealed by the results is that the asymmetric GARCH models provide better fit for NSE than the symmetric models. This proves the presence of leverage effect in the NSE return series.

Robust Parameter Identification Method of Adhesion Model for Heavy Haul Trains

A robust parameter identification method based on Kiencke model was proposed to solve the problem of the parameter identification accuracy being affected by the rail environment change and noise interference for heavy-duty trains. Firstly, a Kiencke stick-creep identification model was constructed, and the parameter identification task was transformed into a quadratic programming problem. Secondly, an iterative algorithm was constructed to solve the problem, into which a time-varying forgetting factor was added to track the change of the rail environment, and to solve the uncertainty problem of the wheel-rail environment. The Granger causality test was adopted to detect the interference, and then the weights of the current data were redistributed to solve the problem of noise interference in parameter identification. Finally, simulations were carried out and the results showed that the proposed method could track the change of the track environment in time, reduce the noise interference in the identification process, and effectively identify the adhesion performance parameters.

Some Explicit Formulae for the Hull and White Stochastic Volatility Model

An explicit formula for the transition probability density function of the Hull and White stochastic volatility model in presence of nonzero correlation between the stochastic differentials of the Wiener processes on the right hand side of the model equations is presented. This formula gives the transition probability density function as a two dimensional integral of an explicitly known integrand. Previously an explicit formula for this probability density function was known only in the case of zero correlation. In the case of nonzero correlation from the formula for the transition probability density function we deduce formulae (expressed by integrals) for the price of European call and put options and closed form formulae (that do not involve integrals) for the moments of the asset price logarithm. These formulae are based on recent results on the Whittaker functions [1] and generalize similar formulae for the SABR and multiscale SABR models [2]. Using the option pricing formulae derived and the least squares method a calibration problem for the Hull and White model is formulated and solved numerically. The calibration problem uses as data a set of option prices. Experiments with real data are presented. The real data studied are those belonging to a time series of the USA S&P 500 index and of the prices of its European call and put options. The quality of the model and of the calibration procedure is established comparing the forecast option prices obtained using the calibrated model with the option prices actually observed in the financial market. The website: http://www.econ.univpm.it/recchioni/finance/w17 contains some auxiliary material including animations and interactive applications that helps the understanding of this paper. More general references to the work of the authors and of their coauthors in mathematical finance are available in the website: http://www.econ.univpm.it/recchioni/finance.

Dynamic Hedging Based on Markov Regime-Switching Dynamic Correlation Multivariate Stochastic Volatility Model

It is important to consider the changing states in hedging.The Markov regime-switching dynamic correlation multivariate stochastic volatility( MRS-DC-MSV) model was proposed to solve this issue. DC-MSV model and MRS-DC-MSV model were used to calculate the time-varying hedging ratios and compare the hedging performance. The Markov chain Monte Carlo( MCMC) method was used to estimate the parameters. The results showed that,there were obviously two economic states in Chinese financial market. Two models all did well in hedging,but the performance of MRS-DCMSV model was better. It could reduce risk by nearly 90%. Thus,in the hedging period,changing states is a factor that cannot be neglected.

The Calibration of Some Stochastic Volatility Models Used in Mathematical Finance

Stochastic volatility models are used in mathematical finance to describe the dynamics of asset prices. In these models, the asset price is modeled as a stochastic process depending on time implicitly defined by a stochastic differential Equation. The volatility of the asset price itself is modeled as a stochastic process depending on time whose dynamics is described by a stochastic differential Equation. The stochastic differential Equations for the asset price and for the volatility are coupled and together with the necessary initial conditions and correlation assumptions constitute the model. Note that the stochastic volatility is not observable in the financial markets. In order to use these models, for example, to evaluate prices of derivatives on the asset or to forecast asset prices, it is necessary to calibrate them. That is, it is necessary to estimate starting from a set of data the values of the initial volatility and of the unknown parameters that appear in the asset price/volatility dynamic Equations. These data usually are observations of the asset prices and/or of the prices of derivatives on the asset at some known times. We analyze some stochastic volatility models summarizing merits and weaknesses of each of them. We point out that these models are examples of stochastic state space models and present the main techniques used to calibrate them. A calibration problem for the Heston model is solved using the maximum likelihood method. Some numerical experiments about the calibration of the Heston model involving synthetic and real data are presented.

Properties of Time-Varying Causality Tests in the Presence of Multivariate Stochastic Volatility

This paper compares the statistical properties of time-varying causality tests when errors of variables have multivariate stochastic volatility (SV). The time-varying causal-ity tests in this paper are based on a logistic smooth transition autoregressive model. The compared time-varying causality tests include asymptotic tests, heteroskedasticity-robust tests, and tests using wild bootstrap. Our simulation results show that asymptotic tests and heteroskedasticity-robust counterparts have size distortions under multivariate SV, whereas tests using wild bootstrap have better size properties regardless of type of error. In particular, the time-varying causality test with first-order Taylor approximation using wild bootstrap has better statistical properties.

Volatility Risk Management of Chinese Stock Grading Market——An Empirical Study of GARCH-VaR Model

This paper mainly through the comparison of GARCH-VaR China stock market board,small board and gem in the United States correspond to the three stock index volatility,volatility between stock indexes obtained U.S.stock market volatility risk multi-level market differences.As a suggestion and reference for investors,it can also provide reference for the supervision department of stock market risk.Based on the empirical research,analyzes the advantages and disadvantages of traditional risk measurement methods,and combined with GARCH model with high degree of complexity and the practice effect analysis,trying to find the objective measure stock model analysis.In the specific study of the volatility of the stock market,through the comparison of China’s three major plates and the market classification mechanism of mature U.S.stock market,combined with the objective situation of the market,draw conclusions and change expectations.From the empirical results,the U.S.stock market has recovered after the financial crisis,and its performance on risk volatility is better than China’s three major plates.From the comparison of the stock market in the same country,the small and medium-sized plates tend to have greater risks,while the risks of the main board and the gem have the characteristics of low average value but frequent fluctuations.

Exploring Apple’s Stock Price Volatility Using Five GARCH Models

The financial market is the core of national economic development,and stocks play an important role in the financial market.Analyzing stock prices has become the focus of investors,analysts,and people in related fields.This paper evaluates the volatility of Apple Inc.(AAPL)returns using five generalized autoregressive conditional heteroskedasticity(GARCH)models:sGARCH with constant mean,GARCH with sstd,GJR-GARCH,AR(1)GJR-GARCH,and GJR-GARCH in mean.The distribution of AAPL’s closing price and earnings data was analyzed,and skewed student t-distribution(sstd)and normal distribution(norm)were used to further compare the data distribution of the five models and capture the shape,skewness,and loglikelihood in Model 4-AR(1)GJR-GARCH.Through further analysis,the results showed that Model 4,AR(1)GJR-GARCH,is the optimal model to describe the volatility of the return series of AAPL.The analysis of the research process is both,a process of exploration and reflection.By analyzing the stock price of AAPL,we reflect on the shortcomings of previous analysis methods,clarify the purpose of the experiment,and identify the optimal analysis model.

Volatility Prediction via Hybrid LSTM Models with GARCH Type Parameters

Since the establishment of financial models for risk prediction,the measurement of volatility at risky market has improved,and its significance has also grown.For high-frequency financial data,the degree of investment risk,which has always been the focus of attention,is measured by the variance of residual sequence obtained following model regression.By integrating the long short-term memory(LSTM)model with multiple generalized autoregressive conditional heteroscedasticity(GARCH)models,a new hybrid LSTM model is used to predict stock price volatility.In this paper,three GARCH models are used,and the model that can best fit the data is determined.