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The purpose of this paper is to perform Shariah review of Brownian motion that is used for prediction of Islamic stock prices and their volatility.

It uses the Shariah compliant development model guidelines to review the Brownian motion and its applications.

The model of Brownian motion does not involve any variable that renders it non-Shariah compliant; neither all applications of Brownian motion are Shariah compliant. Because the model is based on stochastic properties that involve randomness, therefore the issue of gharar takes the utmost important to handle in the applications of the model. The results need to be analyzed strictly in accordance with the Shariah whether they create any element of gharar or uncertainty in case of expected price and volatility estimates.

The research suffers from the limitation that it analyses only one model of physics, i.e. Brownian motion model from Shariah perspective.

The research opens an area for Shariah analysis of results generated from the application of advanced models of physics on matters related to Islamic financial markets.

The originality of this study stems from the fact that to the best of the authors’ knowledge, it is the first study that extends Shariah guidelines into Financial physics for making the foundations of Islamic econophysics.

The time series of the federal funds rate has recently been extended back to 1928, now including several episodes during which interest rates remained near the lower bound of zero. This series is analyzed, using the method of indirect inference, by applying recent research on bounded time series to estimate a set of bounded parametric diffusion models. This combination uncouples the specification of the bounds from the law of motion. Although Louis Bachelier was the first to use arithmetic Brownian motion to model financial time series, he has often been criticized for this proposal, since the process can take on negative values. Most researchers favor processes such as geometric Brownian motion (GBM), which remains positive. Under this framework, Bachelier's proposal remains valid when specified with bounds and is shown to compare favorably when modeling the federal funds rate.

The main purpose of this paper is to compute the appropriate margin level for the stock index futures traded on the Taiwan Futures Exchange (TAIFEX) and, then, to examine the appropriateness of the real margin requirement set by the TAIFEX.

This paper develops a new approach assuming the future's prices follow a geometric Brownian motion process. Compared with the extreme value theory that has been intensively used to determine the appropriate futures margin levels, one of the advantages of the present model is no need to specify the frequency at which extremes are taken.

The evidences indicate that the theoretical margins obtained by the proposed model can provide a more accurate and flexible margin level in accordance with the market volatility.

The main limitation of this approach is that the natural logarithm of the futures prices is assumed to follow a Brownian motion process. However, such an assumption might not be practical for financial returns.

The research is helpful for the clearinghouse to set up its margins policy, especially under various conditions of volatility risks.

This paper proposes a theoretical procedure to set an appropriate futures margin for the TAIFEX. This paper also provides a better understanding of Taiwan's futures market that is newly launched and is useful for investors to hedge and speculate.

It has long been challenged that the distributions of empirical returns do not follow the log-normal distribution upon which many celebrated results of finance are based including the Black–Scholes Option-Pricing model. Borland (2002) succeeds in obtaining alternate closed form solutions for European options based on Tsallis distribution, which allow for statistical feedback as a model of the underlying stock returns. Motivated by this, we simulate two distinct time series based on initial data from NIFTY daily close values, one based on the Gaussian return distribution and the other on non-Gaussian distribution. Using techniques of non-linear dynamics, we examine the underlying dynamic characteristics of both the simulated time series and compare them with the characteristics of actual data. Our findings give a definite edge to the non-Gaussian model over the Gaussian one.

The purpose of this chapter is to study the mathematisation of finance – excessive use of mathematical models in finance – which has been widely blamed for the recent financial and economic crisis. We argue that the problem might actually be the financialisation of mathematics, as evidenced by the gradual embedding of branches of mathematics into financial economics. The concept of embeddedness, originally proposed by Polanyi, is relevant to describe the sociological relationship between fields of knowledge. After exploring the relationship between mathematics, finance and economics since antiquity, we find that theoretical developments in the 1950s and 1970s lead directly to this embedding. The key implication of our findings is the realization that it has become necessary to disembed mathematics from finance and economics, and proposes a number of partial steps to facilitate this process. This chapter contributes to the debate on the mathematisation of finance by uniquely combining a historical approach, which chronicles the evolution of the relation between mathematics and finance, with a sociological approach from the perspective of Polyani’s concept of embedding.

The purpose of this paper is twofold: first, to propose a new robust volatility ratio (RVR) that compares the intraday high–low volatility with that of the intraday open–close volatility estimator; and second, to empirically test the proposed RVR on the cross-sectional (CS) average of the constituent stocks of India’s BSE Sensex and US’s Dow Jones Industrial Average index to find the evidence of “excess volatility.”

The authors model the proposed RVR by assuming the logarithm of the price process to follow the Brownian motion. The authors have theoretically shown that the RVR is unbiased in the case of zero drift parameter. Moreover, the RVR is found to be an even function of the non-zero drift parameter.

The empirical results show that the analysis based on the RVR supports the existence of “excess volatility” in the CS average of the constituent stocks of India’s BSE Sensex and US’s Dow Jones index. In particular, the authors have observed that the CS average of individual constituent stocks of BSE Sensex is found to be more excessively volatile than the US’s Dow Jones index during the period of the study from January 2008 to September 2016, based on multiple k-day time window analysis.

The study has implications for the policy makers and practitioners who would like to understand the volatility behavior in the asset returns based on the RVR of this study. In general, the proposed model can be used as a specification tool to find whether the stock prices follow the random walk behavior or excessively volatile.

The authors contribute to the existing volatility literature in finance by proposing a new RVR based on extreme values of asset prices and absolute returns. The authors implement the bootstrap technique on RVR to find the estimates of mean and standard error for multiple k-day time windows. The RVR can capture the excess volatility by comparing two independent volatility estimators. This is possibly the first study to find the CS average of all the constituent stocks of BSE Sensex based on the RVR.

This paper aims to study fractional Brownian motion and its applications to nonlinear stochastic integral equations. Bernstein polynomials have been applied to obtain the numerical results of the nonlinear fractional stochastic integral equations.

Bernstein polynomials have been used to obtain the numerical solutions of nonlinear fractional stochastic integral equations. The fractional stochastic operational matrix based on Bernstein polynomial has been used to discretize the nonlinear fractional stochastic integral equation. Convergence and error analysis of the proposed method have been discussed.

Two illustrated examples have been presented to justify the efficiency and applicability of the proposed method. The corresponding obtained numerical results have been compared with the exact solutions to establish the accuracy and efficiency of the proposed method.

To the best of the authors’ knowledge, nonlinear stochastic Itô–Volterra integral equation driven by fractional Brownian motion has been for the first time solved by using Bernstein polynomials. The obtained numerical results well establish the accuracy and efficiency of the proposed method.

In this study, the authors investigate methods of sequential analysis to test prospectively for the existence of a unit root against stationary or explosive states in a p-th order autoregressive (AR) process monitored over time. Our sequential sampling schemes use stopping times based on the observed Fisher information of a local-to-unity parameter. In contrast to the Dickey–Fuller (DF) test statistic, the sequential test statistic has asymptotic normality. The authors derive the joint limit of the test statistic and the stopping time, which can be characterized using a 3/2-dimensional Bessel process driven by a time-changed Brownian motion. The authors obtain their limiting joint Laplace transform and density function under the null and local alternatives. In addition, simulations are conducted to show that the theoretical results are valid.