We introduced a new method to compute the European Call (and Put) Option price under the assumption of multifractional Brownian motion (mBm). The reason why we need a procedure for estimating the Option price is due to the absence of a closed formula for this process. To compute the Option price, we first simulated the logarithmic price under mBm and, by using a discount factor, we computed the option’s pay-off. Then, we fitted the best probability distribution associated to the discounted pay-off, computing the European Call Option price as its average.

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Market inefficiency is a latent concept, and it is difficult to be measured by means of a single indicator. In this paper, following both the adaptive market hypothesis (AMH) and the fractal market hypothesis (FMH), we develop a new time-varying measure of stock market inefficiency. The proposed measure, called composite efficiency index (CEI), is estimated as the synthesis of the most common efficiency measures such as the returns’ autocorrelation, liquidity, volatility, and a new measure based on the Hurst exponent, called the Hurst efficiency index (HEI). To empirically validate the indicator, we compare different European stock markets in terms of efficiency over time.

In this paper, we introduce a new method to compute the European Call Option price (ct) under multi-fractional Brownian motion (mBm) with deterministic Hurst function. We build a mathematical framework using a Lebovits et al. study to approximate mBm to fractional Brownian motion (fBm). As a result we obtain ct , through the simulation of the logarithmic price under mBm, using a Vasicek model for the discount factor. Finally, we compare the results with those computed with the Black Scholes model and Call market price (SPX).

The purpose of the present study is to provide further evidence of the weak form efficiency of the Bahrain Bourse. The research methodology is based on daily closing index values of the Bahrain Bourse from 2011 to 2015 in order to test the efficiency of the stock market while runs test, Autocorrelation Function, and advance tools such as ARCH and GARCH models and Hurst Index to provide further evidence of the weak form efficiency of the Bahrain stock market. For instance, a volatile and inefficient stock market has a negative impact on textile and apparel industry in the Kingdom of Bahrain, which is one of the most prosperous and attractive industries in the country. The empirical results revealed that Bahrain stock market does not follow normal distribution and the successive price changes are not independent. Further, ARCH effect is significant and indicative of a time-varying conditional volatility. There is an arbitrage opportunity and extreme mispricing in the Bahrain stock market as indicated by the GARCH (1,1) model. The results of the Hurst exponent also confirm the inefficiency of the market. The results of these tests are consistent indicating that the Bahrain stock market is inefficient