It is widely noted that market capitalisation weighted portfolios are inefficient and underperform an equal weighted portfolio over the long-term. However, at least since 2016, an equal weighted portfolio of stocks in the S&P500 has significantly underperformed the market capitalisation weighted portfolio. In this paper, we analyse this underperformance using stochastic portfolio theory. We show that the equal weighted portfolio does appear to outperform the market capitalisation weighted portfolio over the long-term but with periods of significant short-term underperformance. In addition, we find that concentration in the market capitalisation weighted portfolio has increased in recent years and has contributed to the recent underperformance together with a significantly lower level of diversification benefits. Furthermore, we highlight an approach to improve the performance of a portfolio by dynamically selecting a market cap or an equal weighting using a rudimentary linear regression model.

This paper aims to present a robust computational technique utilizing finite difference schemes for accurately solving time fractional reaction–diffusion models, which are prevalent in chemical and biological phenomena. The time-fractional derivative is treated in the Caputo sense, addressing both linear and nonlinear scenarios. The proposed schemes were rigorously evaluated for stability and convergence. Additionally, the effectiveness of the developed schemes was validated through various linear and nonlinear models, including the Allen–Cahn equation, the KPP–Fisher equation, and the Complex Ginzburg–Landau oscillatory problem. These models were tested in one-, two-, and three-dimensional spaces to investigate the diverse patterns and dynamics that emerge. Comprehensive numerical results were provided, showcasing different cases of the fractional order parameter, highlighting the schemes’ versatility and reliability in capturing complex behaviors in fractional reaction–diffusion dynamics.

The generalized regularized long wave (GRLW) equation is solved numerically by using a distributed approximating functional (DAF) method realized by the regularized Hermite local spectral kernel. Test problems including propagation of single solitons, interaction of two and three solitons, and conservation properties of mass, energy, and momentum of the GRLW equation are discussed to test the efficiency and accuracy of the method. Furthermore, using the Maxwellian initial condition, we show that the number of solitons which are generated can be approximately determined. Comparisons are made between the results of the proposed method, analytical solutions, and numerical methods. It is found that the method under consideration is a viable alternative to existing numerical methods.

In this paper, the pricing performance of the generalised autoregressive conditional heteroskedasticity (GARCH) option pricing model is tested when applied to Bitcoin (BTCUSD). In addition, implied volatility indices (30, 60-and 90-days) of BTCUSD and the Cyptocurrency Index (CRIX) are generated by making use of the symmetric GARCH option pricing model. The results indicate that the GARCH option pricing model produces accurate European option prices when compared to market prices and that the BTCUSD and CRIX implied volatility indices are similar when compared, this is consistent with expectations because BTCUSD is highly weighted when calculating the CRIX. Furthermore, the term structure of volatility indices indicate that short-term volatility (30 days) is generally lower when compared to longer maturities. Furthermore, short-term volatility tends to increase to higher levels when compared to 60 and 90 day volatility when large jumps occur in the underlying asset.

In this paper, two univariate generalised autoregressive conditional heteroskedasticity (GARCH) option pricing models are applied to Bitcoin and the Cryptocurrency Index (CRIX). The first model is symmetric and the other takes asymmetric effects into account. Furthermore, the accuracy of the GARCH option pricing model applied to Bitcoin is tested. Empirical results indicate that asymmetry is not an important factor to consider when pricing options on Bitcoin or CRIX, this is consistent with findings in the literature. In addition, the GARCH option pricing model provides realistic price discovery within the bid-ask spreads suggested by the market.