Volume 31, Issue 4

The permuted congruential generators is a set of pseudorandom number generators released by Melissa E. O’Neill in 2014. The original technical report outlined several lightweight scrambling techniques designed for the linear congruential generator. Each scrambling technique offered some improvement to the quality of the linear congruential generator. However, the real strength of the scrambling techniques was that they could be combined into multiple overall stronger scramblers. The technical report concludes with the creation of the PCG library, a popular pseudorandom number generation library that implements several generators described in the technical report. Starting from the observation that the paper’s work was narrowly focused on implementing their scrambling techniques for specific linear congruential generators, we explore the permuted congruential generator scrambling techniques and their potential for being applied to other pseudorandom number generators by generalizing the scrambling techniques to work across different pseudorandom number generators.

In this note we will discuss and review some recent results concerning well-known financial mathematical problems such as portfolio construction, dynamic trading, option pricing, etc. We will use some Python codes concerning the above and we will compare the results with the existing methods and techniques. We propose also a new type of multi-asset options; the options on correlation. Using this kind of options one can refine more effectively the profit function of his/her portfolio when this contain two or more assets. It is mathematically certain that, in practice, someone will eventually apply the techniques described in this paper. This is because, regarding the portfolio construction problem, we allow the investor to employ any forecasting technique – e.g., statistical methods, machine learning, behavioral finance, intuition, etc. Subsequently, the investor can enhance both the return and safety of their portfolio by incorporating call and put options. In contrast, for the derivative pricing problem, it is evident that there is no room for forecasts (see volatility for example), as pricing involves two counterparties – the seller and the buyer – making it, metaphorically, a dance for two. For this reason, the pricing methodology proposed in this paper is model-free, ensuring that the resulting prices are fully consistent with the market values of available contracts. Moreover, the investor decides at which price to buy or sell a contract based on practical, statically implementable hedging strategies that we propose in this work. That is, any pricing method should justify why an investor ought to buy or sell a derivative at the proposed price. In other words, the method must provide a clear, economically sound rationale-typically grounded in no-arbitrage principles, replication arguments, or explicit hedging strategies – that links the quoted price to actionable, implementable decisions for market participants. What remains to be explored are advanced forecasting techniques that account for events affecting the stocks of interest to the investor, as well as the documentation of hedging strategies for path-dependent options.

We revisit the method of antithetic variates, perhaps the simplest yet profound variance reduction technique among many others, with the aim of comprehensively improving the method in its general formulation in the seminal works A new Monte Carlo technique: Antithetic variates [Hammersley and Morton, Math. Proc. Cambridge Philos. Soc. 52 (1956), 449–475] and Antithetic variates revisited [Fishman and Huang, Commun. ACM 26 (1983), 964–971]. The achieved advancement under this general formulation contrasts with the conventional approach based on equally weighted two variates with negative correlation, commonly referred to as the method of antithetic variates in various contexts. In pursuit of the aim, we investigate its second-order structure in depth and introduce an adaptive algorithm designed to optimize the weights among multiple variates throughout the primary Monte Carlo estimation process. In order to effectively demonstrate the theoretical advancements, we present numerical results throughout the presentation, vividly showcasing the potential effectiveness of the proposed approach and adaptive algorithm in the face of varying weights.

This work suggests different Monte Carlo algorithms for solving large systems of linear algebraic equations arising from the numerical solution of the Dirichlet problem for the Helmholtz equation. Approach based on boundary integral representations, vector randomization algorithm, method of fundamental solutions, stochastic projection algorithm, and randomized singular value decomposition are applied. It is shown that the use of stochastic iterative refinement and preconditioning can significantly improve the accuracy and stability of the computations. Simulation results are presented, demonstrating the effectiveness of the proposed methods.

In logistic regression models, different patterns of data points in observed data can cause large bias in parameter estimates, especially when separation is present in the observed data. In the frequentist approach, maximum likelihood estimates fail to exist when separation occurs in the observed data. In the Bayesian approach, the existence of posterior means is also affected by the presence of separation depending on the form of prior distributions. In this paper, a non-informative G-prior for Bayesian method is proposed to reduce the bias of the parameter estimation when prior distributions of parameters do not have information and separation is present in the data. In this proposed method, the information from observed data and ideas of a normal regression model are implemented to form the mean and standard deviation of the normal prior distributions. The Markov chain Monte Carlo algorithm is then employed by using Metropolis Hasting algorithm to sample for the target posterior distribution. Results show that estimates from the proposed Bayesian method are more accurate and reliable than from the classical approach when separation is present or is not present in the observed data. Moreover, the proposed Bayesian method can provide better estimated results compared to the default Cauchy prior Bayesian approach when the prior distribution does not have information. The proposed method is also validated by applying it to a case study of MROZ data.

This study develops explicit algebraic expressions for the single and product moments of order statistics derived from the generalized Bilal (GB) distribution. These expressions facilitate the computation of means, variances and covariances of order statistics for sample sizes up to n = 10 {n=10} with specified parameter values. The derived moments serve as the foundation for constructing the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the location and scale parameters applicable to both complete and type-II right censored samples. Additionally, the study explores the prediction of unobserved order statistics in type-II right censored samples. The theoretical results are validated through a simulation study, while a real data example highlights their practical utility. These findings establish a robust framework for statistical inference based on order statistics from the GB distribution.

In this paper, we study the class of first order Periodic Generalized Autoregressive Conditional heteroscedasticity processes ( PGARCH ⁢ ( 1 , 1 ) {\mathrm{PGARCH}(1,1)} for short) in which the parameters in the volatility process are allowed to switch between different regimes. First, we establish necessary and sufficient conditions for a PGARCH ⁢ ( 1 , 1 ) {\mathrm{PGARCH}(1,1)} process to have a unique stationary solution (in periodic sense) and for the existence of moments of any order. Next, we are able to estimate the unknown parameters involved in model via the so-called generalized method of moments (GMM). We construct an estimator and establish its asymptotic properties. Specifically, we demonstrate its consistency and asymptotic normality. The GMM estimator in some cases can be more efficient than the least squares estimator (LSE) and the quasi maximum likelihood estimator (QMLE). Some simulation studies are also performed to highlight the impact of our theoretical results.