Accurate pricing of exotic derivatives remains challenging due to complex asset dynamics and restrictive assumptions in classical models. This study prices a European “up-and-out” barrier call option on OPEC oil using two approaches: the parametric Black-Scholes model and a semi-parametric Kernel Density Estimation (KDE) model. All computations are performed under strict no-arbitrage conditions, and model performance is evaluated using an efficiency-based metric that measures deviation from market-consistent prices. Backtesting on 34 years of daily OPEC oil prices, spanning high volatility, structural breaks, and non-normal returns, shows that Black-Scholes performs adequately under log-normality but its pricing error increases sharply for realistic barrier contracts. The KDE model reduces pricing error by ∼2% on average compared to Black-Scholes, with improvements exceeding 5% for strikes >1.5S₀ and barriers >2K where knock-out risk is high. The advantage of KDE is most pronounced during periods of market stress and fat-tailed returns. The contribution of this paper is twofold: it extends non-parametric KDE pricing to commodity barrier options under no-arbitrage conditions, and it provides oil exporters with a data-driven hedging tool that captures multimodality and jumps better than Black-Scholes, thereby improving revenue risk management for oil-dependent economies.
In this paper, we introduce and investigate the N -prime ideal graph of a finite commutative ring with unity. The concept of prime ideal graphs over rings has been studied by many authors. Let R be a finite commutative ring with unity and let J be a proper ideal of R . The N -prime ideal graph of R with respect to J , denoted by G J , is the simple undirected graph whose vertex set consists of the elements of the ring R and the two distinct vertices are adjacent if and only if their product belongs to that ideal and at least one of them belongs to the sum of that ideal and the nilradical of the ring. The aim of this article is to study the structural properties of the N -prime ideal graph and establish the connection between the algebraic properties of ideals and their graph theoretic properties.In particular, the N -prime ideal graphs are characterized as star, pineapple, and complete multipartite graphs. The subgraphs induced by nilpotent and idempotent elements are investigated,along with their connections to zero-divisor and product maximal graphs. Furthermore, the properties of the N -prime ideal graph as a cograph and a threshold graph are studied and illustrated with examples. The study demonstrates how N -prime ideal properties can be effectively represented and analyzed using graph theoretic methods.
In this paper, we discuss the existence and uniqueness of solutions for nonlinear fractional integro-differential equations with the ψ-Caputo fractional derivative under slit-strip integral boundary conditions. Fractional differential equations have been the focus of intense interest, since they can express memory and hereditary properties of various materials and processes in science and engineering. The ψ-Caputo fractional derivative which extends some known fractional operators is a very fruitful instrument for modeling multidimensional dynamical systems and thus applicable to a wide variety of problems. The everything aim of current paper is to get certain sufficient conditions that ensures the multi- solvability as well as uniqueness of the boundary value problem. The study is conducted on the pre-determined space which is a Banach space by using nonlinear functional analysis. In particular, suitable estimates for the nonlinearities involved in the equation are obtained by Hölders inequality and the Mönch fixed-point theorem is the main tool for proving existence of solution. In addition, further conditions are required for ensuring the uniqueness of the obtained solution. The present work contributes to the theory of fractional integro-differential equations with nonlocal boundary conditions and generalizes and improves many results in the literature. The addition of slit-strip integral boundary conditions and the ψ-Caputo fractional derivative results in a more generalized- setting for the study of such types of problems. Finally, a numerical example is provided to illustrate the feasibility and effectiveness of the theoretical results and to check the correctness of the theoretical results.
A multiplicative lattice is a complete lattice endowed with a commutative, associative, join distributive multiplication in which 1 acts as a multiplicative identity, providing an abstract framework for the study of the ideal theory of commutative rings. In this paper, the concept of an S-prime element is introduced, generalizing the notion of an S-prime ideal of a commutative ring with unity to multiplicative lattices. Let L be a multiplicative lattice with k distinct prime elements and S be the multiplicative closed subset of L. Initially, the fundamental characterizations of S-prime elements are investigated. Furthermore, it is shown that every element of a reduced lattice is an idempotent and radical element, respectively. In particular, every proper element of a reduced lattice is an S-prime element. The number of S-prime elements in a given multiplicative is determined and S is the upset of k-1 prime elements in L. Moreover, the sets of non S-prime elements and nilpotent elements have the same cardinality in L. It is further shown that the radical of every nilpotent element is an S-prime element. The uniqueness of an S-prime element in a local lattice is proved and consequently, every zero divisor is a nilpotent element. Finally, several illustrative examples are presented to demonstrate the structural behavior of S-prime elements in L.
Accurate prediction of atmospheric pollutant transport is essential for air quality assessment and environmental management. Classical advection-diffusion models often neglect the combined effects of temperature and humidity, despite their significant influence on pollutant transport and dispersion within the atmospheric boundary layer. This study develops a two-dimensional meteorology-dependent advection-diffusion-reaction model that incorporates temperature-dependent thermodiffusion through the Soret effect and humidity-dependent modification of advective transport. The governing equation is solved numerically using the Explicit Finite Difference (FD) and Crank-Nicolson (CN) schemes to evaluate their accuracy, stability, and computational performance. Von Neumann stability analysis is employed to establish the stability conditions of both numerical methods. The analysis shows that the FD scheme is conditionally stable and requires restrictive time-step selection to satisfy the Courant-Friedrichs-Lewy condition, whereas the CN scheme remains unconditionally stable for the diffusion component and demonstrates superior numerical robustness under practical simulation conditions. Numerical experiments implemented in MATLAB reveal that the CN scheme produces smoother concentration profiles, reduced numerical diffusion, and improved solution accuracy, particularly for long simulation periods. The simulations further indicate that increased wind velocity enhances pollutant transport, higher relative humidity suppresses dispersion and increases pollutant residence time, and elevated temperatures promote stronger atmospheric mixing through thermally induced diffusion. These findings demonstrate that incorporating meteorological variables substantially improves the physical realism of atmospheric dispersion models. The study concludes that the Crank-Nicolson method provides a more reliable and efficient numerical framework for simulating pollutant transport under varying environmental conditions and offers a practical tool for urban air quality assessment, environmental planning, and evidence-based pollution control strategies.
