Search Results (90)

We develop a high-order compact numerical scheme for solving a nonlinear Black–Scholes equation arising in option pricing under transaction costs. By leveraging a Hermite-enhanced Radial Basis Function-Finite Difference (RBF-HFD) method with three-point stencils, we achieve fourth-order spatial accuracy. The fully nonlinear PDE, driven by Gamma-dependent volatility models, is discretized via RBF-HFD in space and integrated using an explicit sixth-order Runge–Kutta scheme. Numerical results confirm the proposed method’s accuracy, stability, and its capability to capture sharp gradient behavior near strike prices. Full article

This study presents a self-contained derivation and solution of the Black and Scholes partial differential equation (PDE), replacing the standard Wiener process with a smoothed Wiener process, which is a differentiable stochastic process constructed via normal kernel smoothing. By presenting a self-contained, Itô-free derivation, this study bridges the gap between heuristic financial reasoning and rigorous mathematics, bringing forth fresh insights into one of the most influential models in quantitative finance. The smoothed Wiener process does not merely simplify the technical machinery but further reaffirms the robustness of the Black and Scholes framework under alternative mathematical formulations. This approach is particularly valuable for instructors, apprentices, and practitioners who may seek a deeper understanding of derivative pricing without relying on the full machinery of stochastic calculus. The derivation underscores the universality of the Black and Scholes PDE, irrespective of the specific stochastic process adopted, under the condition that the essential properties of stochasticity, volatility, and of no arbitrage may be preserved. Full article

Investors face the challenge of how to incorporate economic and financial forecasts into their investment strategy, especially in times of financial crisis. To model this situation, we consider a financial market consisting of a risk-free asset with a constant interest rate as well as a risky asset whose drift and volatility is influenced by a stochastic process indicating the probability of potential market downturns. We use a dynamic portfolio optimization approach in continuous time to maximize the expected utility of terminal wealth and solve the corresponding HJB equations for the general class of HARA utility functions. The resulting optimal strategy can be obtained in closed form. It corresponds to a CPPI strategy with a stochastic multiplier that depends on the information from the crisis indicator. In addition to the theoretical results, a performance analysis of the derived strategy is implemented. The specified model is fitted using historic market data and the performance is compared to the optimal portfolio strategy obtained in a Black–Scholes framework without crisis information. The new strategy clearly dominates the BS-based CPPI strategy with respect to the Sharpe Ratio and Adjusted Sharpe Ratio. Full article

The objective of this investigation is to obtain numerical solutions for a variety of mathematical models in a wide range of disciplines, such as chemical kinetics, neurosciences, nonlinear optics, metallurgical separation/alloying processes, and asset dynamics in mathematical finance. This research features numerical simulations conducted with a remarkably low error measure, providing a visual representation of the examined models in these areas. The proposed method is the double Laplace–Adomian decomposition method, which facilitates the numerical acquisition and analysis of solutions. This paper presents the first report of numerical simulations employing this innovative methodology to address these problems. The findings are expected to benefit the natural sciences, mathematical modeling, and their practical applications, representing the innovative aspect of this article. Additionally, this method can analyze many classes of partial differential equations, whether linear or nonlinear, without the need for linearization or discretization. Full article

In this paper, we study direct and inverse problems for a spatial-fractional Black–Scholes equation with space-dependent volatility. For the direct problem, we provide CN-WSGD (Crank–Nicholson and the weighted and shifted Grünwald difference) scheme to solve the initial boundary value problem. The latter aims to recover the implied volatility via observable option prices. Using a linearization technique, we rigorously derive a mathematical formulation of the inverse problem in terms of a Fredholm integral equation of the first kind. Based on an integral equation, an efficient numerical reconstruction algorithm is proposed to recover the coefficient. Numerical results for both problems are provided to illustrate the validity and effectiveness of proposed methods. Full article

Since the emergence of the Black–Scholes model (BSM) in the early 1970s, models for the pricing of financial options have been developed and evolved with mathematical tools that provide greater efficiency and accuracy in the valuation of these assets. In this research, we have used the generalized conformable derivatives associated with seven obtained conformable models with a closed-form solution that is similar to the traditional Black and Scholes. In addition, an empirical analysis was carried out to test the models with Mexican options contracts listed in 2023. Six foreign options were also tested, in particular three London options and three US options. With this sample, in addition to applying the seven generalized conformable models, we compared the results with the Heston model. We obtained much better results with the conformable models. Similarly, we decided to apply the seven conformable models to the data of the Morales et al. article, and we again determined that the conformable models greatly outperform the approximation of the Black, Scholes (BS), and Merton model with time-varying parameters and the basic Khalil conformable equation. In addition to the base sample, it was decided to test the strength of the seven generalized conformable models on 10 stock options that were out-sampled. In addition to the MSE results, for the sample of six options whose shares were traded in the London and New York stock markets, we tested the positivity and stability of the results. We plotted the values of the option contracts obtained by applying each of the seven generalized conformable models, the values of the contracts obtained by applying the traditional Heston model, and the market value of the contracts. Full article

In this study, we present a novel mathematical framework for pricing financial derivates and modelling asset behaviour by bringing together fractional Brownian motion (fBm), fuzzy logic, and jump processes, all aligned with no-arbitrage principle. In particular, our mathematical developments include fBm defined through Mandelbrot-Van Ness kernels, and advanced mathematical tools such Molchan martingale and BDG inequalities ensuring rigorous theoretical validity. We bring together these different concepts to model uncertainties like sudden market shocks and investor sentiment, providing a fresh perspective in financial mathematics and derivatives pricing. By using fuzzy logic, we incorporate subject factors such as market optimism or pessimism, adjusting volatility dynamically according to the current market environment. Fractal mathematics with the Hurst exponent close to zero reflecting rough market conditions and fuzzy set theory are combined with jumps, representing sudden market changes to capture more realistic asset price movements. We also bridge the gap between complex stochastic equations and solvable differential equations using tools like Feynman-Kac approach and Girsanov transformation. We present simulations illustrating plausible scenarios ranging from pessimistic to optimistic to demonstrate how this model can behave in practice, highlighting potential advantages over classical models like the Merton jump diffusion and Black-Scholes. Overall, our proposed model represents an advancement in mathematical finance by integrating fractional stochastic processes with fuzzy set theory, thus revealing new perspectives on derivative pricing and risk-free valuation in uncertain environments. Full article

It has been previously demonstrated that stochastic volatility emerges as the gauge field necessary to restore local symmetry under changes in stock prices in the Black–Scholes (BS) equation. When this occurs, a Merton–Garman-like equation emerges. From the perspective of manifolds, this means that the Black–Scholes and Merton–Garman (MG) equations can be considered locally equivalent. In this scenario, the MG Hamiltonian is a special case of a more general Hamiltonian, here referred to as the gauge Hamiltonian. We then show that the gauge character of volatility implies a specific functional relationship between stock prices and volatility. The connection between stock prices and volatility is a powerful tool for improving volatility estimations in the stock market, which is a key ingredient for investors to make good decisions. Finally, we define an extended version of the martingale condition, defined for the gauge Hamiltonian. Full article

This paper addresses the portfolio optimisation problem within the jump-diffusion stochastic differential equations (SDEs) framework. We begin by recalling a fundamental theoretical result concerning the existence of solutions to the Black–Scholes–Merton partial differential equation (PDE), which serves as the cornerstone for subsequent analysis. Then, we explore a range of financial applications, spanning scenarios characterised by the absence of jumps, the presence of jumps following a log-normal distribution, and jumps following a distribution of greater generality. Additionally, we delve into optimising more complex portfolios composed of multiple risky assets alongside a risk-free asset, shedding new light on optimal allocation strategies in these settings. Our investigation yields novel insights and potentially groundbreaking results, offering fresh perspectives on portfolio management strategies under jump-diffusion dynamics. Full article

This paper presents a comprehensive investigation into the solution existence of stochastic functional integral equations within real separable Banach spaces, emphasizing the establishment of sufficient conditions. Leveraging advanced mathematical tools including probability measures of noncompactness and Petryshyn’s fixed-point theorem adapted for stochastic processes, a robust analytical framework is developed. Additionally, this paper introduces the Euler–Karhunen–Loève method, which utilizes the Karhunen–Loève expansion to represent stochastic processes, particularly suited for handling continuous-time processes with an infinite number of random variables. By conducting thorough analysis and computational simulations, which also involve implementing the Euler–Karhunen–Loève method, this paper effectively highlights the practical relevance of the proposed methodology. Two specific instances, namely, the Delay Cox–Ingersoll–Ross process and modified Black–Scholes with proportional delay model, are utilized as illustrative examples to underscore the effectiveness of this approach in tackling real-world challenges encountered in the realms of finance and stochastic dynamics. Full article

This paper explores the implications of modifying the canonical Heisenberg commutation relations over two simple systems, such as the free particle and the tunnel effect generated by a step-like potential. The modified commutation relations include position-dependent and momentum-dependent terms analyzed separately. For the position deformation case, the corresponding free wave functions are sinusoidal functions with a variable wave vector $k(x)$. In the momentum deformation case, the wave function has the usual sinusoidal behavior, but the energy spectrum becomes non-symmetric in terms of momentum. Tunneling probabilities depend on the deformation strength for both cases. Also, surprisingly, the quantum mechanical model generated by these modified commutation relations is related to the Black–Scholes model in finance. In fact, by taking a particular form of a linear position deformation, one can derive a Black–Scholes equation for the wave function when an external electromagnetic potential is acting on the particle. In this way, the Scholes model can be interpreted as a quantum-deformed model. Furthermore, by identifying the position coordinate x in quantum mechanics with the underlying asset S, which in finance satisfies stochastic dynamics, this analogy implies that the Black–Scholes equation becomes a quantum mechanical system defined over a random spatial geometry. If the spatial coordinate oscillates randomly about its mean value, the quantum particle’s mass would correspond to the inverse of the variance of this stochastic coordinate. Further, because this random geometry is nothing more than gravity at the microscopic level, the Black–Scholes equation becomes a possible simple model for understanding quantum gravity. Full article

This paper addresses the valuation of European options, which involves the complex and unpredictable dynamics of fractal market fluctuations. These are modeled using the $\alpha$-order time-fractional Black–Scholes equation, where the Caputo fractional derivative is applied with the parameter $\alpha$ ranging from 0 to 1. We introduce a novel, high-order numerical scheme specifically crafted to efficiently tackle the time-fractional Black–Scholes equation. The spatial discretization is handled by a tailored finite point scheme that leverages exponential basis functions, complemented by an L1-discretization technique for temporal progression. We have conducted a thorough investigation into the stability and convergence of our approach, confirming its unconditional stability and fourth-order spatial accuracy, along with $(2-\alpha )$-order temporal accuracy. To substantiate our theoretical results and showcase the precision of our method, we present numerical examples that include solutions with known exact values. We then apply our methodology to price three types of European options within the framework of the time-fractional Black–Scholes model: (i) a European double barrier knock-out call option; (ii) a standard European call option; and (iii) a European put option. These case studies not only enhance our comprehension of the fractional derivative’s order on option pricing but also stimulate discussion on how different model parameters affect option values within the fractional framework. Full article

Since the early 1970s, the study of Black–Scholes (BS) partial differential equations (PDEs) under the Efficient Market Hypothesis (EMH) has been a subject of active research in financial engineering. It has now become obvious, even to casual observers, that the classical BS models derived under the EMH framework fail to account for a number of realistic price evolutions in real-time market data. An alternative approach to the EMH framework is the Fractal Market Hypothesis (FMH), which proposes better and clearer explanations of market behaviours during unfavourable market conditions. The FMH involves non-local derivatives and integral operators, as well as fractional stochastic processes, which provide better tools for explaining the dynamics of evolving market anomalies, something that classical BS models may fail to explain. In this work, using the FMH, we derive a time-fractional Black–Scholes partial differential equation (tfBS-PDE) and then transform it into a heat equation, which allows for ease of implementing a high-order numerical scheme for solving it. Furthermore, the stability and convergence properties of the numerical scheme are discussed, and overall techniques are applied to pricing European put option problems. Full article

The Black–Scholes formula is an important formula for pricing a contingent claim in complete financial markets. This formula can be obtained under the assumption that the investor’s strategy is carried out according to a self-financing criterion; hence, there arise a set of self-financing portfolios corresponding to different contingent claims. The natural questions are: If an investor invests according to self-financing portfolios in the financial market, what are the maximal and minimal distributions of the investor’s wealth on some specific interval at the terminal time? Furthermore, if such distributions exist, how can the corresponding optimal portfolios be constructed? The present study applies the theory of backward stochastic differential equations in order to obtain an affirmative answer to the above questions. That is, the explicit formulations for the maximal and minimal distributions of wealth when adopting self-financing strategies would be derived, and the corresponding optimal (self-financing) portfolios would be constructed. Furthermore, this would verify the benefits of diversified portfolios in financial markets: that is, do not put all your eggs in the same basket. Full article

In this paper, we consider the time-fractional Black–Scholes model with deterministic, time-varying coefficients. These time parametric constituents produce a model with greater flexibility that may capture empirical results from financial markets and their time-series datasets. We make use of transformations to reduce the underlying model to the classical heat transfer equation. We show that this transformation procedure is possible for a specific risk-free interest rate and volatility of stock function. Furthermore, we reverse these transformations and apply one-dimensional optimal subalgebras of the infinitesimal symmetry generators to establish invariant solutions. Full article