Black-Scholes Research Articles

Exploring chaos and sensitivity in the Ivancevic option pricing model through perturbation analysis.

This study explores the Ivancevic Option Pricing Model, a nonlinear wave-based alternative to the Black-Scholes model, using adaptive nonlinear Schrödingerr equations to describe the option-pricing wave function influenced by stock price and time. Our focus is on a comprehensive analysis of this equation from multiple perspectives, including the study of soliton dynamics, chaotic patterns, wave structures, Poincaré maps, bifurcation diagrams, multistability, Lyapunov exponents, and an in-depth evaluation of the model's sensitivity. To begin, a wave transformation is applied to convert the partial differential equation into an ordinary differential equation, from which soliton solutions are derived using the [Formula: see text] method. We explore various forms of the option price function at different time points, including singular-kink, periodic, hyperbolic, trigonometric, exponential, and complex solutions. Furthermore, we simulate 3D surface plots and 2D graphs for the real, imaginary, and modulus components of some of the obtained solutions, assigning specific parameter values to enhance visualization. These graphical representations offer valuable insights into the dynamics and patterns of the solutions, providing a clearer understanding of the model's behavior and potential applications. Additionally, we analyze the system's dynamic behavior when a perturbing force is introduced, identifying chaotic patterns using the Lyapunov exponent, Sensitivity, multistability analysis, RK4 method, wave structures, bifurcation diagrams, and Poincaré maps.

Black–Scholes 50 Years Later: Has the Outperformance of Passive Option Strategies Finally Faded?

Slightly over fifty years ago, the Black–Scholes option pricing model revolutionized investing by enabling a shift from linear to non-linear payoff structures. Myron Scholes later published two papers documenting the performance of passive option strategies that outperformed the underlying index on a risk–return basis. The options market has evolved considerably over the last fifty years from an open outcry trading structure with options being single-listed to a high-frequency computer-based market. This paper re-evaluates the trilogy of foundational studies to determine whether passive-option-enhanced portfolios still produce superior performance in the current high-frequency options market environment.

A Study on the Pricing of Convertible Bonds in the Chinese Market Based on the LSMC Model

Convertible bonds, which possess both stock and option characteristics, currently lack a sufficiently effective pricing method for the complex and specific clauses found in Chinese convertible bonds. This study employs structured model pricing and numerical experiment methods for the pricing of Chinese convertible bonds. The structured model selected is the Black-Scholes model, while the numerical experiment methods include the binomial tree pricing model and the Least Squares Monte Carlo (LSMC) model. A proposal is made to introduce an implied volatility model in the LSMC model pricing method, incorporating the Heston model to consider the impact of the implied volatility of the underlying stock in the real market, and further adjusting the LSMC model by estimating the volatility of the underlying stock using the implied volatility model. Empirical results indicate that numerical methods have higher pricing accuracy than structured models, with the LSMC model demonstrating the highest accuracy. It is also found that there is a certain premium phenomenon in the Chinese market for convertible bonds. Furthermore, the accuracy of the model improves after the volatility adjustment, indicating that implied volatility has explanatory power for the specific clauses contained in convertible bonds, and this influence is reflected in pricing.

On the Distribution of the Mixed-Lognormal-Weibull Option Pricing Model

This paper intends to test whether the Mixed-Lognormal-Weibull Distribution (MLWD) option pricing model comes from the same distribution and whether the model is a good fit in Black-Scholes option pricing model. The data for this study were obtained from Australian Clearing House of Australian Securities Exchange (ASX) which consist of 50 enlisted stock as products of monthly market summary for long term options collected from January, 3rd 2017 to December, 31st 2017, comprising 720 trading days arranged in accordance to 25, 27, 28, 29 and 30 maturity days. Maximum Likelihood Estimate was used to obtain the parameters of both the lognormal and Weibull distributions which were applied in Black-Scholes model. The data were test Wilcoxon Rank Sum test since the mixture model became distribution and Chi-square goodness-of-fit test for the model fit, and the result shows that the mixture model does not follow any of the lognormal or Weibull distributions. The result also shows that the mixture model is a good fit in Black-Scholes option pricing model with the P-value>0.05 when they are shorter maturity days with small sample sizes than longer maturity days with larger sample sizes. Hence, the model is recommended to be used for financial practitioners who are interested in modeling option pricing.

Transfer Learning with Pre-trained ResNet-50: Predicting SPX Option Prices

Due to the volatility and complexity of the options market, it has exacerbated the difficulty of predicting option prices and attracted widespread attention. In recent years, many machine learning (ML) techniques have been explored for this purpose. This paper implements the advanced ML models to boost the accurate prediction of the option price, which includes the comparison between traditional ML techniques Random Forest and Support Vector Regression (SVR), as well as the Convolutional Neural Network (CNN) ResNet-50. Additionally, we further compare the performance of ResNet-50 pre-trained on ImageNet through transfer learning. These methods are used to predict the implied volatility, which is then transformed into option pricing using the Black–Scholes Option Pricing Model (BSOPM). Finally, we compare the performance of these models based on error analysis using RMSE and R². The experimental results show that the pre-trained ResNet-50 reduces the error by 6% compared to the un-pretrained ResNet-50, by 51.2% compared to Decision Trees, and by 51.1% compared to SVR. The predicted option prices align well with the actual values, demonstrating the feasibility of applying pre-trained ResNet-50 on option price prediction.

A Novel Fourth-Order Finite Difference Scheme for European Option Pricing in the Time-Fractional Black–Scholes Model

This paper addresses the valuation of European options, which involves the complex and unpredictable dynamics of fractal market fluctuations. These are modeled using the α-order time-fractional Black–Scholes equation, where the Caputo fractional derivative is applied with the parameter α 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−α)-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.

Optimal approximations for the free boundary problems of the space-time fractional Black-Scholes equations using a combined physics-informed neural network

The combined physics-informed neural network is employed to deal with the free boundary problems of fractional Black-Scholes equations. The solution assumption and the loss function are determined, the transfer learning is borrowed, the combined neural network with data enhancement layer is designed, then the classical Black-Scholes model is numerically solved and the comparative analysis of numerical results under different neural networks is made. For further insight into the long-term memory of fluctuation, the free boundary problems of the space-time Black-Scholes equations under Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivatives are studied. The corresponding empirical analyses are presented and the optimal exercise boundaries of American put option are simulated. The market analysis shows that introducing fractional calculus tools and neural network algorithms into American put option pricing can yield more realistic prediction results. The work provides a viable method for subsequent researchers to study American option pricing using fractional calculus and neural networks combined with true market data and to deal with the free boundary problems in other research fields.

Physical information-guided multidirectional gated recurrent unit network fusing attention to solve the Black-Scholes equation

Reasonable option pricing is crucial in the financial derivatives market. Finding analytical solutions for the Black-Scholes (BS) equation, particularly for American options or with fluctuating volatility and interest rates, is challenging. BS equations exhibit strong time-series characteristics, with asset prices typically adhering to geometric Brownian motion. To address the BS equations, we propose a sequence-to-sequence model guided by physical information (PI), called PiMGA. The PiMGA fuses a multidirectional gated recurrent unit (GRU) network with an attention module, where multidirectional GRU enhances the coding performance of the input sequences and the attention module balances the feature weights of the hidden variables. Prior physical knowledge in BS equations is jointly used as a constraint, forming the penalty function for objective optimization. This allows PiMGA to serve as an efficient approximation function in the learning paradigm of physically informed machine learning to solve BS equations. BS equations with various complexities illustrate the accuracy and feasibility of PiMGA for numerical solutions. Furthermore, the out-of-distribution generalization ability of PiMGA is verified by predicting the Nasdaq 100 index.

Option Pricing Based on Neural Stochastic Differential Equations

Firstly, based on the Black-Scholes stock price model, the neural stochastic differential equation (NSDE) model was established by parameterizing the asset return rate and volatility as a drift network and a diffusion network, respectively. Secondly, in the empirical analysis, the underlying asset as a single stock option was used as the research object, and real stock data was used for the network training and testing. The experimental results show that the NSDE model can overcome the defects of the constant assumption of the Black-Scholes model. Finally, for the case where the price of the underlying asset of the option was unobservable, we proposed that the price of any target option and the price of a known option could be constrained within the Wasserstein distance of their risk-neutral equivalent martingale measure, and theoretically proved the method.

A Robust numerical technique based on the chromatic polynomials for the European options regulated by the time-fractional Black–Scholes equation

AbstractRisk mitigation and control are critical for investors in the finance sector. Purchasing significant instruments that eliminate the risk of price fluctuation helps investors manage these risks. In theory and practice, option pricing is a substantial issue among many financial derivatives. In this scenario, most investors adopt the Black–Scholes model to describe the behavior of the underlying asset in option pricing. The exceptional memory effect prevalent in fractional derivatives makes it easy to understand and explain the approximation of financial options in terms of their inherited characteristics prompted by the given reason. Finding numerical solutions that are both successful and suitably precise is crucial when working with financial fractional differential equations. Hence, this paper proposes an innovative method, designated the Chromatic polynomial collocation method (CPM), for the theoretical study of the Time fractional Black–Scholes equation (TFBSE) that regulates European call options. The newly developed numerical algorithm CPM is on a functional basis of the Chromatic polynomials of Complete graphs (Kn) and operational matrices of the basis polynomials. The CPM transforms the TFBSE into a framework of nonlinear algebraic equations with the help of operational matrices and equispaced collocation points. The fractional orders in the PDE are concerned in the Caputo sense. The CPM findings further corroborate the results of the most recent numerical schemes to show the effectiveness of the suggested numerical algorithm.

Time-consistent asset allocation for risk measures in a Lévy market

Focusing on gains & losses relative to a risk-free benchmark instead of terminal wealth, we consider an asset allocation problem to maximize time-consistently a mean-risk reward function with a general risk measure which is (i) law-invariant, (ii) cash- or shift-invariant, and (iii) positively homogeneous, and possibly plugged into a general function. Examples include (relative) Value at Risk, coherent risk measures, variance, and generalized deviation risk measures. We model the market via a generalized version of the multi-dimensional Black–Scholes model using α-stable Lévy processes and give supplementary results for the classical Black–Scholes model. The optimal solution to this problem is a Nash subgame equilibrium given by the solution of an extended Hamilton–Jacobi–Bellman equation. Moreover, we show that the optimal solution is deterministic under appropriate assumptions.

A Comprehensive Methodology for Analysing Options Trading: Utilizing Historical Data and the Black-Scholes Model

This paper presents a systematic methodology for analysing options trading, integrating user inputs with historical stock data to facilitate informed decision-making. The process begins with the collection of user-specified stock tickers and expiration dates, followed by the retrieval of historical price data to assess past stock performance. Key components of the analysis include calculating historical volatility, fetching the option chain, and filtering options into put and call categories. The methodology employs the Black-Scholes model to estimate fair option prices, which are further evaluated by calculating expected returns. The analysis culminates in the ranking of the top three put and call options based on their expected returns, providing users with actionable insights. Additionally, the methodology incorporates visual representations of stock prices and option data, alongside calculations of option Greeks to assess risk sensitivity. This comprehensive approach empowers traders to navigate the complexities of options trading with a clear understanding of potential outcomes and associated risks.

Stock Return Autocorrelations and Expected Option Returns

We show that the return autocorrelation of underlying stock is an important determinant of expected equity option returns. Using an extended Black-Scholes model incorporating the presence of stock return autocorrelation, we demonstrate that expected returns of both call and put options are increasing in the return autocorrelation coefficient of the underlying stock. Consistent with this insight, we find strong empirical support in the cross-section of average returns of equity options. Our paper highlights the necessity to control for stock return autocorrelation when studying option return predictability. This paper was accepted by Agostino Capponi, finance. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.03071 .

Further Reflections Based on the Black–Scholes (B-S) Model

Abstract The Black–Scholes (B-S) model is considered by the academic environment one of the greatest achievements of financial economics. Yet it brings with it some conundrums. The model is often used in a manner that contradicts one of its assumptions, and its predictions are not supported by market reality. Here we address, from the perspective of philosophy of science, an additional issue related to this model: the distinction between its explanatory and predictive capabilities.

Series form solutions of time–space fractional Black–Scholes model via extended He-Aboodh algorithm

The objective of the current study is analyze linear and nonlinear time–space fractional Black–Scholes models via modified homotopy perturbation method (m-HPM). In current investigation, memory effects in financial markets are explored through fractional derivative in Caputo sense. The effectiveness of proposed methodology is checked numerically by finding residual errors and presented in tables. These tables also provide a benchmark for the comparison with already existing results in literature. Furthermore, solutions are graphically analyzed via 3D and contour plots across a range of parameters under varying market conditions. Analysis confirms the efficiency of m-HPM for predicting solutions of time–space fractional Black–Scholes models. The current study can contribute in understanding the applications of fractional calculus in finance, and can be a valuable computational tool for pricing financial derivatives in fractional environments.

Current Developments and Limitations for Pricing Options

Valuing options holds significant importance within the derivative market, serving as a cornerstone for investors, traders, and financial institutions. Over time, economists have dedicated efforts to develop models that accurately reflect the complexities of this market environment. The evolution of economic theories can be observed through the examination of key option pricing models, which have undergone refinements and enhancements over the decades. By exploring models such as the Black-Scholes model, the Merton model, and the Cox-Ross-Rubinstein model, it is possible to gain valuable insights into the progression of economic thought in this field. Each of these models offers unique perspectives on option pricing, illustrating various principles and methodologies employed to assess and value options. Moreover, understanding the extensions and limitations of these models provides crucial insights into their applicability and effectiveness in real-world scenarios. By conducting an in-depth examination of these models, the objective of this article is to offer a nuanced comprehension of how economic theories have developed and adjusted to the ever-changing conditions of the derivative market.

Extreme ATM skew in a local volatility model with discontinuity: joint density approach

This paper concerns a local volatility model in which the volatility takes two possible values, and the specific value depends on whether the underlying price is above or below a given threshold. The model is known, and a number of results have been obtained for it. In particular, option pricing formulas and a power-law behaviour of the implied volatility skew have been established in the case when the threshold is taken at the money. In this paper, we derive an alternative representation of option pricing formulas. In addition, we obtain an approximation of option prices by the corresponding Black–Scholes prices. Using this approximation streamlines obtaining the aforementioned behaviour of the skew. Our approach is based on the natural relationship of the model with skew Brownian motion and consists of the systematic use of the joint distribution of this stochastic process and some of its functionals.

Convertible Bond Pricing in Chinese Transportation Industry : A Comparison Methods Between Binomial Tree model and Black-Scholes Model

Convertible Bond Pricing in Chinese Transportation Industry : A Comparison Methods Between Binomial Tree model and Black-Scholes Model

Analysis of Put and Call Option Pricing on BCA Stock Using the Black-Scholes Model: Financial and Risk Management Perspective

Analysis of Put and Call Option Pricing on BCA Stock Using the Black-Scholes Model: Financial and Risk Management Perspective

The Impact of Stochastic Volatility on Option Pricing Using the Black-Scholes Model: Empirical Evidence from NVIDIA

This paper investigates the impact of volatility on option pricing using the Black-Scholes model. Utilizing historical stock and option data from NVIDIA, this study demonstrates the limitations of the Black-Scholes model in capturing dynamic market conditions. The empirical analysis reveals significant discrepancies between model predictions and actual market prices, highlighting the importance of accurate volatility estimation. The findings suggest that incorporating better volatility measures can improve the accuracy of option pricing models. Additionally, the implications of these findings for traders and risk managers are discussed, emphasizing the need for more sophisticated approaches to volatility estimation in financial markets. This study adds to the body of knowledge by presenting actual data regarding the efficiency of different volatility metrics in option pricing. Moreover, it underscores the necessity of continuous improvement in financial modeling techniques to adapt to changing market dynamics. The study's results align with previous research that has shown the limitations of the Black-Scholes model and the benefits of alternative volatility estimation methods. This paper also discusses the potential application of these findings to improving trading strategies and risk management practices. The practical implications are significant, suggesting that traders and risk managers should consider incorporating advanced volatility measures into their models to enhance decision-making processes and financial outcomes. The conclusions drawn from this study highlight the critical role of accurate volatility estimation in achieving more reliable option pricing and underscore the need for ongoing research and innovation in this field.