Black-Scholes Research Articles

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.

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.

Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model

We present a unified, market-complete model that integrates both Bachelier and Black–Scholes–Merton frameworks for asset pricing. The model allows for the study, within a unified framework, of asset pricing in a natural world that experiences the possibility of negative security prices or riskless rates. Unlike the classical Black–Scholes–Merton, we show that option pricing in the unified model differs depending on whether the replicating, self-financing portfolio uses riskless bonds or a single riskless bank account. We derive option price formulas and extend our analysis to the term structure of interest rates by deriving the pricing of zero-coupon bonds, forward contracts, and futures contracts. We identify a necessary condition for the unified model to support a perpetual derivative. Discrete binomial pricing under the unified model is also developed. In every scenario analyzed, we show that the unified model simplifies to the standard Black–Scholes–Merton pricing under specific limits and provides pricing in the Bachelier model limit. We note that the Bachelier limit within the unified model allows for positive riskless rates. The unified model prompts us to speculate on the possibility of a mixed multiplicative and additive deflator model for risk-neutral option pricing.

Quantum computational finance for martingale asset pricing in incomplete markets

A derivative is a financial asset whose future payoff is a function of underlying assets. Pricing a financial derivative involves setting up a market model, finding a martingale (“fair game”) probability measure for the model from the existing asset prices, and using that probability measure to price the derivative. When the number of underlying assets and/or the number of market outcomes in the model is large, pricing can be computationally demanding. In this work, we first formulate the pricing problem in a linear algebra setting, including the realistic setting of incomplete markets where derivatives do not have a unique price. We show that the problem can be solved with a variety of quantum techniques such as quantum linear programming and the quantum linear systems algorithm. While in previous works the martingale measure is assumed to be given, here it is extracted from market variables akin to bootstrapping, a common practice among financial institutions. We discuss the quantum zero-sum game algorithm and the quantum simplex algorithm as viable subroutines. For quantum linear systems solvers, we formalize a new market assumption milder than market completeness, which allows the potential for large speedups. Towards prototype use cases, we conduct numerical experiments motivated by the Black–Scholes–Merton model.

A High-Order Numerical Method Based on a Spatial Compact Exponential Scheme for Solving the Time-Fractional Black–Scholes Model

This paper investigates a high-order numerical method based on a spatial compact exponential scheme for solving the time-fractional Black–Scholes model. Firstly, the original time-fractional Black–Scholes model is converted into an equivalent time-fractional advection–diffusion reaction model by means of a variable transformation technique. Secondly, a novel high-order numerical method is constructed with (2−α) accuracy in time and fourth-order accuracy in space based on a spatial compact exponential scheme, where α is the fractional derivative. The uniqueness of solvability of the derived numerical method is rigorously discussed. Thirdly, the unconditional stability and convergence of the derived numerical method are rigorously analyzed using the Fourier analysis technique. Finally, numerical examples are presented to test the effectiveness of the derived numerical method. The proposed numerical method is also applied to solve the time-fractional Black–Scholes model, whose exact analytical solution is unknown; numerical results are illustrated graphically.

An Efficient Numerical Scheme for a Time-Fractional Black–Scholes Partial Differential Equation Derived from the Fractal Market Hypothesis

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.