Black-Scholes Research Articles

A study on the fractional Black–Scholes option pricing model of the financial market via the Yang-Abdel-Aty-Cattani operator

PurposeFinancial mathematics is one of the most rapidly evolving fields in today’s banking and cooperative industries. In the current study, a new fractional differentiation operator with a nonsingular kernel based on the Robotnov fractional exponential function (RFEF) is considered for the Black–Scholes model, which is the most important model in finance. For simulations, homotopy perturbation and the Laplace transform are used and the obtained solutions are expressed in terms of the generalized Mittag-Leffler function (MLF).Design/methodology/approachThe homotopy perturbation method (HPM) with the help of the Laplace transform is presented here to check the behaviours of the solutions of the Black–Scholes model. HPM is well known for its accuracy and simplicity.FindingsIn this attempt, the exact solutions to a famous financial market problem, namely, the BS option pricing model, are obtained using homotopy perturbation and the LT method, where the fractional derivative is taken in a new YAC sense. We obtained solutions for each financial market problem in terms of the generalized Mittag-Leffler function.Originality/valueThe Black–Scholes model is presented using a new kind of operator, the Yang-Abdel-Aty-Cattani (YAC) operator. That is a new concept. The revised model is solved using a well-known semi-analytic technique, the homotopy perturbation method (HPM), with the help of the Laplace transform. Also, the obtained solutions are compared with the exact solutions to prove the effectiveness of the proposed work. The different characteristics of the solutions are investigated for different values of fractional-order derivatives.

Lie Symmetries and the Invariant Solutions of the Fractional Black–Scholes Equation under Time-Dependent Parameters

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.

The Discussion of Convertible Bond Valuation

With the development of financial analysis techniques, computer technology, and artificial intelligence, the valuation methods for convertible bonds in the financial markets are constantly evolving and developing. This article discusses various approaches to valuing convertible bonds, including the Black-Scholes model, the Binomial model, the Monte Carlo model, and artificial intelligence techniques like machine learning. The article provides a detailed overview of each valuation method and analyzes the strengths and weaknesses of each approach.

A tempered subdiffusive Black–Scholes model

A tempered subdiffusive Black–Scholes model

A high-order numerical scheme and its analysis for Caputo temporal-fractional Black-Scholes model: European double barrier knock-out option

A high-order numerical scheme and its analysis for Caputo temporal-fractional Black-Scholes model: European double barrier knock-out option

Using a Mix of Finite Difference Methods and Fractional Differential Transformations to Solve Modified Black–Scholes Fractional Equations

This paper discusses finding solutions to the modified Fractional Black–Scholes equation. As is well known, the options theory is beneficial in the stock market. Using call-and-pull options, investors can theoretically decide when to sell, hold, or buy shares for maximum profits. However, the process of forming the Black–Scholes model uses a normal distribution, where, in reality, the call option formula obtained is less realistic in the stock market. Therefore, it is necessary to modify the model to make the option values obtained more realistic. In this paper, the method used to determine the solution to the modified Fractional Black–Scholes equation is a combination of the finite difference method and the fractional differential transformation method. The results show that the combined method of finite difference and fractional differential transformation is a very good approximation for the solution of the Fractional Black–Scholes equation.

Accelerated numerical solutions for discretized Black–Scholes equations

Abstract Accepted by: Aris Syntetos This study thoroughly investigates the efficiency of advanced numerical extrapolation methods aimed at enhancing the convergence of vector sequences in the realm of mathematical finance. Our focus lies in the application of polynomial extrapolation techniques to calculate finite difference solutions for the Black–Scholes (BS) equation—an indispensable model in options pricing. The performance of our algorithms undergoes rigorous evaluation through a comprehensive analysis involving both simulated and real-world data. Notably, our experiments uncover that a stochastic scheme, incorporating two extrapolation strategies and a random relaxation parameter, outperforms other proposed methods, excelling in both convergence and stability metrics. Our findings underscore the potential of this numerical extrapolation method to enhance the efficiency of financial calculations, particularly in the realm of option pricing. This innovation holds promise for refining financial models and addressing specific challenges within the field of mathematical programming, providing effective solutions to the primary computational bottlenecks commonly encountered in financial decision-making.

European Option Pricing Based on FSDE Driven by Fractional Brownian Motion

In the actual financial market, the classical Black-Scholes (B-S) model cant perfectly describe the process of stock price. Besides, memory effect is an important phenomenon in financial systems. Thus, in this paper, we establish a fractional order stochastic differential equations (FSDE) which is driven by fractional Brownian motion (fBm) to describe the effect of noise memory and trend memory in financial pricing. Finally, we derive a European option pricing formula based on the established model. After conducting an empirical analysis based on the SSE 50ETF, we find that the established model performs better than the traditional one.

ANALYTICALLY PRICING EUROPEAN OPTIONS UNDER A TWO-FACTOR STOCHASTIC INTEREST RATE MODEL WITH A STOCHASTIC LONG-RUN EQUILIBRIUM LEVEL

AbstractWe construct a new stochastic interest rate model with two stochastic factors, by introducing a stochastic long-run equilibrium level into the Vasicek interest rate model which follows another Ornstein–Uhlenbeck process. With the interest rate under the Black–Scholes model being assumed to follow the newly proposed model, a closed-form representation of European option prices is successfully presented, when the analytical characteristic function of the underlying log-price under a forward measure is derived. To assess the model performance, a preliminary empirical study is conducted using S&P 500 index and its options, with the Vasicek model and an alternative two-factor Vasicek model taken as benchmarks.

THE PAUL-PAINLEVÉ APPROACH OF THE BLACK SCHOLES MODEL AND ITS EXACT AND NUMERICAL SOLUTIONS

In this article, we employ the Black Scholes model which plays a vital role in economic operation and financial market management. The Paul-Painlevé approach is used for the first time to achieve the exact wave solution to this equation. Furthermore, the numerical solution to this equation has been constructed by using the variational iteration method.

Pricing European Call Options with Visualization Based on the Binomial Model, Monte-Carlo Simulation, and Classical Black-Scholes Model

Pricing European Call Options with Visualization Based on the Binomial Model, Monte-Carlo Simulation, and Classical Black-Scholes Model

Calibration of European option pricing model in uncertain environment: Valuation of uncertainty implied volatility

Uncertain differential equations have been widely used in modeling financial markets, and option pricing formulae have been obtained by employing these equations. However, according to the existing literature, the parameter estimation of the option pricing model driven by the uncertain differential equation has not been evaluated so far, and the parameters have been assumed to be arbitrary numbers for such problems. Based on the mentioned gap, this study presents an approach for parameter estimation of European option pricing model in the uncertain environment for the first time. For this purpose, some of the most common uncertain models are considered, and the corresponding empirical examples are presented. Since implied volatility as an instrument provides traders a range of prices that securities change among them, this study evaluates the implied volatility of the European option under the geometric Liu process for the first time. Empirical comparison for both in sample and out of sample dataset based on the uncertainty framework and Black–Scholes model demonstrates that the uncertain models are more accurate for evaluating the European option prices and the implied volatilities.

Hedging with physical or cash settlement under transient multiplicative price impact

We solve the superhedging problem for European options in an illiquid extension of the Black–Scholes model, in which transactions have transient price impact and the costs and strategies for hedging are affected by physical or cash settlement requirements at maturity. Our analysis is based on a convenient choice of reduced effective coordinates of magnitudes at liquidation for geometric dynamic programming. The price impact is transient over time and multiplicative, ensuring nonnegativity of underlying asset prices while maintaining an arbitrage-free model. The basic (log-)linear example is a Black–Scholes model with a relative price impact proportional to the volume of shares traded, where the transience for impact on log-prices is modelled like in Obizhaeva and Wang (J. Financ. Mark. 16:1–32, 2013) for nominal prices. More generally, we allow nonlinear price impact and resilience functions. The viscosity solutions describing the minimal superhedging price are governed by the transient character of the price impact and by the physical or cash settlement specifications. The pricing equations under illiquidity extend no-arbitrage pricing à la Black–Scholes for complete markets in a non-paradoxical way (cf. Çetin et al. (Finance Stoch. 14:317–341, 2010)) even without additional frictions, and can recover it in base cases.

Stochastic Calculus and Its Impact On Analyzing Option Pricing

This research study the behaviour of the correction of Black-Scholes portfolios based on historical stock price data. We build a model that simulates one sample path of the stock price stochastic process at discrete time steps and track the correction over a time interval as it relates to the change in stock price over time. We also study the effect that relaxing this assumption has on the self-financing property of the replicating portfolio. We show that as the frequency of the time steps increases, the correction is more likely to be close to 0. We also show that the majority of historical stock return series that studied have caused the replicated portfolio to have a positive correction. We conclude that the Black-Scholes model can be used to find the no-arbitrage rational price for an option, a financial instrument that derives its value from the value of an underlying asset.

"Black-Scholes-Artificial Neural Network": A novel option pricing model

Purpose: This study conducts a comparative study of various options pricing models and introduces a new model. Research methodology: This paper reviews eight option pricing models, including the Black-Scholes-Merton model (BSM), Monte Carlo simulation (MC), Heston, GARCH, Lattice, Jump Diffusion models (JDM), Normal Inverse Gaussian-Cox-Ingersoll-Ross Model, and a novel model called Black-Scholes-Artificial Neural Network (BSANN). The objective is to predict the European call and put options using a payoff calculation. The underlying asset is Khodro, a famous automobile producer company in Iran, for the last year. The daily prices were also used as historical data. The primary software used for the calculations and plots was MATLAB. An Excel option pricing toolbox was used to obtain more accurate and improved results. Results: Based on the results, it can be concluded that the proposed model, BS-ANN, provides the most accurate estimation with the lowest standard deviation. Limitations: There are several limitations to be considered when choosing an underlying asset. An important factor is the availability of sufficient data on the number of shared transactions. Another limitation of this study is the absence of trading halts. Additionally, caution is crucial when selecting an appropriate number of estimated parameters. Contribution: By utilizing the presented model, researchers, individuals, investors, and stock market analysts interested in trading can enhance their estimations. Novelty: The most significant novelty of this study is the presentation of a hybrid model incorporating unique features.

A Prediction of Option Price via Two Volatility Computations with Application to a 50ETF Option

Financial derivative trading is integral to stock markets, leading to high option price volatility due to increased trading volume. Determining a reasonable option price is complex and requires extensive research in fields like Economics, Applied Mathematics, and Finance Engineering. The Black-Scholes (BS) equation provides a scientific pricing tool for options by considering five parameters: stock price (S), option strike price (K), risk-free interest rate (r), time to expiration (τ), and volatility (σ). Notably, all model parameters except volatility σ can be directly observed from market data, necessitating the determination of this parameter from historical data when applying the BS model in practice.In this report, we examine two widely used computation methods for volatility. The first method involves a simple statistical calculation of historical data, resulting in the historical volatility (HV). The second method utilizes the BS model in a "backward" manner: given any previous option price and four other parameters, we solve the BS equation to obtain the implied volatility (IV). To determine such an IV parameter, we propose a generalized fixed-point iterative solver for solving a complex nonlinear equation. By employing a well-designed initial guess, we demonstrate that this fixed-point solver achieves global and rapid convergence.The two volatilities lead to different predictions of the option price. This report examines if the BS model accurately predicts the option price using these volatilities. It presents an empirical study on a 50ETF option, where we use t-tests to check the hypothesis and determine the relationship between predicted and actual option prices. The study finds that using implied volatility in the BS model provides significantly more accurate predictions compared to historical volatility.

The Analysis of Black-Scholes Model of Option Pricing with Time-Varying Parameters on Share Prices for Capital Market

This paper studied the perception of European option which is geared towards valuation of financial assets at a prescribed time in the future. In particular, the analytic formula of Black-Scholes model was considered for share prices of Fidelity and Access banks which gave closed form prices of call options. The explicit price on the variations of maturity days is found accordingly. The closed form prices of both banks were compared. The simulation results show that: the share price determines the value of call option prices. An increase on the maturity days dominantly increases the value of call option for both Fidelity Bank, Access Bank and their future merger Bank. Merging of the two banks improved the value of call option prices. Fidelity bank has a good maximum value of call option prices during the period of investments, Kolmogorov ?Smirnov (KS) test shows that the two call option prices (Fidelity and Access) do not come from a common distribution, the normality test of both banks are not significant. The results presents to the Banks management, a basis for taking vital decisions, depending on the levels of their investments.

Effectiveness of deterministic option pricing models: new evidence from Nifty and Bank Nifty Index options

AbstractThis research delves into the empirical performance of deterministic option pricing models in the dynamic financial landscape of India. The primary focus is on uncovering pricing discrepancies and discerning whether these disparities arise from inherent limitations in the theoretical foundations of the models or are influenced by the trading behaviors of market participants. The investigation centers on the analysis of call and put option contracts for the Nifty Index and Bank Nifty Index, both extensively traded on the National Stock Exchange (NSE) of India. The study’s findings highlight that models developed to address the theoretical constraints of the benchmark Black–Scholes model demonstrate noteworthy performance. However, the complexity of these models does not consistently translate into enhanced pricing efficiency. Notably, the Black–Scholes and Practitioner Black–Scholes models exhibit superior performance across various moneyness-maturity categories. Furthermore, the research underscores the substantial impact of option contract liquidity on the efficiency of the pricing models. Specifically, highly traded at-the-money and out-of-the-money option contracts exhibit a higher level of pricing accuracy.

Option Pricing with the Logistic Return Distribution

The Black–Scholes model and many of its extensions imply a log-normal distribution of stock total returns over any finite holding period. However, for a holding period of up to one year, empirical stock return distributions (both conditional and unconditional) are not log-normal, but rather much closer to the logistic distribution. This paper derives analytic option pricing formulas for an underlying asset with a logistic return distribution. These formulas are simple and elegant and employ exactly the same parameters as B&S. The logistic option pricing formula fits empirical option prices much better than B&S, providing explanatory power comparable to much more complex models with a larger number of parameters.

Review of the Fractional Black-Scholes Equations and Their Solution Techniques

The pioneering work in finance by Black, Scholes and Merton during the 1970s led to the emergence of the Black-Scholes (B-S) equation, which offers a concise and transparent formula for determining the theoretical price of an option. The establishment of the B-S equation, however, relies on a set of rigorous assumptions that give rise to several limitations. The non-local property of the fractional derivative (FD) and the identification of fractal characteristics in financial markets have paved the way for the introduction and rapid development of fractional calculus in finance. In comparison to the classical B-S equation, the fractional B-S equations (FBSEs) offer a more flexible representation of market behavior by incorporating long-range dependence, heavy-tailed and leptokurtic distributions, as well as multifractality. This enables better modeling of extreme events and complex market phenomena, The fractional B-S equations can more accurately depict the price fluctuations in actual financial markets, thereby providing a more reliable basis for derivative pricing and risk management. This paper aims to offer a comprehensive review of various FBSEs for pricing European options, including associated solution techniques. It contributes to a deeper understanding of financial model development and its practical implications, thereby assisting researchers in making informed decisions about the most suitable approach for their needs.