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Artykuły w czasopismach na temat „Black and Scholes options pricing model”

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Data publikacji: 3 czerwca 2025

Data aktualizacji: 31 lipca 2025

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1

Li, Chenwei. "A Study of Option Pricing Models with Market Price Adjustments: Empirical Analysis Beyond the Black-Scholes Model." Advances in Economics, Management and Political Sciences 137, no. 1 (2024): 94–98. https://doi.org/10.54254/2754-1169/2024.18702.

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In 1973, Fischer Black and Myron Scholes unveiled the Black-Scholes option pricing model, a groundbreaking contribution that profoundly influenced the domain of option pricing theory. The introduction of the Black-Scholes pricing formula has garnered substantial acclaim across both academic and industrial spheres, leading to its widespread dissemination and application. This formula not only underscores its vital significance but also exemplifies its unique position as a cornerstone of financial theory, reshaping how options are valued and traded in markets worldwide. However, in the real fina

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Kim, Sol. "The Best Option Pricing Model for KOSPI 200 Weekly Options." Korean Journal of Financial Studies 51, no. 5 (2022): 499–521. http://dx.doi.org/10.26845/kjfs.2022.10.51.5.499.

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This study finds the best option pricing model for KOSPI 200 weekly options. It examines the in-sample pricing, out-of-sample pricing and hedging performances of the short-term options with a maximum maturity of seven days or less, which have not been analyzed in previous studies. The Black and Scholes (1973) model, Ad Hoc Black-Scholes model, and stochastic volatility and jumps models are compared. As a result, one of the Ad Hoc BlackScholes models, the absolute smile model using the strike price as an independent variable shows the best performance. However, its performance is not significan

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3

Rani,, Dr Pushpa. "Analysis of Option Prices Using Black Scholes Model." INTERANTIONAL JOURNAL OF SCIENTIFIC RESEARCH IN ENGINEERING AND MANAGEMENT 08, no. 05 (2024): 1–5. http://dx.doi.org/10.55041/ijsrem34488.

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A mathematical formula used in finance to calculate the theoretical price of an option and ascertain its option premium is known as the Black Scholes option pricing model, which aids option traders in making informed decisions. This article estimates the option premium of various call and put options using the Black Scholes Model. The three distinct option chains chosen for this essay are all Mid-Cap companies that are listed on the Indian National Stock Exchange. The companies are Suzlon Energy, Kalyan jewellers India, and Exide Industries Ltd. The analysis demonstrates that the options are e

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4

Chauhan, Arun, and Ravi Gor. "COMPARISON OF THREE OPTION PRICING MODELS FOR INDIAN OPTIONS MARKET." International Journal of Engineering Science Technologies 5, no. 4 (2021): 54–64. http://dx.doi.org/10.29121/ijoest.v5.i4.2021.203.

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  Black-Scholes option pricing model is used to decide theoretical price of different Options contracts in many stock markets in the world. In can find many generalizations of BS model by modifying some assumptions of classical BS model. In this paper we compared two such modified Black-Scholes models with classical Black-Scholes model only for Indian option contracts. We have selected stock options form 5 different sectors of Indian stock market. Then we have found call and put option prices for 22 stocks listed on National Stock Exchange by all three option pricing models. Finall

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5

Wu, Yawei. "Options Pricing Comparison between the Black-Scholes Model and the Binomial Tree Model: A Case Study of American Equity Option and European-style Index Option." BCP Business & Management 32 (November 22, 2022): 168–77. http://dx.doi.org/10.54691/bcpbm.v32i.2885.

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In recent years, quantitative researchers used a wide range of models to price options, from the Black-Scholes model to more complex models such as the Heston model. This paper aims to analyze the effectiveness of the Black-Scholes model and the Binomial Tree model by using them to price Berkshire Hathaway’s equity options and European-style S&P 100 index options. The method used in this paper is gathering the market data of the options first. Second, using the data gathered to price the options by applying the Black-Scholes and Binomial Tree models. Third, comparing the derived theoretica

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6

SHOKROLLAHI, FOAD. "THE VALUATION OF EUROPEAN OPTION UNDER SUBDIFFUSIVE FRACTIONAL BROWNIAN MOTION OF THE SHORT RATE." International Journal of Theoretical and Applied Finance 23, no. 04 (2020): 2050022. http://dx.doi.org/10.1142/s0219024920500223.

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In this paper, we propose an extension of the Merton model. We apply the subdiffusive mechanism to analyze European option in a fractional Black–Scholes environment, when the short rate follows the subdiffusive fractional Black–Scholes model. We derive a pricing formula for call and put options and discuss the corresponding fractional Black–Scholes equation. We present some features of our model pricing model for the cases of [Formula: see text] and [Formula: see text].

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7

Blake, D. "Option pricing models." Journal of the Institute of Actuaries 116, no. 3 (1989): 537–58. http://dx.doi.org/10.1017/s0020268100036696.

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S., Dakurah, Odoi F.N.D., Kongyir B.K., Ampaw-Asiedu M.O., and K. Dedu V. "A Model for Pricing Insurance Using Options." Journal of Research in Business, Economics and Management 10, no. 3 (2018): 1971–88. https://doi.org/10.5281/zenodo.3956116.

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Traditional Expected Value and Bayesian Methods of pricing insurance products are not robust both under minimal data and frequent portfolio adjustments. Deriving a partial di_erential equation for the price of a an insurance put, parallel is struck with the reverse Black Scholes partial di_erential equation for pricing call options. With appropriate parameter translation of the Black Scholes model, a Pure Premium valuation function that is an improvement over the traditional methods of pricing insurance products results. Its robustness is illustrated with the pricing of a third party insurance

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9

Alp, Özge Sezgin. "The Performance of Skewness and Kurtosis Adjusted Option Pricing Model in Emerging Markets." International Journal of Finance & Banking Studies (2147-4486) 5, no. 3 (2016): 70–84. http://dx.doi.org/10.20525/ijfbs.v5i3.285.

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In this study, the option pricing performance of the adjusted Black-Scholes model proposed by Corrado and Su (1996) and corrected by Brown and Robinson (2002), is investigated and compared with original Black Scholes pricing model for the Turkish derivatives market. The data consist of the European options written on BIST 30 index extends from January 02, 2015 to April 24, 2015 for given exercise prices with maturity April 30, 2015. In this period, the strike prices are ranging from 86 to 124. To compare the models, the implied parameters are derived by minimizing the sum of squared deviations

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10

BUCKLEY, JAMES J., and ESFANDIAR ESLAMI. "PRICING STOCK OPTIONS USING BLACK-SCHOLES AND FUZZY SETS." New Mathematics and Natural Computation 04, no. 02 (2008): 165–76. http://dx.doi.org/10.1142/s1793005708001008.

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We use the basic Black-Scholes equation for pricing European stock options but we allow some of the parameters in the model to be uncertain and we model this uncertainty using fuzzy numbers. We compute the fuzzy number for the call value of option with and without uncertain dividends. This fuzzy set displays the uncertainty in the option's value due to the uncertainty in the input values to the model. We also correct an error in a recent paper which also fuzzified the Black-Scholes equation.

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11

Sun, Yesen, Wenxiu Gong, Hongliang Dai, and Long Yuan. "Numerical Method for American Option Pricing under the Time-Fractional Black–Scholes Model." Mathematical Problems in Engineering 2023 (February 20, 2023): 1–17. http://dx.doi.org/10.1155/2023/4669161.

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The fractional Black–Scholes model has had limited applications in financial markets. Instead, the time-fractional Black–Scholes equation has attracted much research interest. However, it is difficult to obtain the analytic expression for American option pricing under the time-fractional Black–Scholes model. This paper will present an operator-splitting method to price the American options under the time-fractional Black–Scholes model. The fractional partial differential complementarity problem (FPDCP) that the American option price satisfied is split into two subproblems: a linear boundary va

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12

Wu, Shujin, and Shiyu Wang. "European Option Pricing Formula in Risk-Aversive Markets." Mathematical Problems in Engineering 2021 (July 31, 2021): 1–17. http://dx.doi.org/10.1155/2021/9713521.

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In this study, using the method of discounting the terminal expectation value into its initial value, the pricing formulas for European options are obtained under the assumptions that the financial market is risk-aversive, the risk measure is standard deviation, and the price process of underlying asset follows a geometric Brownian motion. In particular, assuming the option writer does not need the risk compensation in a risk-neutral market, then the obtained results are degenerated into the famous Black–Scholes model (1973); furthermore, the obtained results need much weaker conditions than t

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13

Lin, Wensheng. "Black-Scholes Model’s application in rainbow option pricing." BCP Business & Management 32 (November 22, 2022): 500–507. http://dx.doi.org/10.54691/bcpbm.v32i.2988.

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In this paper, we use excel as a tool to explore the pricing of rainbow options and their advantages based on the Black-Scholes Model. Two-color rainbow options are mainly explored in the paper, in which the underlying stocks are Apple and ExxonMobil. Simulating the price of two stocks is performed through Excel. Return on the corresponding European options and rainbow options is obtained after that. Next, the differences between the return on rainbow options and European options and pricing on rainbow option are analyzed. Finally, sensitivity analysis is carried out to further explore rainbow

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14

Alfajriyah, Aimmatul Ummah, Endah R. M. Putri, Daryono Budi Utomo, and Moch Taufik Hakiki. "Stock Option Pricing Using Binomial Trees with Implied Volatility." Jurnal Matematika, Statistika dan Komputasi 20, no. 3 (2024): 724–42. http://dx.doi.org/10.20956/j.v20i3.34476.

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The Black-Scholes model provides an analytical solution in option pricing and has been widely used in finance. This model assumes constant volatility. Pricing option incorporating implied volatility is conducted using implied binomial tree. This study aims to simulate the prices of put options and call options using implied binomial trees, binomial trees and the Black-Scholes model and determine the factors that influence option prices. The simulation was conducted using Matlab. The option price resulted from implied binomial tree and binomial tree are compared with the option prices of the Bl

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15

Lee, Jung-Kyung. "On a Free Boundary Problem for American Options Under the Generalized Black–Scholes Model." Mathematics 8, no. 9 (2020): 1563. http://dx.doi.org/10.3390/math8091563.

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We consider the problem of pricing American options using the generalized Black–Scholes model. The generalized Black–Scholes model is a modified form of the standard Black–Scholes model with the effect of interest and consumption rates. In general, because the American option problem does not have an exact closed-form solution, some type of approximation is required. A simple numerical method for pricing American put options under the generalized Black–Scholes model is presented. The proposed method corresponds to a free boundary (also called an optimal exercise boundary) problem for a partial

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16

Song, Lina, and Weiguo Wang. "Solution of the Fractional Black-Scholes Option Pricing Model by Finite Difference Method." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/194286.

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This work deals with the put option pricing problems based on the time-fractional Black-Scholes equation, where the fractional derivative is a so-called modified Riemann-Liouville fractional derivative. With the aid of symbolic calculation software, European and American put option pricing models that combine the time-fractional Black-Scholes equation with the conditions satisfied by the standard put options are numerically solved using the implicit scheme of the finite difference method.

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17

Dr., M. Tulasinadh* Dr.R. Mahesh. "THE GREEKS & BLACK AND SCHOLE MODEL" TO EVALUATE OPTIONS PRICING & SENSITIVITY IN INDIAN OPTIONS MARKET." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 4, no. 5 (2017): 74–78. https://doi.org/10.5281/zenodo.801245.

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Derivatives’ trading is a core part of the Indian Stock Market in the Current Scenario. Trading volumes in stock options have grown up tremendously during recent years. This also leads to be high volatility in the options prices Options Pricing is crucial factor for hedging and Speculative activities. Pricing plays a vital role for option writers. In this paper we have tried to find out the price of an option in the future and its sensitivity through the Greek & Black and Scholes Option pricing model. Many option traders rely on the “Greeks” to evaluate option positions and to determine op

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18

Yan, Runze. "Option pricing and risk hedging for Visa." BCP Business & Management 32 (November 22, 2022): 203–10. http://dx.doi.org/10.54691/bcpbm.v32i.2889.

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As the core of the option transaction, the option price changing with the supply and demand in the market is a variable which affects the profit and loss of both trading sides directly. In the 20th century, multitudinous econometric pricing models proposed lacked universal recognition until the Black Scholes Merton model came out. This paper focuses on the stocks and options from Visa Inc. to do the article consisting of calibration, option pricing and hedging using fundamental Black Scholes Merton model and the extensive jump model mainly under the seldom used method. The article demonstrates

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19

SOBEHART, JORGE R., and SEAN C. KEENAN. "UNCERTAINTY IN PRICING TRADABLE OPTIONS." International Journal of Theoretical and Applied Finance 06, no. 02 (2003): 103–17. http://dx.doi.org/10.1142/s0219024903001864.

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In this paper we introduce an options pricing model consistent with the level of uncertainty observed in the options market. By assuming that the price at which an option can be traded is intrinsically uncertain, either because of the inability to hedge continuously or because of errors in the estimation of the security's volatility and interest rates, random delays in the execution of orders or information deficiencies, we show that the Black-Scholes model produces a biased estimate of the expected value of tradable options. Information deficiencies lead to a call-put relationship that reduce

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20

Antwi Baafi, Joseph. "The Nexus Between Black-Scholes-Merton Option Pricing and Risk: A Case of Ghana Stock Exchange." Archives of Business Research 10, no. 5 (2022): 140–52. http://dx.doi.org/10.14738/abr.105.12350.

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Even though option pricing and its market activities are not new, in Ghana the idea of trading options is yet to be realized. One popular method in pricing options is known as Black-Scholes-Merton option pricing model. Even though option pricing activities are not currently happening on the Ghana Stock Exchange, authors looked at the possibilities and preparedness of the GES to start trading such financial instrument. The main objective of this study therefore was to know how Black-Scholes-Merton model could be used to help in appropriate option value and undertake a risk assessment of stocks

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21

Wang, Lujian, Minqing Zhang, and Zhao Liu. "The Progress of Black-Scholes Model and Black-Scholes-Merton Model." BCP Business & Management 38 (March 2, 2023): 3405–10. http://dx.doi.org/10.54691/bcpbm.v38i.4314.

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Black-Scholes (BS) model was first proposed in 1973, which has been modified by Robert Merton as the Black-Scholes-Merton (BSM) model subsequently. Contemporarily, these two models have been widely used and praised by financial scholars as well as employees. Plenty of scholars have tried to verify the accuracy of the and expressed their views on the existing defects in above models. Based on the existing literature, this article first introduces and derives the two models step by step and discusses the basic assumptions for these models. Subsequently, the applications of the two models are dem

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22

Li, Yuxuan. "Review and Practice of Option Pricing Research." Advances in Economics, Management and Political Sciences 7, no. 1 (2023): 320–28. http://dx.doi.org/10.54254/2754-1169/7/20230250.

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This paper reviews the past research on option pricing, including the Bachelier option pricing formula, Black-Scholes model, and the Cox-Ross-Rubinstein (CRR) binomial option pricing model. Then some practical calculations are performed on the Black-Scholes model and CRR binomial model and the differences between the results of these two models are analyzed for investors to choose the appropriate model for option prices. From the practical calculation results, it is verified that the Black-Scholes model can give out the analytical solutions of option prices and decrease the calculation of solu

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23

LO, C. F., P. H. YUEN, and C. H. HUI. "PRICING BARRIER OPTIONS WITH SQUARE ROOT PROCESS." International Journal of Theoretical and Applied Finance 04, no. 05 (2001): 805–18. http://dx.doi.org/10.1142/s021902490100122x.

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The square root constant elasticity of variance (CEV) process has been paid little attention in previous research on valuation of barrier options. In this paper we derive analytical option pricing formulae of up-and-out options with this process using the eigenfunction expansion technique. We develop an efficient algorithm to compute the eigenvalues where the basis functions in the formulae are the confluent hypergeometric functions. The numerical results obtained from the formulae are compared with the corresponding model prices under the Black–Scholes model. We find that the differences in t

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24

Dere, Zainab Olabisi, Gbeminiyi Musibau Sobamowo, and Antonio Marcos de Oliveira Siqueira. "Analytical Solutions of Black-Scholes Partial Differential Equation of Pricing for Valuations of Financial Options using Hybrid Transformation Methods." Journal of Engineering and Exact Sciences 8, no. 1 (2022): 15223–01. http://dx.doi.org/10.18540/jcecvl8iss1pp15223-01i.

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Black–Scholes partial differential equation is a generally acceptable model in financial markets for option pricing. However, without variable transformations, the provision of symbolic solutions to the variable coefficient partial differential equation is not a straight-forward task. Moreover, the coefficients of the Black–Scholes can depend on the time and the asset price which makes the analytical solution of the Black–Scholes model very difficult to develop. In this paper, analytical solutions of the model of valuations of financial options are presented using Laplace and differential tran

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25

Lindgren, Jussi. "A Generalized Model for Pricing Financial Derivatives Consistent with Efficient Markets Hypothesis—A Refinement of the Black-Scholes Model." Risks 11, no. 2 (2023): 24. http://dx.doi.org/10.3390/risks11020024.

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This research article provides criticism and arguments why the canonical framework for derivatives pricing is incomplete and why the delta-hedging approach is not appropriate. An argument is put forward, based on the efficient market hypothesis, why a proper risk-adjusted discount rate should enter into the Black-Scholes model instead of the risk-free rate. The resulting pricing equation for derivatives and, in particular, the formula for European call options is then shown to depend explicitly on the drift of the underlying asset, which is following a geometric Brownian motion. It is conjectu

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26

Bhat, Aparna, and Kirti Arekar. "Empirical Performance of Black-Scholes and GARCH Option Pricing Models during Turbulent Times: The Indian Evidence." International Journal of Economics and Finance 8, no. 3 (2016): 123. http://dx.doi.org/10.5539/ijef.v8n3p123.

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Exchange-traded currency options are a recent innovation in the Indian financial market and their pricing is as yet unexplored. The objective of this research paper is to empirically compare the pricing performance of two well-known option pricing models – the Black-Scholes-Merton Option Pricing Model (BSM) and Duan’s NGARCH option pricing model – for pricing exchange-traded currency options on the US dollar-Indian rupee during a recent turbulent period. The BSM is known to systematically misprice options on the same underlying asset but with different strike prices and maturities resulting in

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Zapart, Christopher A. "Beyond Black–Scholes: A Neural Networks-Based Approach to Options Pricing." International Journal of Theoretical and Applied Finance 06, no. 05 (2003): 469–89. http://dx.doi.org/10.1142/s0219024903002006.

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The paper presents two alternative schemes for pricing European and American call options, both based on artificial neural networks. The first method uses binomial trees linked to an innovative stochastic volatility model. The volatility model is based on wavelets and artificial neural networks. Wavelets provide a convenient signal/noise decomposition of the volatility in the non-linear feature space. Neural networks are used to infer future volatility levels from the wavelets feature space in an iterative manner. The bootstrap method provides the 95% confidence intervals for the options price

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MA, CHENGHU. "PREFERENCES, LÉVY JUMPS AND OPTION PRICING." Annals of Financial Economics 03, no. 01 (2007): 0750001. http://dx.doi.org/10.1142/s2010495207500017.

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This paper derives an equilibrium formula for pricing European options and other contingent claims which allows incorporating impacts of several important economic variable on security prices including, among others, representative agent preferences, future volatility and rare jump events. The derived formulae is general and flexible enough to include some important option pricing formulae in the literature, such as Black–Scholes, Naik–Lee, Cox–Ross and Merton option pricing formulae. The existence of jump risk as a potential explanation of the moneyness biases associated with the Black–Schole

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Xu, Lingling, Hongjie Zhang, and Fu Lee Wang. "Pricing of Arithmetic Average Asian Option by Combining Variance Reduction and Quasi-Monte Carlo Method." Mathematics 11, no. 3 (2023): 594. http://dx.doi.org/10.3390/math11030594.

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Financial derivatives have developed rapidly over the past few decades due to their risk-averse nature, with options being the preferred financial derivatives due to their flexible contractual mechanisms, particularly Asian options. The Black–Scholes stock option pricing model is often used in conjunction with Monte Carlo simulations for option pricing. However, the Black–Scholes model assumes that the volatility of asset returns is constant, which does not square with practical financial markets. Additionally, Monte Carlo simulation suffers from slow error convergence. To address these issues

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Xu, Feng. "Bifractional Black-Scholes Model for Pricing European Options and Compound Options." Journal of Systems Science and Information 8, no. 4 (2020): 346–55. http://dx.doi.org/10.21078/jssi-2020-346-10.

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AbstractRecent empirical studies show that an underlying asset price process may have the property of long memory. In this paper, it is introduced the bifractional Brownian motion to capture the underlying asset of European options. Moreover, a bifractional Black-Scholes partial differential equation formulation for valuing European options based on Delta hedging strategy is proposed. Using the final condition and the method of variable substitution, the pricing formulas for the European options are derived. Furthermore, applying to risk-neutral principle, we obtain the pricing formulas for th

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Singh, Shivam, and Vipul . "Performance of Black-Scholes model with TSRV estimates." Managerial Finance 41, no. 8 (2015): 857–70. http://dx.doi.org/10.1108/mf-06-2014-0177.

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Purpose – The purpose of this paper is to test the pricing performance of Black-Scholes (B-S) model, with the volatility of the underlying estimated with the two-scale realised volatility measure (TSRV) proposed by Zhang et al. (2005). Design/methodology/approach – The ex post TSRV is used as the volatility estimator to ensure efficient volatility estimation, without forecasting error. The B-S option prices, thus obtained, are compared with the market prices using four performance measures, for the options on NIFTY index, and three of its constituent stocks. The tick-by-tick data are used in t

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Purwandari, Diana. "PENGARUH PEMBAGIAN DIVIDEN MELALUI MODEL BLACK-SCHOLES." Jurnal Lebesgue : Jurnal Ilmiah Pendidikan Matematika, Matematika dan Statistika 2, no. 3 (2021): 351–54. http://dx.doi.org/10.46306/lb.v2i3.111.

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Stock trading has a risk that can be said to be quite large due to fluctuations in stock prices. In stock trading, one alternative to reduce the amount of risk is options. The focus of this research is on European options which are financial contracts by giving the holder the right, not the obligation, to sell or buy the principal asset from the writer when it expires at a predetermined price. The Black-Scholes model is an option pricing model commonly used in the financial sector. This study aims to determine the effect of dividend distribution through the Black-Scholes model on stock prices.

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Јањић, Драган. "Практични оквир Блек-Шолсовог модела вредновања европске кол опције: економска и математичка интерпретација // The practical framework of the Black-Scholes model of pricing a european call option: economical and mathematical interpretation". ACTA ECONOMICA 12, № 21 (2014): 141. http://dx.doi.org/10.7251/ace1421141j.

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Резиме: Још 1973. године и објављивањем рада под називом „Oдређивање цијене опција и корпоративних обавеза“ (The Pricing of Options and Corporate Liabilities), Блек Фишер и Мајрон Шолс су направили револуцију у свијету финансија. Дакле, они су развили модел за вредновање опција, познатији као Блек-Шолсов модел. Као такав, модел се данас налази у самом средишту економске теорије и савремених финансија. Такође, неопходно је истаћи и рад Роберта Мертона који је дао математичко објашњење датог проблема. За допринос развоју економске теорије Мајрон Шoлс и Роберт Мертон су 1997. године добили Нобело

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SOBEHART, J. R., and S. C. KEENAN. "A PARADOX OF INTUITION: HEDGING THE LIMIT OR HEDGING IN THE LIMIT?" International Journal of Theoretical and Applied Finance 05, no. 07 (2002): 729–36. http://dx.doi.org/10.1142/s0219024902001705.

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Here we review the notion of covergence in Itô calculus and its application to the Black-Scholes options pricing model and its extensions. The concept of covergence is fundamental to the development of the differential calculus of stochastic processes. It is also the key to understanding the validity of the no arbitrage condition imposed by Black and Scholes (1973) that leads to their options pricing equation.

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Luo, YuLin, and ZhaoYu Wang. "A Review of the Option Pricing Model and Further Development." Advances in Economics, Management and Political Sciences 64, no. 1 (2023): 96–103. http://dx.doi.org/10.54254/2754-1169/64/20231499.

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The Black Scholes model and binomial tree model have been the main research objects of scholars in the past fifty years. This article summarizes the optimization process of these two classic option pricing models to understand the different directions of optimizing the models and to provide ideas for future model improvement. Improvements from scholars to the Black-Scholes model mainly focus on the basic model and the relevant variables involved in option pricing, while the optimization of the binomial tree model focuses on the reduction of pricing errors as well as the improvement of model fi

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Achchab, B., A. Cheikh Maloum, and A. Qadi El Idrissi. "Pricing European and American Options by SPH Method." International Journal of Computational Methods 17, no. 08 (2019): 1950043. http://dx.doi.org/10.1142/s0219876219500439.

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In this paper, the meshless smoothed particle hydrodynamic (SPH) method is applied for solving the Black–Scholes model for European and American options, which are governed by a generalized Black–Scholes partial differential equation. We use the [Formula: see text]-method and SPH for discretizing the governing equation in time variable and option pricing, respectively. To validate our SPH method, we compare it with the analytical solution and also the finite difference method. The numerical tests demonstrate the accuracy and robustness of our method.

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Sisodia, Neha, and Ravi Gor. "A STUDY OF OPTION PRICING MODELS WITH DISTINCT INTEREST RATES." International Journal of Engineering Science Technologies 6, no. 2 (2022): 90–104. http://dx.doi.org/10.29121/ijoest.v6.i2.2022.310.

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This paper analyses the effect of different interest rates on Black-Scholes’ and Heston Option Pricing Model. We discuss the concept of interest rate in the two Models. We compare the two models for the parameter –‘Interest Rate’. A mathematical tool, UMBRAE (Unscaled Mean Bounded Relative Absolute Error) is used to compare the two models for pricing European call options. NSE (National Stock Exchange) is used for real market data and comparison is done through Moneyness (which is defined as the percentage difference of stock price and strike price) and Time-To-Maturity. Mathematical software

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Lee, Min-Ku, and Kyu-Hwan Jang. "Pricing Parisian Option under a Stochastic Volatility Model." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/956454.

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We study the pricing of a Parisian option under a stochastic volatility model. Based on the manipulation problem that barrier options might create near barriers, the Parisian option has been designed as an extended barrier option. A stochastic volatility correction to the Black-Scholes price of the Parisian option is obtained in a partial differential equation form and the solution is characterized numerically.

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Behera, Prashanta kumar, and Dr Ramraj T. Nadar. "Dynamic Approach for Index Option Pricing Using Different Models." Journal of Global Economy 13, no. 2 (2017): 105–20. http://dx.doi.org/10.1956/jge.v13i2.460.

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Option pricing is one of the exigent and elementary problems of computational finance. Our aims to determine the nifty index option price through different valuation technique.Â In this paper, we illustrate the techniques for pricing of options and extracting information from option prices. We also describe various ways in which this information has been used in a number of applications. When dealing with options, we inevitably encounter the Black-Scholes-Merton option pricing formula, which has revolutionized the way in which options are priced in modern time. Black and Scholes (1973) and Mer

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B. S. Mohungoo, Aaisha, and Jason Narsoo. "Option valuation using Garch-type models : Empirical evidence using USD/INR data." Journal of Statistics and Management Systems 28, no. 3 (2025): 483–504. https://doi.org/10.47974/jsms-1359.

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The aim of this paper1 is to evaluate the option pricing performance of three GARCHtype models against the Black-Scholes model. The standard GARCH model and the asymmetric GJR-GARCH and EGARCH models are employed with the Gaussian innovation distribution. The models are assessed on two sample sets of USD/INR currency European call options data obtained from the National Stock Exchange of India. A Monte Carlo simulation approach is implemented on the estimated GARCH models for the valuation of European call options. An accuracy analysis of the option prices is then performed across different mo

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Zhang, Hongji. "Option Pricing and Risk Hedging in the Current Financial Market: A Case for Google." Advances in Economics, Management and Political Sciences 13, no. 1 (2023): 106–14. http://dx.doi.org/10.54254/2754-1169/13/20230684.

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In the past few decades, financial derivative securities have been developing rapidly around the world, and the issue of options and investment consumption has attracted more and more attention from mathematicians and financiers at home and abroad. In this paper, option pricing models are constructed and calibrated based on the Black Scholes Merton model, binomial tree model, historical data model and Monte Carlo diffusion model. The differences between different option pricing models for options and stock hedging of the same company in a short period of time are discussed and analyzed. In thi

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ALGHALITH, MOAWIA. "PRICING OPTIONS UNDER STOCHASTIC INTEREST RATE AND THE FRASCA–FARINA PROCESS: A SIMPLE, EXPLICIT FORMULA." Annals of Financial Economics 16, no. 01 (2021): 2150003. http://dx.doi.org/10.1142/s2010495221500032.

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Assuming a stochastic interest rate, we introduce a simple formula for pricing European options. In doing so, we provide a complete closed-form formula that does not require any numerical/computational methods. Furthermore, the model and formula are far simpler than the previous models/formulas. Our formula is as simple as the classical Black–Scholes pricing formula. Moreover, it removes the theoretical limitation of the original Black–Scholes model without any added practical complexity.

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Umeorah, Nneka, Phillip Mashele, Onyecherelam Agbaeze, and Jules Clement Mba. "Barrier Options and Greeks: Modeling with Neural Networks." Axioms 12, no. 4 (2023): 384. http://dx.doi.org/10.3390/axioms12040384.

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This paper proposes a non-parametric technique of option valuation and hedging. Here, we replicate the extended Black–Scholes pricing model for the exotic barrier options and their corresponding Greeks using the fully connected feed-forward neural network. Our methodology involves some benchmarking experiments, which result in an optimal neural network hyperparameter that effectively prices the barrier options and facilitates their option Greeks extraction. We compare the results from the optimal NN model to those produced by other machine learning models, such as the random forest and the pol

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ZHAO, JINSHI, and JIAZHEN HUO. "COORDINATION MECHANISM COMBINING SUPPLY CHAIN OPTIMIZATION AND RULE IN EXCHANGE." Asia-Pacific Journal of Operational Research 30, no. 05 (2013): 1350015. http://dx.doi.org/10.1142/s0217595913500152.

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There are two kinds of option pricing. The option pricing in exchange follows the Black–Scholes rule but does not consider the optimizing of supply chain. The traditional supply chain option contract can optimize supply chain but does not meet the Black–Scholes rule. We integrate the assumption of above two kinds of option pricing, and design a model to combine the Black–Scholes rule and traditional option contract of optimizing in a supplier-led supply chain. Our combined model can guide the enterprises to write or buy option considering both option pricing rule in financial market and the op

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Li, Yu. "A mean bound financial model and options pricing." International Journal of Financial Engineering 04, no. 04 (2017): 1750047. http://dx.doi.org/10.1142/s2424786317500475.

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Most of financial models, including the famous Black–Scholes–Merton options pricing model, rely upon the assumption that asset returns follow a normal distribution. However, this assumption is not justified by empirical data. To be more concrete, the empirical observations exhibit fat tails or heavy tails and implied volatilities against the strike prices demonstrate U-shaped curve resembling a smile, which is the famous volatility smile. In this paper we present a mean bound financial model and show that asset returns per time unit are Pareto distributed and assets are log Gamma distributed u

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Morales-Bañuelos, Paula, Sebastian Elias Rodríguez Bojalil, Luis Alberto Quezada-Téllez, and Guillermo Fernández-Anaya. "A General Conformable Black–Scholes Equation for Option Pricing." Mathematics 13, no. 10 (2025): 1576. https://doi.org/10.3390/math13101576.

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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 als

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FEDOTOV, SERGEI, and ABBY TAN. "LONG MEMORY STOCHASTIC VOLATILITY IN OPTION PRICING." International Journal of Theoretical and Applied Finance 08, no. 03 (2005): 381–92. http://dx.doi.org/10.1142/s0219024905003013.

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The aim of this paper is to present a stochastic model that accounts for the effects of a long-memory in volatility on option pricing. The starting point is the stochastic Black–Scholes equation involving volatility with long-range dependence. We define the stochastic option price as a sum of classical Black–Scholes price and random deviation describing the risk from the random volatility. By using the fact that the option price and random volatility change on different time scales, we derive the asymptotic equation for this deviation involving fractional Brownian motion. The solution to this

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Lee, Eun-Kyung, and Yoon-Dong Lee. "Understanding Black-Scholes Option Pricing Model." Communications for Statistical Applications and Methods 14, no. 2 (2007): 459–79. http://dx.doi.org/10.5351/ckss.2007.14.2.459.

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Xue, Kexuan. "Option Pricing Models: A Study of the Black-Scholes-Merton Model." SHS Web of Conferences 215 (2025): 01005. https://doi.org/10.1051/shsconf/202521501005.

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Financial Derivatives refer to financial instruments whose value depends on or is derived from other underlying assets such as stocks, bonds, commodities, exchange rates, and interest rates Examples include futures and options. This paper first introduces the early theories of option pricing. It then focuses on the Black-Scholes model, discussing the academic modifications and expansions made to this model due to its theoretical assumptions not aligning with reality. These adjustments aim to explore more efficient and practical methods for calculating option prices.

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Xiang, Shang. "Research on Option Pricing Method Based on the Black-Scholes Model." Economics and Management Innovation 2, no. 1 (2025): 69–77. https://doi.org/10.71222/k8mkc798.

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Option pricing is one of the core research topics in the field of financial engineering. As a classical option pricing method, the Black-Scholes model provides a significant foundation for both theory and practice in modern financial markets. This paper first elaborates on the theoretical foundation and mathematical derivation of the Black-Scholes model, analyzes its practical applications in option pricing, and explores its limitations, including the strict assumptions about market conditions and its applicability in environments with fluctuating volatility or sudden market jumps. To address

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