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Journal articles on the topic 'Multidimensional Black Scholes market'

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Published: 4 June 2021

Last updated: 8 February 2022

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1

ALBEVERIO, SERGIO, ALEX POPOVICI, and VICTORIA STEBLOVSKAYA. "A NUMERICAL ANALYSIS OF THE EXTENDED BLACK–SCHOLES MODEL." International Journal of Theoretical and Applied Finance 09, no. 01 (February 2006): 69–89. http://dx.doi.org/10.1142/s0219024906003469.

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In this article some numerical results regarding the multidimensional extension of the Black–Scholes model introduced by Albeverio and Steblovskaya [1] (a multidimensional model with stochastic volatilities and correlations) are presented. The focus lies on aspects concerning the use of this model for the practice of financial derivatives. Two parameter estimation methods for the model using historical data from the market and an analysis of the corresponding numerical results are given. Practical advantages of pricing derivatives using this model compared to the original multidimensional Black–Scholes model are pointed out. In particular the prices of vanilla options and of implied volatility surfaces computed in the model are close to those observed on the market.

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2

Dolinsky, Yan. "Shortfall Risk Approximations for American Options in the Multidimensional Black-Scholes Model." Journal of Applied Probability 47, no. 04 (December 2010): 997–1012. http://dx.doi.org/10.1017/s0021900200007312.

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We show that shortfall risks of American options in a sequence of multinomial approximations of the multidimensional Black-Scholes (BS) market converge to the corresponding quantities for similar American options in the multidimensional BS market with path-dependent payoffs. In comparison to previous papers we consider the multiassets case for which we use the weak convergence approach.

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3

Dolinsky, Yan. "Shortfall Risk Approximations for American Options in the Multidimensional Black-Scholes Model." Journal of Applied Probability 47, no. 4 (December 2010): 997–1012. http://dx.doi.org/10.1239/jap/1294170514.

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We show that shortfall risks of American options in a sequence of multinomial approximations of the multidimensional Black-Scholes (BS) market converge to the corresponding quantities for similar American options in the multidimensional BS market with path-dependent payoffs. In comparison to previous papers we consider the multiassets case for which we use the weak convergence approach.

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4

Bernard, Carole, Mateusz Maj, and Steven Vanduffel. "Improving the Design of Financial Products in a Multidimensional Black-Scholes Market." North American Actuarial Journal 15, no. 1 (January 2011): 77–96. http://dx.doi.org/10.1080/10920277.2011.10597610.

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5

Zimbidis, Alexandros A. "Optimal Management of a Variable Annuity Invested in a Black–Scholes Market Driven by a Multidimensional Fractional Brownian Motion." Stochastic Analysis and Applications 29, no. 1 (December 27, 2010): 61–77. http://dx.doi.org/10.1080/07362994.2011.532021.

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6

Guillaume, Tristan. "On the multidimensional Black–Scholes partial differential equation." Annals of Operations Research 281, no. 1-2 (August 11, 2018): 229–51. http://dx.doi.org/10.1007/s10479-018-3001-1.

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7

Munn, Luke. "From the Black Atlantic to Black-Scholes." Cultural Politics 16, no. 1 (March 1, 2020): 92–110. http://dx.doi.org/10.1215/17432197-8017284.

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Rather than being unprecedented, contemporary technologies are the most sophisticated instances of a long-standing dream: if space could be more comprehensively captured and coded, it could be more intensively capitalized. Two moments within this lineage are explored: maritime insurance of slave ships in the eighteenth century, and the Black-Scholes model of option pricing from the twentieth century. Maritime insurance rendered the unknown space of the ocean knowable and therefore profitable. By collecting information at Lloyds, merchants developed a map of threat within the Atlantic, and by writing a 10 percent buffer into slave-ship contracts they internalized contingency. This codification of risk pressured captains and established a logic for the violence enacted on the ship’s human “cargo.” The Black-Scholes formula of option pricing sought to codify the ocean of risk represented by the financial market. The formula mapped stock movements into a knowable stochastic equation. Traders could quantify and hedge against the unpredictable, rendering the stock market a space of riskless profit. However, the 2008 financial crash demonstrated the limits of spatial calculation. Taken together, these two moments demonstrate the historical continuity of a core imperative to exhaustively capitalize space. This historicization also foregrounds the racialized inequalities coded within these informatic logics. Against the bright innovation narratives of technology, this article stresses a longer and darker lineage based on inequality and dispossession.

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8

Dixit, Alok, and Shivam Singh. "Ad-Hoc Black–Scholes vis-à-vis TSRV-based Black–Scholes: Evidence from Indian Options Market." Journal of Quantitative Economics 16, no. 1 (February 15, 2017): 57–88. http://dx.doi.org/10.1007/s40953-017-0078-3.

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9

Özer, H. Ünsal, and Ahmet Duran. "The source of error behavior for the solution of Black–Scholes PDE by finite difference and finite element methods." International Journal of Financial Engineering 05, no. 03 (September 2018): 1850028. http://dx.doi.org/10.1142/s2424786318500287.

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Black–Scholes partial differential equation (PDE) is one of the most famous equations in mathematical finance and financial industry. In this study, numerical solution analysis is done for Black–Scholes PDE using finite element method with linear approach and finite difference methods. The numerical solutions are compared with Black–Scholes formula for option pricing. The numerical errors are determined for the finite element and finite difference applications to Black–Scholes PDE. We examine the error behavior and find the source of the corresponding errors under various market situations.

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10

Kermiche, Lamya. "Too Much Of A Good Thing? A Review Of Volatility Extensions In Black-Scholes." Journal of Applied Business Research (JABR) 30, no. 4 (June 30, 2014): 1171. http://dx.doi.org/10.19030/jabr.v30i4.8662.

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Since the seminal Black and Scholes model was introduced in the 1970s, researchers and practitioners have been continuously developing new models to enhance the original. All these models aim to ease one or more of the Black and Scholes assumptions, but this often results in a set of equations that is difficult if not impossible to use in practice. Nevertheless, in the wake of the financial crisis, an understanding of the various pricing models is essential to calm investors nerves. This paper reviews the models developed since Black and Scholes, giving the advantages and disadvantages of each type. It focuses on the main variable for which Black and Scholes gives results that differ widely from market data: implied volatility. This variable also forms the basis for the development of a new type of models, called market models.

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11

Levy, Haim, and James A. Yoder. "Applying the Black–Scholes model after large market shocks." Journal of Portfolio Management 16, no. 1 (October 31, 1989): 103–6. http://dx.doi.org/10.3905/jpm.1989.409230.

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12

Xidonas, Panos, Christos E. Kountzakis, Christis Hassapis, and Christos Staikouras. "A use of Black-Scholes model in market risk." International Journal of Financial Engineering and Risk Management 2, no. 3 (2016): 200. http://dx.doi.org/10.1504/ijferm.2016.082983.

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13

Takaoka, Koichiro. "A Complete-Market Generalization of the Black-Scholes Model." Asia-Pacific Financial Markets 11, no. 4 (December 2004): 431–44. http://dx.doi.org/10.1007/s10690-006-9021-x.

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14

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 those of the Black–Scholes model. As a by-product, the obtained results show that the value of European option depends on the drift coefficient μ of its underlying asset, which does not display in the Black–Scholes model only because μ = r in a risk-neutral market according to the no-arbitrage opportunity principle. At last, empirical analyses on Shanghai 50 ETF options and S&P 500 options show that the fitting effect of obtained pricing formulas is superior to that of the Black–Scholes model.

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15

DENI, PUTU AYU, KOMANG DHARMAWAN, and G. K. GANDHIADI. "PENENTUAN HARGA OPSI DAN NILAI HEDGE MENGGUNAKAN PERSAMAAN NON-LINEAR BLACK-SCHOLES." E-Jurnal Matematika 5, no. 1 (January 30, 2016): 27. http://dx.doi.org/10.24843/mtk.2016.v05.i01.p117.

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Option are contracts that give the right to sell and buy the asset at a price and a certain period of time. In addition investors use option as a means of hedge against asset owned. Many methods are used to determine the price of option, one of them by using the Black-Scholes equation. But its use these in the assumption that the value for the constant volatility. On market assumption are not appropriates, so many researchers proposed using a volatility calculation option that is non-constant Black-Scholes equation modelled using the volatility is not constant in the range so as to produce a non-linear equation of Black-Scholes. In addition to determine the value of hedge ratio. On completions of this study, for the numerical solution of non-linear Black-Scholes equation using method of explicit finite difference scheme. Option use in research us a stock YAHOO!inc. as the underlying asset. The result showed that the price of the option is calculated using non-linear Black-Scholes equation price close on the market. Therefore, it can produce hedge ration for a risk-free portfolio containing of the option and stock.

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16

Sinkala, Winter, and Tembinkosi F. Nkalashe. "Lie Symmetry Analysis of a First-Order Feedback Model of Option Pricing." Advances in Mathematical Physics 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/361785.

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A first-order feedback model of option pricing consisting of a coupled system of two PDEs, a nonliner generalised Black-Scholes equation and the classical Black-Scholes equation, is studied using Lie symmetry analysis. This model arises as an extension of the classical Black-Scholes model when liquidity is incorporated into the market. We compute the admitted Lie point symmetries of the system and construct an optimal system of the associated one-dimensional subalgebras. We also construct some invariant solutions of the model.

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17

Golembiovsky, D., and I. Baryshnikov. "VOLATILITY SMILE AT THE RUSSIAN OPTION MARKET." Journal of Business Economics and Management 7, no. 1 (March 31, 2006): 9–15. http://dx.doi.org/10.3846/16111699.2006.9636116.

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The main derivative exchange in Russia is FORTS (Futures and Options in RTS) which is a division of Russian Trade System (RTS). The underlying assets of option contracts are futures on Russian companies’ shares: OJSC “EES"1, OJPC “Lukoil"2 and OJSC “Gazprom"3. A basic model for estimation of fair option price is Black‐Scholes model, developed in the beginning of 70‐s’ years of the last century. This model defines the option premium as a cost of its hedging by underlying asset. It uses a number of assumptions: prices of underlying assets follow log‐normal distribution; hedging is accomplished continuously; an underlying asset is infinitely divisible; a volatility is constant on all period of option life. However, according to practice, prices of shares and futures do not follow normal or log‐normal distribution, a volatility can change during a life of option, and hedging is a discrete process. Thus, Black‐Scholes model can yield inexact results in real markets, especially it concerns deeply “in the money” or deeply “out of the money” options. The basic purpose of the paper is to investigate opportunities to apply Black‐Scholes model for an estimation of option premiums in the Russian market.

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18

Yavuz, M., and N. Özdemir. "A different approach to the European option pricing model with new fractional operator." Mathematical Modelling of Natural Phenomena 13, no. 1 (2018): 12. http://dx.doi.org/10.1051/mmnp/2018009.

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In this work, we have derived an approximate solution of the fractional Black-Scholes models using an iterative method. The fractional differentiation operator used in this paper is the well-known conformable derivative. Firstly, we redefine the fractional Black-Scholes equation, conformable fractional Adomian decomposition method (CFADM) and conformable fractional modified homotopy perturbation method (CFMHPM). Then, we have solved the fractional Black-Scholes (FBS) and generalized fractional Black-Scholes (GFBS) equations by using the proposed methods, which can analytically solve the fractional partial differential equations (FPDE). In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of these two option pricing problems by using in pricing the actual market data. Also, we have found out that the proposed models are very efficient and powerful techniques in finding approximate solutions of the fractional Black-Scholes models which are considered in conformable sense.

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19

Almgren, Robert, and Tianhui Michael Li. "Option Hedging with Smooth Market Impact." Market Microstructure and Liquidity 02, no. 01 (June 2016): 1650002. http://dx.doi.org/10.1142/s2382626616500027.

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We consider intraday hedging of an option position, for a large trader who experiences temporary and permanent market impact. We formulate the general model including overnight risk, and solve explicitly in two cases which we believe are representative. The first case is an option with approximately constant gamma: the optimal hedge trades smoothly towards the classical Black–Scholes delta, with trading intensity proportional to instantaneous mishedge and inversely proportional to illiquidity. The second case is an arbitrary non-linear option structure but with no permanent impact: the optimal hedge trades toward a value offset from the Black–Scholes delta. We estimate the effects produced on the public markets if a large collection of traders all hedge similar positions. We construct a stable hedge strategy with discrete time steps.

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20

Chang, Mou-Hsiung, and Roger K. Youree. "Spectral Approximation of Infinite-Dimensional Black-Scholes Equations with Memory." Journal of Applied Mathematics and Stochastic Analysis 2009 (January 14, 2009): 1–37. http://dx.doi.org/10.1155/2009/782572.

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This paper considers the pricing of a European option using a -market in which the stock price and the asset in the riskless bank account both have hereditary price structures described by the authors of this paper (1999). Under the smoothness assumption of the payoff function, it is shown that the infinite dimensional Black-Scholes equation possesses a unique classical solution. A spectral approximation scheme is developed using the Fourier series expansion in the space for the Black-Scholes equation. It is also shown that the th approximant resembles the classical Black-Scholes equation in finite dimensions.

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21

BORDAG, L. A., and A. Y. CHMAKOVA. "EXPLICIT SOLUTIONS FOR A NONLINEAR MODEL OF FINANCIAL DERIVATIVES." International Journal of Theoretical and Applied Finance 10, no. 01 (February 2007): 1–21. http://dx.doi.org/10.1142/s021902490700407x.

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Families of explicit solutions are found to a nonlinear Black–Scholes equation which incorporates the feedback-effect of a large trader in case of market illiquidity. The typical solution of these families will have a payoff which approximates a strangle. These solutions were used to test numerical schemes for solving a nonlinear Black–Scholes equation.

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22

Simonelli, M. R. "Black-Scholes Fuzzy Numbers as Indexes of Performance." Applied Computational Intelligence and Soft Computing 2010 (2010): 1–7. http://dx.doi.org/10.1155/2010/607214.

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We use the set of propositions of some previous papers to define a fuzzy version of the Black-Scholes value where the risk free instantaneous interest intensity, the volatility and the initial stock price are fuzzy numbers whose parameters are built with statistical financial data. With our Black-Scholes fuzzy numbers we define indexes of performance varing in time. As an example, with data of the Italian Stock Exchange on MIB30, we see that in 2004 and 2006 our indexes are negative, that is, they are indexes of the refuse to invest and this refuse increased. So, on November 11, 2006 we could forecast that the market will become with more risk: the risk of loss will increase. Now, on January 25, 2010, we know that this forecast has happened. Obviously, the parameters of our Black-Scholes fuzzy numbers can be valued also with incomplete, possibilistic data. With respect to the probabilistic one, our fuzzy method is more simple and immediate to have a forecast on the financial market.

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23

Meng, Li, and Mei Wang. "Comparison of Black–Scholes Formula with Fractional Black–Scholes Formula in the Foreign Exchange Option Market with Changing Volatility." Asia-Pacific Financial Markets 17, no. 2 (October 27, 2009): 99–111. http://dx.doi.org/10.1007/s10690-009-9102-8.

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24

Wahyuni, Elsa, Riri Lestari, and Mahdhivan Syafwan. "MODEL BLACK-SCHOLES OPSI CALL DAN OPSI PUT TIPE EROPA DENGAN DIVIDEN PADA KEADAAN CONSTANT MARKET." Jurnal Matematika UNAND 6, no. 2 (July 11, 2017): 43. http://dx.doi.org/10.25077/jmu.6.2.43-49.2017.

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Opsi tipe Eropa adalah suatu bentuk perjanjian berupa kontrak yang memberikan pemegang opsi suatu hak tetapi bukan suatu kewajiban untuk membeli atau menjual aset tertentu dengan harga tertentu pada waktu jatuh tempo. Opsi call memberikan hak kepada pemegang opsi untuk membeli saham pada waktu jatuh tempo. Sementara opsi put memberikan hak untuk menjual saham. Metode Black-Scholes merupakan salah satu metode untuk menentukan harga opsi. Asumsi yang digunakan pada model ini adalah adanya pembagian dividen. Dividen dibayarkan pada keadaan constant market. Harga saham yang berubah secara acak menurut waktu diasumsikan sebagai proses stokastik. Prediksi harga saham diasumsikan hanya dipengaruhi oleh harga saham saat ini dan tidak dipengaruhi oleh harga saham di masa lampau. Perhitungan harga opsi saham chevron corporation pada tanggal 16 November 2016 dengan mengaplikasikan model Black-Scholes. Hasil yang diperoleh menunjukkan bahwa pada keadaan constant market sebaiknya investor membeli opsi put di pasar saham dengan harga opsi yang lebih kecil dari harga opsi model Black-Scholes yaitu pada harga pelaksanaan 101, 102, 104, 105, 106, 107 dan 108, sedangkan untuk opsi call sebaiknya investor membeli opsi call di pasar saham untuk harga pelaksanaan 100 dan 101.Kata Kunci: Opsi tipe Eropa, Opsi call, Opsi put, Proses Stokastik, Dividen, Constant Market, Model Black-Scholes

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25

Vaidya, Tushar, Carlos Murguia, and Georgios Piliouras. "Learning agents in Black–Scholes financial markets." Royal Society Open Science 7, no. 10 (October 2020): 201188. http://dx.doi.org/10.1098/rsos.201188.

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Black–Scholes (BS) is a remarkable quotation model for European option pricing in financial markets. Option prices are calculated using an analytical formula whose main inputs are strike (at which price to exercise) and volatility. The BS framework assumes that volatility remains constant across all strikes; however, in practice, it varies. How do traders come to learn these parameters? We introduce natural agent-based models, in which traders update their beliefs about the true implied volatility based on the opinions of other agents. We prove exponentially fast convergence of these opinion dynamics, using techniques from control theory and leader-follower models, thus providing a resolution between theory and market practices. We allow for two different models, one with feedback and one with an unknown leader.

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26

Sawangtong, Panumart, Kamonchat Trachoo, Wannika Sawangtong, and Benchawan Wiwattanapataphee. "The Analytical Solution for the Black-Scholes Equation with Two Assets in the Liouville-Caputo Fractional Derivative Sense." Mathematics 6, no. 8 (July 25, 2018): 129. http://dx.doi.org/10.3390/math6080129.

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It is well known that the Black-Scholes model is used to establish the behavior of the option pricing in the financial market. In this paper, we propose the modified version of Black-Scholes model with two assets based on the Liouville-Caputo fractional derivative. The analytical solution of the proposed model is investigated by the Laplace transform homotopy perturbation method.

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27

Tehranchi, Michael R. "A Black–Scholes inequality: applications and generalisations." Finance and Stochastics 24, no. 1 (October 18, 2019): 1–38. http://dx.doi.org/10.1007/s00780-019-00410-6.

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Abstract The space of call price curves has a natural noncommutative semigroup structure with an involution. A basic example is the Black–Scholes call price surface, from which an interesting inequality for Black–Scholes implied volatility is derived. The binary operation is compatible with the convex order, and therefore a one-parameter sub-semigroup gives rise to an arbitrage-free market model. It is shown that each such one-parameter semigroup corresponds to a unique log-concave probability density, providing a family of tractable call price surface parametrisations in the spirit of the Gatheral–Jacquier SVI surface. An explicit example is given to illustrate the idea. The key observation is an isomorphism linking an initial call price curve to the lift zonoid of the terminal price of the underlying asset.

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28

ZHAO, JINSHI, and JIAZHEN HUO. "COORDINATION MECHANISM COMBINING SUPPLY CHAIN OPTIMIZATION AND RULE IN EXCHANGE." Asia-Pacific Journal of Operational Research 30, no. 05 (October 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 optimization of supply chain. Then we simulate and verify the model in Zinc industry of China. It is proved that our option pricing model is equalized and optimal to supply chain and consistent with Black–Scholes rule.

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29

Chen, Shen-Yuan. "Valuation of Covered Warrant Subject to Default Risk." Review of Pacific Basin Financial Markets and Policies 06, no. 01 (March 2003): 21–44. http://dx.doi.org/10.1142/s0219091503001018.

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There is no margin settlement mechanism for existing covered warrants in Taiwan, thus the credit risk of the warrant issuer must be considered when investors evaluate the price of a covered warrant. This paper applies the vulnerable option valuation model to empirically study the difference in the theoretical value of a vulnerable warrant, Black–Scholes option price and the market price of warrant by using the Taiwan warrant data. Empirical results show that the theoretical value of a vulnerable warrant is lower than the Black–Scholes non-vulnerable option value and its market value.

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30

BUFFINGTON, JOHN, and ROBERT J. ELLIOTT. "AMERICAN OPTIONS WITH REGIME SWITCHING." International Journal of Theoretical and Applied Finance 05, no. 05 (August 2002): 497–514. http://dx.doi.org/10.1142/s0219024902001523.

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A Black-Scholes market is considered in which the underlying economy, as modeled by the parameters and volatility of the processes, switches between a finite number of states. The switching is modeled by a hidden Markov chain. European options are priced and a Black-Scholes equation obtained. The approximate valuation of American options due to Barone-Adesi and Whaley is extended to this setting.

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31

Lindgren, Jussi. "Efficient Markets and Contingent Claims Valuation: An Information Theoretic Approach." Entropy 22, no. 11 (November 12, 2020): 1283. http://dx.doi.org/10.3390/e22111283.

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This research article shows how the pricing of derivative securities can be seen from the context of stochastic optimal control theory and information theory. The financial market is seen as an information processing system, which optimizes an information functional. An optimization problem is constructed, for which the linearized Hamilton–Jacobi–Bellman equation is the Black–Scholes pricing equation for financial derivatives. The model suggests that one can define a reasonable Hamiltonian for the financial market, which results in an optimal transport equation for the market drift. It is shown that in such a framework, which supports Black–Scholes pricing, the market drift obeys a backwards Burgers equation and that the market reaches a thermodynamical equilibrium, which minimizes the free energy and maximizes entropy.

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32

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 (April 21, 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 between the observed and theoretical option prices. The implied distribution of BIST 30 index does not significantly deviate from normal distribution. In addition, pricing performance of Black Scholes model performs better in most of the time. Black Scholes pricing Formula, Carrado-Su pricing Formula, Implied Parameters

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33

GÖNCÜ, AHMET, and ERDINC AKYILDIRIM. "STATISTICAL ARBITRAGE IN THE MULTI-ASSET BLACK–SCHOLES ECONOMY." Annals of Financial Economics 12, no. 01 (March 2017): 1750004. http://dx.doi.org/10.1142/s201049521750004x.

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In this study, we consider the statistical arbitrage definition given in Hogan, S, R Jarrow, M Teo and M Warachka (2004). Testing market efficiency using statistical arbitrage with applications to momentum and value strategies, Journal of Financial Economics, 73, 525–565 and derive the statistical arbitrage condition in the multi-asset Black–Scholes economy building upon the single asset case studied in Göncü, A (2015). Statistical arbitrage in the Black Scholes framework. Quantitative Finance, 15(9), 1489–1499. Statistical arbitrage profits can be generated if there exists at least one asset in the economy that satisfies the statistical arbitrage condition. Therefore, adding a no-statistical arbitrage condition to no-arbitrage pricing models is not realistic if not feasible. However, with an example we show that what excludes statistical arbitrage opportunities in the Black–Scholes economy, and possibly in other complete market models, is the presence of uncertainty or stochasticity in the model parameters. Furthermore, we derive analytical formulas for the expected value and probability of loss of the statistical arbitrage portfolios and compute optimal boundaries to sell the risky assets in the portfolio by maximizing the expected return with a constraint on the probability of loss.

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34

Singh, Shivam, and Vipul . "Performance of Black-Scholes model with TSRV estimates." Managerial Finance 41, no. 8 (August 10, 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 this study for price comparisons. Findings – The B-S model shows significantly negative pricing bias for all the options, which is dependent on the moneyness of the option and the volatility of the underlying. Research limitations/implications – The negative pricing bias of B-S model, despite the use of the more efficient TSRV estimate, and post facto volatility values, confirms its inadequacy. It also points towards the possible existence of volatility risk premium in the Indian options market. Originality/value – The use of tick-by-tick data obviates the nonsynchronous error. TSRV, used for estimating the volatility, is a significantly improved estimate (in terms of efficiency and bias), as compared to the estimates based on closing data. The use of ex post realised volatility ensures that the forecasting error does not vitiate the test results. The sample is selected to be large and varied to ensure the robustness of the results.

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Xu, Song, and Yujiao Yang. "Fractional Black-Scholes Model and Technical Analysis of Stock Price." Journal of Applied Mathematics 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/631795.

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In the stock market, some popular technical analysis indicators (e.g., Bollinger bands, RSI, ROC, etc.) are widely used to forecast the direction of prices. The validity is shown by observed relative frequency of certain statistics, using the daily (hourly, weekly, etc.) stock prices as samples. However, those samples are not independent. In earlier research, the stationary property and the law of large numbers related to those observations under Black-Scholes stock price model and stochastic volatility model have been discussed. Since the fitness of both Black-Scholes model and short-range dependent process has been questioned, we extend the above results to fractional Black-Scholes model with Hurst parameterH>1/2, under which the stock returns follow a kind of long-range dependent process. We also obtain the rate of convergence.

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JOSEPHY, NORMAN, LUCIA KIMBALL, and VICTORIA STEBLOVSKAYA. "ON THE NUMERICAL ASPECTS OF OPTIMAL OPTION HEDGING WITH TRANSACTION COSTS." International Journal of Theoretical and Applied Finance 20, no. 01 (February 2017): 1750002. http://dx.doi.org/10.1142/s0219024917500029.

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We present a numerical study of non-self-financing hedging of European options under proportional transaction costs. We describe an algorithmic approach based on a discrete time financial market model that extends the classical binomial model. We review the analytical basis for our algorithm and present a variety of empirical results using real market data. The performance of the algorithm is evaluated by comparing to a Black–Scholes delta hedge with transaction costs incorporated. We also evaluate the impact of recalibrating the hedging strategy one or more times during the life of the option using the most recent market data. These results are compared to a recalibrated Black–Scholes delta hedge modified for transaction costs.

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Loeper, Gregoire. "Option pricing with linear market impact and nonlinear Black–Scholes equations." Annals of Applied Probability 28, no. 5 (October 2018): 2664–726. http://dx.doi.org/10.1214/17-aap1367.

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Kumar, Sunil, Devendra Kumar, and Jagdev Singh. "Numerical computation of fractional Black–Scholes equation arising in financial market." Egyptian Journal of Basic and Applied Sciences 1, no. 3-4 (December 2014): 177–83. http://dx.doi.org/10.1016/j.ejbas.2014.10.003.

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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 (August 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 prices. In the second approach neural networks are trained with genetic algorithms in order to reverse-engineer the Black–Scholes formulae. The standard Black–Scholes model provides a starting point for an evolutionary training process, which yields improved options prices. Market options prices as quoted on the Chicago Board Options Exchange are used for performance comparison between the Black–Scholes model and the proposed options pricing schemes. The proposed models produce as good as and often better options prices than the conventional Black–Scholes formulae.

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Ďuriš, Karol, Shih-Hau Tan, Choi-Hong Lai, and Daniel Ševčovič. "Comparison of the Analytical Approximation Formula and Newton's Method for Solving a Class of Nonlinear Black–Scholes Parabolic Equations." Computational Methods in Applied Mathematics 16, no. 1 (January 1, 2016): 35–50. http://dx.doi.org/10.1515/cmam-2015-0035.

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AbstractMarket illiquidity, feedback effects, presence of transaction costs, risk from unprotected portfolio and other nonlinear effects in PDE-based option pricing models can be described by solutions to the generalized Black–Scholes parabolic equation with a diffusion term nonlinearly depending on the option price itself. In this paper, different linearization techniques such as Newton's method and the analytic asymptotic approximation formula are adopted and compared for a wide class of nonlinear Black–Scholes equations including, in particular, the market illiquidity model and the risk-adjusted pricing model. Accuracy and time complexity of both numerical methods are compared. Furthermore, market quotes data was used to calibrate model parameters.

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HU, YAOZHONG, and BERNT ØKSENDAL. "FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE." Infinite Dimensional Analysis, Quantum Probability and Related Topics 06, no. 01 (March 2003): 1–32. http://dx.doi.org/10.1142/s0219025703001110.

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The purpose of this paper is to develop a fractional white noise calculus and to apply this to markets modeled by (Wick–) Itô type of stochastic differential equations driven by fractional Brownian motion BH(t); 1/2 < H < 1. We show that if we use an Itô type of stochastic integration with respect to BH(t) (as developed in Ref. 8), then the corresponding Itô fractional Black–Scholes market has no arbitrage, contrary to the situation when the pathwise integration is used. Moreover, we prove that our Itô fractional Black–Scholes market is complete and we compute explicitly the price and replicating portfolio of a European option in this market. The results are compared to the classical results based on standard Brownian motion B(t).

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Thanompolkrang, Sirunya, Wannika Sawangtong, and Panumart Sawangtong. "Application of the Generalized Laplace Homotopy Perturbation Method to the Time-Fractional Black–Scholes Equations Based on the Katugampola Fractional Derivative in Caputo Type." Computation 9, no. 3 (March 12, 2021): 33. http://dx.doi.org/10.3390/computation9030033.

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In the finance market, the Black–Scholes equation is used to model the price change of the underlying fractal transmission system. Moreover, the fractional differential equations recently are accepted by researchers that fractional differential equations are a powerful tool in studying fractal geometry and fractal dynamics. Fractional differential equations are used in modeling the various important situations or phenomena in the real world such as fluid flow, acoustics, electromagnetic, electrochemistry and material science. There is an important question in finance: “Can the fractional differential equation be applied in the financial market?”. The answer is “Yes”. Due to the self-similar property of the fractional derivative, it can reply to the long-range dependence better than the integer-order derivative. Thus, these advantages are beneficial to manage the fractal structure in the financial market. In this article, the classical Black–Scholes equation with two assets for the European call option is modified by replacing the order of ordinary derivative with the fractional derivative order in the Caputo type Katugampola fractional derivative sense. The analytic solution of time-fractional Black–Scholes European call option pricing equation with two assets is derived by using the generalized Laplace homotopy perturbation method. The used method is the combination of the homotopy perturbation method and generalized Laplace transform. The analytic solution of the time-fractional Black–Scholes equation is carried out in the form of a Mittag–Leffler function. Finally, the effects of the fractional-order in the Caputo type Katugampola fractional derivative to change of a European call option price are shown.

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Ampun, Sivaporn, and Panumart Sawangtong. "The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative." Mathematics 9, no. 3 (January 21, 2021): 214. http://dx.doi.org/10.3390/math9030214.

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In the finance market, it is well known that the price change of the underlying fractal transmission system can be modeled with the Black-Scholes equation. This article deals with finding the approximate analytic solutions for the time-fractional Black-Scholes equation with the fractional integral boundary condition for a European option pricing problem in the Katugampola fractional derivative sense. It is well known that the Katugampola fractional derivative generalizes both the Riemann–Liouville fractional derivative and the Hadamard fractional derivative. The technique used to find the approximate analytic solutions of the time-fractional Black-Scholes equation is the generalized Laplace homotopy perturbation method, the combination of the generalized Laplace transform and homotopy perturbation method. The approximate analytic solution for the problem is in the form of the generalized Mittag-Leffler function. This shows that the generalized Laplace homotopy perturbation method is one of the most effective methods to construct approximate analytic solutions of the fractional differential equations. Finally, the approximate analytic solutions of the Riemann–Liouville and Hadamard fractional Black-Scholes equation with the European option are also shown.

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Nyakinda, Joseph Otula. "A LOGISTIC NONLINEAR BLACK-SCHOLES-MERTON PARTIAL DIFFERENTIAL EQUATION: EUROPEAN OPTION." International Journal of Research -GRANTHAALAYAH 6, no. 6 (June 30, 2018): 480–87. http://dx.doi.org/10.29121/granthaalayah.v6.i6.2018.1393.

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Nonlinear Black-Scholes equations provide more accurate values by taking into account more realistic assumptions, such as transaction costs, illiquid markets, risks from an unprotected portfolio or large investor's preferences, which may have an impact on the stock price, the volatility, the drift and the option price itself. Most modern models are represented by nonlinear variations of the well-known Black-Scholes Equation. On the other hand, asset security prices may naturally not shoot up indefinitely (exponentially) leading to the use of Verhulst’s Logistic equation. The objective of this study was to derive a Logistic Nonlinear Black Scholes Merton Partial Differential equation by incorporating the Logistic geometric Brownian motion. The methodology involves, analysis of the geometric Brownian motion, review of logistic models, process and lemma, stochastic volatility models and the derivation of the linear and nonlinear Black-Scholes-Merton partial differential equation. Illiquid markets have also been analyzed alongside stochastic differential equations. The result of this study may enhance reliable decision making based on a rational prediction of the future asset prices given that in reality the stock market may depict a nonlinear pattern.

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BOYARCHENKO, SVETLANA I., and SERGEI Z. LEVENDORSKIǏ. "OPTION PRICING FOR TRUNCATED LÉVY PROCESSES." International Journal of Theoretical and Applied Finance 03, no. 03 (July 2000): 549–52. http://dx.doi.org/10.1142/s0219024900000541.

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A general class of truncated Lévy processes is introduced, and possible ways of fitting parameters of the constructed family of truncated Lévy processes to data are discussed. For a market of a riskless bond and a stock whose log-price follows a truncated Lévy process, TLP-analogs of the Black–Scholes equation, the Black–Scholes formula, the Dynkin derivative and the Leland's model are obtained, a locally risk-minimizing portfolio is constructed, and an optimal exercise price for a perpetual American put is computed.

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Ayekple, Yao Elikem, Charles Kofi Tetteh, and Prince Kwaku Fefemwole. "Markov Chain Monte Carlo Method for Estimating Implied Volatility in Option Pricing." Journal of Mathematics Research 10, no. 6 (November 29, 2018): 108. http://dx.doi.org/10.5539/jmr.v10n6p108.

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Using market covered European call option prices, the Independence Metropolis-Hastings Sampler algorithm for estimating Implied volatility in option pricing was proposed. This algorithm has an acceptance criteria which facilitate accurate approximation of this volatility from an independent path in the Black Scholes Model, from a set of finite data observation from the stock market. Assuming the underlying asset indeed follow the geometric brownian motion, inverted version of the Black Scholes model was used to approximate this Implied Volatility which was not directly seen in the real market: for which the BS model assumes the volatility to be a constant. Moreover, it is demonstrated that, the Implied Volatility from the options market tends to overstate or understate the actual expectation of the market. In addition, a 3-month market Covered European call option data, from 30 different stock companies was acquired from Optionistic.Com, which was used to estimate the Implied volatility. This accurately approximate the actual expectation of the market with low standard errors ranging between 0.0035 to 0.0275.

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Zhou, Li Li. "China's Carbon Emissions Pricing Options: Based on Black-Scholes Model Testing." Advanced Materials Research 573-574 (October 2012): 1010–16. http://dx.doi.org/10.4028/www.scientific.net/amr.573-574.1010.

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In this paper, by analyzing the cause of the weak power for China's carbon emissions, we design the trading and OTC options contracts of China's carbon emissions. By testing we found that the pricing method can indeed improve the pricing power of carbon emissions and acquire more transactions in negotiations .The policy implications of this article: Firstly, China should combine with domestic and international progress to plan the framework of China's carbon emissions trading as soon as possible. Secondly, we also launch the research of carbon emissions trading market and experimental work; the third, establishing trading center with the market-oriented to guide China's implementation of greenhouse gas emission reduction projects and to achieve Low-cost emission.

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Dong, Yihang, Pengyun Wang, Weixin Zhang, and Yixuan He. "Real Estate Exotic Options based on Black-Scholes model (BSM)." Proceedings of Business and Economic Studies 4, no. 3 (June 18, 2021): 36–40. http://dx.doi.org/10.26689/pbes.v4i3.2187.

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This paper analyzed the issue of high housing prices in China in view of exotic options using the traditional BSM and improved it while applying it to the current situation in the real estate market. A certain set time frame in the purchase of the options with real estate prices was designed in the implementation of exotic option pricings to ease the speculative pressures caused by high housing prices.

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Ronnie Sircar, K., and George Papanicolaou. "General Black-Scholes models accounting for increased market volatility from hedging strategies." Applied Mathematical Finance 5, no. 1 (March 1998): 45–82. http://dx.doi.org/10.1080/135048698334727.

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Vulandari, Retno Tri, and Sutrima Sutrima. "Black-Scholes Model of European Call Option Pricing in Constant Market Condition." International Journal of Computing Science and Applied Mathematics 6, no. 2 (August 17, 2020): 46. http://dx.doi.org/10.12962/j24775401.v6i2.5828.

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