Study of Black-Scholes Model and its Applications

Shinde, Akanksha S.; Takale, Kalyanrao

doi:10.1016/j.proeng.2012.06.035

Cited by 26 publications

(18 citation statements)

References 3 publications

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“…Data set 1: For approximating call option value we chose the parameters as follows [6] Now substituting the parameters given in data set 1 into equation , we get Data set 2: For approximating put option value we chose the parameters as follows [4] Now substituting the parameters given in data set 2 into equation , we get Data set 3: [14] Now substituting the parameters given in data set 3 into equation , we get where,…”

Section: Numerical Experiments and Results Discussionmentioning

confidence: 99%

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2020

International Journal of Mathematical Research

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Black-Scholes equation European call optionEuropean put option Du Fort-Frankel finite difference method (DF3DM) Galerkin weighted residual method (GWRM) Modified legendre polynomials.The main objective of this paper is to find the approximate solutions of the Black-Scholes (BS) model by two numerical techniques, namely, Du Fort-Frankel finite difference method (DF3DM), and Galerkin weighted residual method (GWRM) for both (call and put) type of European options. Since both DF3DM and GWRM are the most familiar numerical techniques for solving partial differential equations (PDE) of parabolic type, we estimate options prices by using these techniques. For this, we first convert the Black-Scholes model into a modified parabolic PDE, more specifically, in DF3DM, the first temporal vector is calculated by the Crank-Nicolson method using the initial boundary conditions and then the option price is evaluated. On the other hand, in GWRM, we use piecewise modified Legendre polynomials as the basis functions of GWRM which satisfy the homogeneous form of the boundary conditions. We may observe that the results obtained by the present methods converge fast to the exact solutions. In some cases, the present methods give more accurate results than the earlier results obtained by the adomian decomposition method [14]. Finally, all approximate solutions are shown by the graphical and tabular representations.Contribution/Originality: The paper's primary contribution is finding that the approximate results of Black-Scholes model by DF3DM, and GWRM with modified Legendre polynomials as basis functions. INTRODUCTIONOptions are treated as the most important part of the security markets from the beginning of the Chicago Board Options Exchange (CBOE) in 1973, which is the largest options market in the world [1]. During last decades, the valuation of options has become important problem for both financial and mathematical point of view. Details about options are available in HullThere are many models for calculating the value of options but among all of those models, the Black-Scholes model is a suitable way to find the European options price.The discovery of the Black-Scholes model took long time. Fishers Black took the first step to make a model for valuation of stock. Afterward Myron Scholes added with Black and today we use their result for finding the value of different kinds of stocks. In 1973, the concept of the Black-Scholes model was first disclosed in the paper entitled, "The pricing of options and corporate liabilities" in the Journal of political economy by Black and Scholes [3] and then advanced in "Theory of rational option pricing" by Robert Merton. In 2003, Chawla, et al. [4] approximate European put option value by using Generalized trapezoidal formula and found better approximation than Crank-Nicolson method especially near the strike price. In Hackmann [5] Crank-Nicolson method has been used for evaluation of European options price with accuracy up to 3 decimal places. In 2012, Shinde and Takale [6] have

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Section: Numerical Experiments and Results Discussionmentioning

confidence: 99%

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2020

International Journal of Mathematical Research

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“…We also compare the explicit method with the result obtained by another work using semi-implicit method [5]. This section consists of the basic assumptions in the Black-Scholes model [6].…”

Section: Introductionmentioning

confidence: 93%

A Study on Numerical Solution of Black-Scholes Model

Anwar¹,

Andallah²

2018

JMF

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In the history of option pricing, Black-Scholes model is one of the most significant models. In this article, the main concern is the numerical solution of the Black-Scholes model (a.k.a. Black/Scholes/Merton) for the European call option in a different way. The model is described and an explicit difference scheme was used for the numerical approximation. The stability condition of the scheme is established through convex combination. A different way was used to obtain the numerical value of the model. Estimation of the relative error was calculated in L 1-norm in order to test the accuracy of the scheme. Finally, a comparison of the numerical outcomes with the value obtained by another scheme is given.

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“…This model measures the ability of banks to meet their obligations when debt holders will exercise their options to obtain repayment. Options pricing is obtained from the Black-Scholes equation [24].…”

Section: 𝑧 = (𝑘 − 𝜌) 𝜎mentioning

confidence: 99%