• Title/Summary/Keyword: Black-Scholes equation

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ON THE PARAMETIC INTEREST OF THE BLACK-SCHOLES EQUATION

  • Kananthai, Amnuay
    • Journal of applied mathematics & informatics
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    • v.28 no.3_4
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    • pp.923-929
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    • 2010
  • We have discovered some parametics $\lambda$ in the Black-Scholes equation which depend on the interest rate $\gamma$ and the Volatility $\sigma$ and later is named the parametic interest. On studying the parametic interest $\lambda$, we found that such $\lambda$ gives the sufficient condition for the existence of solutions of the Black-Scholes equation which is either weak or strong solutions.

ADAPTIVE NUMERICAL SOLUTIONS FOR THE BLACK-SCHOLES EQUATION

  • Park, H.W.;S.K. Chung
    • Journal of applied mathematics & informatics
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    • v.12 no.1_2
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    • pp.335-349
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    • 2003
  • Almost all business are affected by the weather so that weather derivatives has been traded to hedge weather risk. Since the weather itself is not an asset with a market price, some analysts believe that the Black-Scholes equation could not be used appropriately to price weather derivative options. But some weather derivatives can be considered as an Asian option, we revisit the Black-scholes model. Numerical solution of the Black-Scholes equation has a significant error at the money option or around the money option, it is necessary to adopt adaptive mesh near to the strike value. Here we propose a numerical method with an adaptive grid refinement.

COMPARISON OF NUMERICAL METHODS (BI-CGSTAB, OS, MG) FOR THE 2D BLACK-SCHOLES EQUATION

  • Jeong, Darae;Kim, Sungki;Choi, Yongho;Hwang, Hyeongseok;Kim, Junseok
    • The Pure and Applied Mathematics
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    • v.21 no.2
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    • pp.129-139
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    • 2014
  • In this paper, we present a detailed comparison of the performance of the numerical solvers such as the biconjugate gradient stabilized, operator splitting, and multigrid methods for solving the two-dimensional Black-Scholes equation. The equation is discretized by the finite difference method. The computational results demonstrate that the operator splitting method is fastest among these solvers with the same level of accuracy.

AN ADAPTIVE FINITE DIFFERENCE METHOD USING FAR-FIELD BOUNDARY CONDITIONS FOR THE BLACK-SCHOLES EQUATION

  • Jeong, Darae;Ha, Taeyoung;Kim, Myoungnyoun;Shin, Jaemin;Yoon, In-Han;Kim, Junseok
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.4
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    • pp.1087-1100
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    • 2014
  • We present an accurate and efficient numerical method for solving the Black-Scholes equation. The method uses an adaptive grid technique which is based on a far-field boundary position and the Peclet condition. We present the algorithm for the automatic adaptive grid generation: First, we determine a priori suitable far-field boundary location using the mathematical model parameters. Second, generate the uniform fine grid around the non-smooth point of the payoff and a non-uniform grid in the remaining regions. Numerical tests are presented to demonstrate the accuracy and efficiency of the proposed method. The results show that the computational time is reduced substantially with the accuracy being maintained.

FINITE DIFFERENCE METHOD FOR THE TWO-DIMENSIONAL BLACK-SCHOLES EQUATION WITH A HYBRID BOUNDARY CONDITION

  • HEO, YOUNGJIN;HAN, HYUNSOO;JANG, HANBYEOL;CHOI, YONGHO;KIM, JUNSEOK
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.23 no.1
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    • pp.19-30
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    • 2019
  • In this paper, we develop an accurate explicit finite difference method for the two-dimensional Black-Scholes equation with a hybrid boundary condition. In general, the correlation term in multi-asset options is problematic in numerical treatments partially due to cross derivatives and numerical boundary conditions at the far field domain corners. In the proposed hybrid boundary condition, we use a linear boundary condition at the boundaries where at least one asset is zero. After updating the numerical solution by one time step, we reduce the computational domain so that we do not need boundary conditions. To demonstrate the accuracy and efficiency of the proposed algorithm, we calculate option prices and their Greeks for the two-asset European call and cash-or-nothing options. Computational results show that the proposed method is accurate and is very useful for nonlinear boundary conditions.

AN ADAPTIVE MULTIGRID TECHNIQUE FOR OPTION PRICING UNDER THE BLACK-SCHOLES MODEL

  • Jeong, Darae;Li, Yibao;Choi, Yongho;Moon, Kyoung-Sook;Kim, Junseok
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.17 no.4
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    • pp.295-306
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    • 2013
  • In this paper, we consider the adaptive multigrid method for solving the Black-Scholes equation to improve the efficiency of the option pricing. Adaptive meshing is generally regarded as an indispensable tool because of reduction of the computational costs. The Black-Scholes equation is discretized using a Crank-Nicolson scheme on block-structured adaptively refined rectangular meshes. And the resulting discrete equations are solved by a fast solver such as a multigrid method. Numerical simulations are performed to confirm the efficiency of the adaptive multigrid technique. In particular, through the comparison of computational results on adaptively refined mesh and uniform mesh, we show that adaptively refined mesh solver is superior to a standard method.

DOMAIN OF INFLUENCE OF LOCAL VOLATILITY FUNCTION ON THE SOLUTIONS OF THE GENERAL BLACK-SCHOLES EQUATION

  • Kim, Hyundong;Kim, Sangkwon;Han, Hyunsoo;Jang, Hanbyeol;Lee, Chaeyoung;Kim, Junseok
    • The Pure and Applied Mathematics
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    • v.27 no.1
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    • pp.43-50
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    • 2020
  • We investigate the domain of influence of the local volatility function on the solutions of the general Black-Scholes model. First, we generate the sample paths of underlying asset using the Monte Carlo simulation. Next, we define the inner and outer domains to find the effective volatility region. To confirm the effect of the inner domain, we use the root mean square error for the European call option prices, and then change the values of volatility in the proposed domain. The computational experiments confirm that there is an effective region which dominates the option pricing.

열방정식 입장에서 바라본 세 방정식

  • 송종철
    • Journal for History of Mathematics
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    • v.15 no.3
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    • pp.59-64
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    • 2002
  • This paper investigates a history of Fourier Series for the heat equation and how deeply it is related to modern famous three equations, Navier-Stokes equations in fluid dynamics, drift-diffusion equations in semiconductor, and Black-Scholes equation in finance. We also propose improved models for the heat equation with finite propagation speeds.

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HEDGING OPTION PORTFOLIOS WITH TRANSACTION COSTS AND BANDWIDTH

  • KIM, SEKI
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.4 no.2
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    • pp.77-84
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    • 2000
  • Black-Scholes equation arising from option pricing in the presence of cost in trading the underlying asset is derived. The transaction cost is chosen precisely and generalized to reflect the trade in the real world. Furthermore the concept of the bandwidth is introduced to obtain the better rehedging. The model with bandwidth derived in this paper can be used to calculate the more accurate option price numerically even if it is nonlinear and more complicated than the models shown before.

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