• 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.

BARRIER OPTION PRICING UNDER THE VASICEK MODEL OF THE SHORT RATE

  • Sun, Yu-dong;Shi, Yi-min;Gu, Xin
    • Journal of applied mathematics & informatics
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    • v.29 no.5_6
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    • pp.1501-1509
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    • 2011
  • In this study, assume that the stock price obeys the stochastic differential equation driven by mixed fractional Brownian motion, and the short rate follows the Vasicek model. Then, the Black-Scholes partial differential equation is held by using fractional Ito formula. Finally, the pricing formulae of the barrier option are obtained by partial differential equation theory. The results of Black-Scholes model are generalized.

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.

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.

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.

COMPARISON OF NUMERICAL SCHEMES ON MULTI-DIMENSIONAL BLACK-SCHOLES EQUATIONS

  • Jo, Joonglee;Kim, Yongsik
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.6
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    • pp.2035-2051
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    • 2013
  • In this paper, we study numerical schemes for solving multi-dimensional option pricing problem. We compare the direct solving method and the Operator Splitting Method(OSM) by using finite difference approximations. By varying parameters of the Black-Scholes equations for the maximum on the call option problem, we observed that there is no significant difference between the two methods on the convergence criterion except a huge difference in computation cost. Therefore, the two methods are compatible in practice and one can improve the time efficiency by combining the OSM with parallel computation technique. We show numerical examples including the Equity-Linked Security(ELS) pricing based on either two assets or three assets by using the OSM with the Monte-Carlo Simulation as the benchmark.

Understanding Black-Scholes Option Pricing Model

  • Lee, Eun-Kyung;Lee, Yoon-Dong
    • Communications for Statistical Applications and Methods
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    • v.14 no.2
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    • pp.459-479
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    • 2007
  • Theories related to financial market has received big attention from the statistics community. However, not many courses on the topic are provided in statistics departments. Because the financial theories are entangled with many complicated mathematical and physical theories as well as ambiguously stated financial terminologies. Based on our experience on the topic, we try to explain the rather complicated terminologies and theories with easy-to-understand words. This paper will briefly cover the topics of basic terminologies of derivatives, Black-Scholes pricing idea, and related basic mathematical terminologies.