• 한국수학사학회지
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• 제20권3호
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• pp.127-146
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• 2007

미국수학사에서 가장 중요한 시기로 여겨지는 1890년에서 1950년 사이의 미국수학계의 발전과정을 당시의 미국수학 연구에 있어서 혁신적 발전의 계기를 제공한 시카고 대학의 초대 수학과장 E.H. Moore 의 역할을 중심으로 고찰한다. 19세기말 아직 낙후되었던 미국 수학계는 시카고 대학의 핵심 학과였던 수학과는 총장의 비전을 같이 할 우수교수를 확보하고, 새로운 제도 하에서 선발된 우수 대학원 학생들을 탐구지향 교수법으로 지도하며 미국 수학연구의 초장기에 우수한 인재를 공급하기 시작한다. 이를 통하여 미국은 인재양성과 새로운 연구 분야 및 연구방법의 개척에 성공하고, 1950년 국제수학자대회(ICM)를 미국에서 개최하며, 당당히 세계수학의 주류에 진입한다. 본 원고는 위의 발전과정이 현재 한국에 주는 의미를 분석한다.

• Journal of applied mathematics & informatics
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• 제31권5_6호
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• pp.769-784
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• 2013

We consider the modulus-based successive overrelaxation method for the linear complementarity problems from the discretization of Black-Scholes American options model. The $H_+$-matrix property of the system matrix discretized from American option pricing which guarantees the convergence of the proposed method for the linear complementarity problem is analyzed. Numerical experiments confirm the theoretical analysis, and further show that the modulus-based successive overrelaxation method is superior to the classical projected successive overrelaxation method with optimal parameter.

• 한국수학사학회지
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• 제19권2호
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• pp.77-88
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• 2006

미국 버지니아대학 수학과 교수, 보험회사 계리인, 변호사를 거쳐, 영국 육군사관학교 교관으로 55세에 정년을 한 유태계 영국 수학자 J. J. 실베스터는 61세의 나이로 1876년 미국 최초의 연구중심대학인 존스홉킨스대학에 초대 수학과장으로 초빙되어 연구 인력을 배출하고 미국 최초의 수학연구저널을 발간하며 미국에 현대수학의 연구 여건을 마련 해 준다. 본 논문은 그와 그가 후임으로 추천한 F. 클라인이 19세기 후반 미국수학계에 끼친 역할을 분석한다. 우리는 실베스터와 클라인과 미국인 수학자 E. H. 무어가 100여년 전 낙후된 미국 수학을 당시 유럽 중심의 수학계 주류에 진입시키는 과정에서의 역할과 이 과정이 한국에서 갖는 의미를 생각한다.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제12권4호
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• pp.271-287
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• 2008

This paper deals with American call and put options. Determining the fair price and the free boundary of an American option is a very difficult problem since they depends on each other. This paper presents numerical algorithms of finite element method based on the three-level scheme to compute both the price and the free boundary. One algorithm is designed for American call options and the other one for American put options. These algorithms are formulated on the system of the Jamshidian equation for the option price and the free boundary. Here, the Jamshidian equation is of a kind of the nonhomogeneous Black-Scholes equations. We prove the existence and uniqueness of the numerical solution by the Lax-Milgram lemma and carried out extensive numerical experiments to compare with various methods.

• Korean Journal of Mathematics
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• 제25권2호
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• pp.229-246
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• 2017

This paper concerns barrier option of American type where the underlying price is monitored during only part of the option's life. Analytic valuation formulas of the American partial barrier options are obtained by approximation method. This approximation method is based on barrier options along with exponential early exercise policies. This result is an extension of Jun and Ku [10] where the exercise policies are constant.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제4권2호
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• pp.11-20
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• 2000

This paper presents a formula that relates the optimal exercise boundaries of American call and put options on futures contract. It is shown that the geometric mean of the optimal exercise boundaries for call and put written on the same futures contract with the same exercise price is equal to the exercise price which is time invariant. The paper also investigates the properties of American calls and puts on futures contract.

• Journal of applied mathematics & informatics
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• 제6권2호
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• pp.569-574
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• 1999

In this paper we obtain relation between the heding prices in European and American option and apply to call option.

• 대한수학회보
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• 제60권2호
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• pp.361-388
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• 2023

In this study, we deal with American lookback option prices on dividend-paying assets under a stochastic volatility (SV) model. By using the asymptotic analysis introduced by Fouque et al. [17] and the Laplace-Carson transform (LCT), we derive the explicit formula for the option prices and the free boundary values with a finite expiration whose volatility is driven by a fast mean-reverting Ornstein-Uhlenbeck process. In addition, we examine the numerical implications of the SV on the American lookback option with respect to the model parameters and verify that the obtained explicit analytical option price has been obtained accurately and efficiently in comparison with the price obtained from the Monte-Carlo simulation.

• 대한수학회보
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• 제48권4호
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• pp.791-812
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• 2011

This is a survey on American options. An American option allows its owner the privilege of early exercise, whereas a European option can be exercised only at expiration. Because of this early exercise privilege American option pricing involves an optimal stopping problem; the price of an American option is given as a free boundary value problem associated with a Black-Scholes type partial differential equation. Up until now there is no simple closed-form solution to the problem, but there have been a variety of approaches which contribute to the understanding of the properties of the price and the early exercise boundary. These approaches typically provide numerical or approximate analytic methods to find the price and the boundary. Topics included in this survey are early approaches(trees, finite difference schemes, and quasi-analytic methods), an analytic method of lines and randomization, a homotopy method, analytic approximation of early exercise boundaries, Monte Carlo methods, and relatively recent topics such as model uncertainty, backward stochastic differential equations, and real options. We also provide open problems whose answers are expected to contribute to American option pricing.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제19권1호
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• pp.1-21
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• 2015

This paper presents two algorithms based on the Jamshidian equation which is from the Black-Scholes partial differential equation. The first algorithm is for American call options and the second one is for American put options. They compute numerically free boundary and then option price, iteratively, because the free boundary and the option price are coupled implicitly. By the upwind finite-difference scheme, we discretize the Jamshidian equation with respect to asset variable s and set up a linear system whose solution is an approximation to the option value. Using the property that the coefficient matrix of this linear system is an M-matrix, we prove several theorems in order to formulate a bisection method, which generates a sequence of intervals converging to the fixed interval containing the free boundary value with error bound h. These algorithms have the accuracy of O(k + h), where k and h are step sizes of variables t and s, respectively. We prove that they are unconditionally stable. We applied our algorithms for a series of numerical experiments and compared them with other algorithms. Our algorithms are efficient and applicable to options with such constraints as r > d, $r{\leq}d$, long-time or short-time maturity T.