• 대한수학회지
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• 제43권4호
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• pp.845-858
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• 2006

We present two approaches of the stochastic interest rate European option pricing model. One is a bond numeraire approach which is applicable to a nonzero value asset. In this approach, we assume log-normality of returns of the asset normalized by a bond whose maturity is the same as the expiration date of an option instead that of an asset itself. Another one is the expectation hypothesis approach for value zero asset which has futures-style margining. Bond numeraire approach allows us to calculate volatilities implied in options even though stochastic interest rate is considered.

• Journal of Applied and Pure Mathematics
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• 제6권3_4호
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• pp.141-154
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• 2024

This article offers a thorough exploration of a modified Black-Scholes model featuring two assets. The determination of option prices is accomplished through the Black-Scholes partial differential equation, leveraging the variational iteration method. This approach represents a semi-analytical technique that incorporates the use of Lagrange multipliers. The Lagrange multiplier emerges as a beacon of efficiency, adeptly streamlining the computational intricacies, and elevating the model's efficacy to unprecedented heights. For better understanding of the presented system, a graphical and tabular interpretation is presented with the help of Maple software.

• 대한수학회보
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• 제48권2호
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• pp.413-426
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• 2011

Two types of new methods with variable time steps are proposed in order to valuate binary options efficiently. Type I changes adaptively the size of the time step at each time based on the magnitude of the local error, while Type II combines two uniform meshes. The new methods are hybrid finite difference methods, namely starting the computation with a fully implicit finite difference method for a few time steps for accuracy then performing a ${\theta}$-method during the rest of computation for efficiency. Numerical experiments for standard European vanilla, binary and American options show that both Type I and II variable time step methods are much more efficient than the fully implicit method or hybrid methods with uniform time steps.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제27권1호
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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.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제23권1호
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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.

• East Asian mathematical journal
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• 제35권1호
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• pp.59-66
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• 2019

In this paper we study implicit-explicit (IMEX) methods combined with a semi-Lagrangian scheme to evaluate the prices of fixed strike arithmetic Asian options under jump-diffusion models. An Asian option is described by a two-dimensional partial integro-differential equation (PIDE) that has no diffusion term in the arithmetic average direction. The IMEX methods with the semi-Lagrangian scheme to solve the PIDE are discretized along characteristic curves and performed without any fixed point iteration techniques at each time step. We implement numerical simulations for the prices of a European fixed strike arithmetic Asian put option under the Merton model to demonstrate the second-order convergence rate.

• Journal of the Korean Data and Information Science Society
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• 제24권4호
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• pp.669-677
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• 2013

본 연구에서는 다중자산 옵션 가격의 추정에 있어 자산의 수, 상관계수, 자산의 값들과 표준편차의 여러 조합에 대한 시뮬레이션을 통하여 저불일치 수열에 따르는 준난수 몬테칼로 방법들을 비교하였다. 결과적으로 준난수와 모로 역변환을 이용하는 것이 기본적인 몬테칼로 방법보다 정확하였으며 자산의 수와 관계없이 준난수 방법들 중 혼합법들이 더욱 효과적임을 알 수 있었다.

• Journal of the Korean Data and Information Science Society
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• 제25권3호
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• pp.513-522
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• 2014

옵션가격의 결정에 있어서 실제 변동성은 사후에 알 수 있는 정보이므로 대용값으로 내재변동성을 가장 많이 사용하는데 본 연구에서는 동일한 기초자산을 가진 옵션의 잔존만기와 행사가격을 이용하여 내재변동성을 추정하고자 한다. KOSPI200 옵션 데이터와 서포트벡터회귀, 나무모형 및 회귀모형을 통해 모형의 설명력을 평균제곱근오차 (RMSE)와 평균절대오차 (MAE)를 사용하여 살펴보았다. 그 결과 서포트벡터회귀와 MART의 성능이 최소제곱회귀보다 우수한 것으로 나타났으며, 서포트벡터회귀와 MART의 성능은 거의 비슷하였다.

• 자원ㆍ환경경제연구
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• 제14권4호
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• pp.943-964
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• 2005

본 연구는 CKLS (1992)와 Nowman and Wang (2001)을 참고하여 다양한 형태의 확률과정 모형들을 추정하였다. 실증분석에서는 1996년 1월부터 2005년 1월까지의 월간 브렌트(Brent) 유가를 대상으로 일반적 적률법(GMM)을 적용하였다. 또한, 시뮬레이션된 시계열자료를 활용하여 유럽행 콜옵션의 가치를 산정하고, 확률과정 모형별로 비교하였다. 실증분석 결과에 의하면, 원유가격의 경우 가격 수준에 따라 변동성이 크게 좌우된다는 것을 알 수 있다. 하지만, 기존 관련 연구의 결과들과 달리 유가의 평균회귀 성향은 약한 것으로 나타났다. 이와 함께, 본 연구에서 채택한 상이한 확률과정 모형에 따라 원유를 기초자산으로 하는 파생상품의 가치가 달라진다는 것을 알 수 있다.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제26권4호
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• pp.296-309
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• 2022

Timer options are one of the contingent claims that, for given the variance budget, its payoff depends on a random maturity in terms of the realized variance unlike the standard European vanilla option with a fixed time maturity. Since it was first launched by Société Générale Corporate and Investment Banking in 2007, the valuation of the timer options under several stochastic environment for the volatility has been conducted by many researches. In this study, we propose the pricing of timer power options combined with standard timer options and the index of the power to the underlying asset for the investors to actualize lower risks and higher returns at the same time under the uncertain markets. By using the asymptotic analysis, we obtain the first-order approximation of timer power options. Moreover, we demonstrate that our solution has been derived accurately by comparing it with the solution from the Monte-Carlo method. Finally, we analyze the impact of the stochastic volatility with regards to various parameters on the timer power options numerically.