• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제25권3호
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• pp.93-106
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• 2021

In this paper, an efficient hybrid numerical method for solving two-asset option pricing problem is presented based on the Crank-Nicolson and the radial basis function methods. For this purpose, the two-asset Black-Scholes partial differential equation is considered. Also, the convergence of the proposed method are proved and implementation of the proposed hybrid method is specifically studied on Exchange and Call on maximum Rainbow options. In addition, this method is compared to the explicit finite difference method as the benchmark and the results show that the proposed method can achieve a noticeably higher accuracy than the benchmark method at a similar computational time. Furthermore, the stability of the proposed hybrid method is numerically proved by considering the effect of the time step size to the computational accuracy in solving these problems.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제14권3호
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• pp.175-187
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• 2010

This paper presents the numerical valuation of the two-asset step-down equitylinked securities (ELS) option by using the operator-splitting method (OSM). The ELS is one of the most popular financial options. The value of ELS option can be modeled by a modified Black-Scholes partial differential equation. However, regardless of whether there is a closedform solution, it is difficult and not efficient to evaluate the solution because such a solution would be represented by multiple integrations. Thus, a fast and accurate numerical algorithm is needed to value the price of the ELS option. This paper uses a finite difference method to discretize the governing equation and applies the OSM to solve the resulting discrete equations. The OSM is very robust and accurate in evaluating finite difference discretizations. We provide a detailed numerical algorithm and computational results showing the performance of the method for two underlying asset option pricing problems such as cash-or-nothing and stepdown ELS. Final option value of two-asset step-down ELS is obtained by a weighted average value using probability which is estimated by performing a MC simulation.

• Journal of applied mathematics & informatics
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• 제38권5_6호
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• pp.439-461
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• 2020

A step-up option is a newly developed financial instrument that simultaneously provides higher security and profitability. This paper introduces two step-up options: step-up type1 and step-up type2 options, and derives the option pricing formulas using the Laplace transform. We assume that the underlying equity price follows a regime-switching model that reflects the long-term maturity of these options. The option prices are calculated for the two types of funds, a pure stock fund composed of risky assets only and a mixed fund composed of stocks and bonds, to reflect possible variety in the fund underlying asset mix. The impact of changes in the model parameters on the option prices is analyzed. This paper provides information crucial to product developments.

• 대한수학회보
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• 제53권5호
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• pp.1497-1530
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• 2016

External barrier options are two-asset options with stochastic variables where the payoff depends on one underlying asset and the barrier depends on another state variable. The barrier state variable determines whether the option is knocked in or out when the value of the variable is above or below some prescribed barrier level. This paper derives the explicit analytic solution of the chained option with an external single or double barrier by utilizing the probabilistic methods - the reflection principle and the change of measure. Before we do this, we examine the closed-form solution of the external barrier option with a single or double-curved barrier using the methods of image and double Mellin transforms. The exact solution of the external barrier option price enables us to obtain the pricing formula of the chained option with the external barrier more easily.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제26권2호
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• pp.121-137
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• 2022

In this paper, we investigate an efficient hybrid method for solving two-asset time fractional Black-Scholes partial differential equations. The proposed method is based on the Crank-Nicolson the radial basis functions methods. We show that, this method is convergent and we obtain good approximations for solution of our problems. The numerical results show high accuracy of the proposed method without needing high computational cost.

• 자원ㆍ환경경제연구
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• 제13권4호
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• pp.735-761
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• 2004

해외자원개발사업은 성공할 경우 높은 수익률을 보장하지만 장기적인 투자기간과 높은 시장위험부담으로 인하여 사업의 가치분석에 있어서 사업기간 동안의 여러 가지 변수들을 분석할 수 있는 유연성을 요구하고 있다. 기업의 투자 의사결정과정에서 가장 널리 이용되는 평가방법인 전통적 기존의 현금흐름할인법의 단점을 보완할 대안으로서 제시된 옵션가격 결정모형(Option Pricing Model)을 여타의 다른 자산의 평가 및 사업성 평가에 응용하고자 하는 연구 분야인 실물옵션(Real Options)은 특히 위험도가 큰 자원개발사업의 가치를 평가할 좋은 방법론으로 주목받아왔으나, 다양한 현실적 상황을 도입하게 되면 확률과정이 난해한 형태로 변하여 수학적 처리가 용이하지 않아 실용화에 가장 큰 걸림돌로 작용하고 있다. 따라서 기존의 연구들은 확률과정의 선정과정에서 자원개발사업의 특성이나 실용성을 고려하여 확률과정을 선정하지 않고 기초적인 확률과정을 적용하여 왔다. 본 연구에서는 해외자원개발사업을 대상으로 옵션가격 결정모형을 활용하는 경우를 산정하여, 해외자원개발사업의 평가에 쉽게 활용될 수 있는 단순화된 함수의 형태로 표현된 옵션가격 결정모형을 제시해 보았다. 즉, 이론적인 정교한 확률과정을 도출하기보다는 자원개발사업의 특징을 충분히 반영하면서도 사업평가실무에 손쉽게 이용될 수 있는 현실적이면서도 단순한 확률과정을 선정하고자 하였다. 이를 위하여 구리, 연, 아연의 국제시장가격의 특성과 연-아연광 개발사업의 사례를 활용하여 기존의 모형연구들과 달리 실제의 위험을 모두 분석하되, 분석하는 모형을 최대한 단순화하여가는 과정을 통하여 Gibson-Schwartz가 제안한 Two-Factor Model과 Long-Term Asset Model을 적절한 모형으로 선정하고, 이를 바탕으로 운영옵션과 투자개시옵션의 두 가지 경영옵션을 분석하여 그 결과를 제시하였다. 본 연구에서 분석, 제안한 단순화 과정은 앞으로 옵션가격 결정이론을 바탕으로 한 가치평가모형의 실제사례 적용연구에서 활용될 수 있을 것으로 기대한다.

• 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 the Korean Society for Industrial and Applied Mathematics
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• 제18권1호
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• pp.61-74
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• 2014

This paper presents accurate and efficient numerical methods for calculating the sensitivities of two-asset European options, the Greeks. The Greeks are important financial instruments in management of economic value at risk due to changing market conditions. The option pricing model is based on the Black-Scholes partial differential equation. The model is discretized by using a finite difference method and resulting discrete equations are solved by means of an operator splitting method. For Delta, Gamma, and Theta, we investigate the effect of high-order discretizations. For Rho and Vega, we develop an accurate and robust automatic algorithm for finding an optimal value. A cash-or-nothing option is taken to demonstrate the performance of the proposed algorithm for calculating the Greeks. The results show that the new treatment gives automatic and robust calculations for the Greeks.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제26권4호
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• pp.343-352
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• 2022

We present the user-friendly graphical user interface design and implementation of Monte Carlo simulation (MCS) for computing option price of the four-underlying asset step-down equity linked securities (ELS) using the Android platform. The ELS has been one of the most important and influential financial products in South Korea. Most ELS products are based on one-, two-, and three-underlying assets. However, currently there is a demand for higher coupon payment from ELS products because of the increased interest rate in financial market. In order to allow the investors to have higher coupon payment, it is necessary to design a multi-asset ELS such as four-asset step-down ELS. We conduct the computational experiments to demonstrate the performance of the Android platform for pricing four-asset step-down ELS. Furthermore, we perform a comparison test with a three-asset step-down ELS.

• 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.