
 이항석, 이가은, 송성주 (2021) : Multistep Reflection Principle and Barrier Options
 This paper examines a class of barrier options–multistep barrier options, which can have any finite number of barriers of any level. We obtain a general, explicit expression of option prices of this type under the BlackScholes model. Multistep barrier options are not only useful in that they can handle barriers of different levels and time steps, but can also approximate options with arbitrary barriers. Moreover, they can be embedded in financial products such as deposit insurances based on jump models with simple barriers. Along the way, we derive multistep reflection principle, which generalizes the reflection principle of Brownian motion.

 작성일 20210804
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 이항석, 하홍준, 이민하 (2021) : Valuation of piecewise linear barrier options
 Abstract This paper discusses the valuation of piecewise linear barrier options that generalize classical barrier options. We establish formulas for joint probabilities of the logarithmic returns of the underlying asset and its partial running maxima when the process has a piecewise constant drift. In particular, we show that our results embrace the famous reflection principle as a special case, and that our established proposition delivers useful scalability for computing desired probabilities related to various types of barriers. We derive the closedform prices of piecewise linear barrier options under the Black–Scholes framework, which are obtainable with little effort by relying on the derived probabilities. In addition, we provide numerical examples and discuss how option prices respond to several types of piecewise linear barriers.

 작성일 20210804
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 이항석, 하홍준, 이태원 (2021) : Decrement rates and a numerical method under competing risks
 Abstract Modeling the interactions of competing risks that affect the occurrence of various decrements such as death or disease is an essential issue in survival analysis and actuarial science. Popular assumptions for the construction of decrement models are uniform distributions of decrements in a multiple decrement table (mUDD) and associated single decrement tables (sUDD), respectively. Even though there are many theoretical generalizations to relaxing mUDD assumption, it is not clear how to obtain theoretical and numerical methods for modeling relationships of competing risks under a general assumption in associated single decrement tables beyond the sUDD. We fill this gap in the literature by discussing the conversion between probabilities of decrement and absolute rates under a general form of a distribution of fractional ages. In particular, we show that extracting absolute rates from the probabilities under the general competing risk assumption boils down to solving a system of nonlinear equations and propose a novel numerical algorithm for its solution. Extensive numerical experiments relying on the algorithm verify that the algorithm delivers reliable results in terms of efficiency and accuracy.

 작성일 20210122
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 이항석, 김은채, 송성주 (2020) : Pricing twoasset alternating barrier options with icicles and their variations
 Abstract This paper introduces a new class of barrier options and its variations. We call the new class of options as twoasset alternating barrier options, since we consider alternating barrier levels for two underlying assets. The alternating barrier levels are placed in the subperiods of the option’s lifetime; each being applied to one of the two underlying assets. We also consider vertical branches of the barrier, which are termed as icicles. The alternating barrier with icicles can be often seen as an embedded form in various equitylinked financial products. To price such new options, we obtain the joint distribution of two underlying asset prices at an intermediate time point and the maturity, along with their partial maximums under the Black–Scholes model. This joint distribution plays a critical role in the derivation of the pricing formulas for alternating barrier options and their variants. As in ordinary barrier options, we consider eight types of alternating barrier options and derive their explicit option pricing formulas. To our knowledge, the pricing formulas for these options have never been obtained explicitly in the literature even under the Black–Scholes model. We also examine an auto callable equitylinked investment product to derive its explicit pricing formula. Our results are illustrated with numerical examples, showing the effect of different barrier levels and different values of correlation coefficient between two underlying asset prices.

 작성일 20210111
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 이항석, 손지훈 (2020) : Rederivation of Gertler's model and analysis of the Korean economy
 Abstract This paper makes a theoretical contribution by providing clear and detailed derivation of economic agents’ decision problems including elastic labor supply in Gertler’s overlapping generation (OLG) model. We apply the model to the Korean economy by calibration based on Korean economic data. It also analyzes the impact of current social issues such as aging and extension of retirement age, on the Korean economy in a longrun equilibrium. Subsequently, we also discuss the implications of the analysis. Aging has prolonged the period of retirement; therefore, population structure changes by the increase in the proportion of retirees, the total consumptiontoGDP ratio decreases, and capital stock increases due to reduced propensity to consume out of wealth in preparation for an individual’s retirement life. The implementation of retirement age extension increases the proportion of retirees relatively less and alleviates fluctuations in labor supply and the share of financial assets for both economic agents. However, the decrements in consumptiontoGDP ratio is larger than before, and this leads to a larger rise in the capital stock compared to when there is only an aging effect. Keywords: overlapping generation model, elastic labor supply, aging, life expectancy, retirement age extension, steadystate

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 이항석, 최형석, 하홍준 (2020) : A sharing mechanism of investment outcome for interestsensitive life insurance products
 Abstract As growing sales of insurance contracts with a saving feature, an issue of sharing investment outcome gets the attention of insurers and policyholders. This paper focuses on a systematic way of finding the sharing mechanism for an optimal contract design in such a way that a policyholder and an insurer maximize their expected utilities. We adopt the policyholder and the insurer as a principal and an agent, respectively, and regard a share of the investment performance as an incentive for the insurer to elicit efforts. As a result of this setting, the moral hazard issue generated from the insurer is unavoidable. For the purpose, the Holmström (1979)’s principalagent model with limited observability of the insurer’s action plays a leading role in resolving a piecutting problem. Under our model assumption, the sharing mechanism states that a portion of the outcome belonging to the insurer is a nondecreasing function of the excess of the portfolio return over a benchmark return when the two parties are riskaverse. In particular, the sensitivity of the sharing portion has an Sshape curve which is consistent with the insurer’s risk propensity. An empirical study based on companies’ portfolio attributes and crediting rates verifies that our theoretical findings are consistent with statistically significant results. In particular, we confirm that the bargaining power of the insurer has a considerable impact on the sharing mechanism as it is theoretically important.

 작성일 20210108
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 이항석, 백혜연 (2020) : 계리 리스크관리 연구 동향과 향후의 연구 과제
 저널 : 한국리스크관리학회(Vol31 No.2) 본 연구는 지난 30년간(1990년부터 2019년 상반기) 한국리스크관리학회지인 「리스크관 리연구」 논문에 게재된 계리 리스크관리 연구 논문 총 78편에 대해 논문의 주요 내용들과 시사점들을 리뷰한 것이다. 계리 리스크관리 연구 논문들의 연구 동향을 살펴보기 위해 시 대별·주제별로 그동안의 연구 성과를 주로 다루었다. 계리 리스크관리 연구는 전통적인 계 리 업무인 상품개발 관련 연구, 재무건전성에 관한 연구, 재보험과 ALM, ERM 등과 같은 계리 리스크관리 기법에 관한 연구들로 구분해 볼 수 있었다. 이러한 연구 논문들을 살펴본 결과, 계리 리스크관리 분야의 연구 주제가 2000년 이후 매우 세분화된 것에 비해 「리스크 관리연구」에 게재된 논문들은 일부 주제에 편중되어 있는 편이었고, 실증분석과 같이 수리 적 접근 방법의 논문들이 제도와 정책 연구 논문들에 비해 좀 더 많은 비중을 차지하고 있 었다. 이 결과들을 바탕으로 향후 새로운 분야나 다양한 연구방법론들을 다룬 논문들이 균 형 있게 「리스크관리연구」에서 다뤄지기를 기대해 본다.

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 이항석, 안재윤, 고방원 (2019) : Construction of multiple decrement tables under generalized fractional age assumptions
 In this paper, we intend to develop a consistent methodology for constructing multiple decrement tables under generalized fractional age assumptions. Assuming that decrements have a common distribution at fractional ages, we derive conversion formulas to split or merge given multiple decrement tables in order to obtain a new multiple decrement table of interest. The assumptions that we consider are quite general, with a wide range of fractional age assumptions including the uniform distribution of decrements or the constant forces of decrement.Our proposed approaches allow us to directly obtain multiple decrement tables without the need for the associated single rates of decrement. They will also enable us to avoid potential in consistency under the uniform distribution assumption so run natural nessarising from the constant forces assumption. In addition, as they navigate through a larger window, they will deepen our understanding of the classical results under the uniform distribution assumptions. Although our methodology is based on a common distribution function assumption, knowing the specific form of the function is unnecessary,since our conversion formulas do not depend upon it. Finally, numerical examples are illustrated where we investigate the main factors of the errors induced by the discrepancy between the true and assumed distributions. The numerical result shows that the relative errors under our approaches are practically negligible for moderate ranges of multiple decrement probabilities.

 작성일 20210804
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 이항석, 고방원 (2019) : Valuing equityindexed annuities with icicled barrier options
 Abstract Inspired by the recent popularity of autocallable structured products, this paper intends to enhance equityindexed annuities (EIAs) by introducing a new class of barrier options, termed icicled barrier options. The new class of options has a vertical (icicled) barrier along with the horizontal one of the ordinary barrier options, which may act as an additional knockin or knockout trigger. To improve the crediting method of EIAs, we propose a new EIA design, termed autocallable EIA, with payoff structure similar to the autocallable products except for the minimum guarantee, and further investigate the possibility of embedding various icicled barrier options into the plain pointtopoint or the ratchet EIAs. Explicit pricing formulas for the proposed EIAs and the icicled barrier options are obtained under the Black–Scholes model. To the purpose, we derive the joint distribution of the logarithmic returns at the icicled time and the maturity, and their running maximum. As an application of the wellknown reflection principle, the derivation itself is an interesting probability problem and the joint distribution plays a key role in the subsequent pricing stage. Our option pricing result can be easily transferred to EIAs or other equitylinked products. The pricing formulas for the EIAs and the options are illustrated through numerical examples.

 작성일 20210122
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