
 이항석, 하홍준, 이민하 (2021) : Piecewise linear double barrier options
 A piecewise linear double barrier option generalizes classical double barrier options because of its versatility in designing various double boundaries. This paper discusses how to price piecewise linear double barrier options. To this purpose, we derive the probability that an underlying process does not cross a given piecewise linear double barrier, where the underlying process follows the Brownian motion of piecewise constant drift. Using the established noncrossing probability, we provide the explicit pricing formulas of piecewise linear double barrier options and show how the shape of a double barrier affects the option prices through extensive numerical experiments.

 작성일 20220901
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 이항석, 이가은, 최양호 (2021) : Outside barrier lookback options with floating strike
 This paper introduces a new class of exotic options, lookback outside barrier options with two underlying assets, which combine lookback and barrier options by incorporating active barrier conditions into lookback payoff. Bermin (Essays on lookback and barrier options: A Malliavin calculus approachs, Lund University, Department of Economics, PhD thesis, 1998) proposed lookbackbarrier options to compensate for the shortcomings of the highpriced lookback option, and later several applications were made by either mitigating the conditions or adding other conditions. In this paper, we extend the existing idea to lookbackbarrier options with two underlying assets. To reduce the burden of the computational process and simplify the conditions, the life of the options was divided into two nonoverlapping intervals. The structure of the lookback outside barrier option with two underlying assets is as follows: whether the barrier conditions are satisfied or not in the first subinterval, and the amount of payoff is determined in the second subinterval if the condition is met. In traditional lookbackbarrier options, barrier conditions and payoff are determined by one asset, while in a lookbackbarrier option with two underlying assets, one of the two assets determines the barrier conditions and the other is responsible for payoff. Our main study develops a complete valuation framework that allows for closedform pricing formulas under the BlackScholes model. This closed pricing formula for lookback outside barrier option with two underlying assets offers significant advantages over Monte Carlo simulation methods, as a large number of simulations may be required for accurate computation. Complexities involved in the derivation process would be resolved by the Esscher transform and the reflection principle of Brownian motion. We illustrate our results with numerical examples, showing the effects of different values of correlation coefficients between two underlying asset prices.

 작성일 20220901
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 이항석, 류두진, 손지훈 (2021) : Riskadjusted valuation in the worker's economic decision making
 We suggest a riskadjusted valuation approach for workers at retirement risk to make decisions in an overlapping generations economy. The riskaverse workers use greater weights than expected, socalled riskadjusted probability, on their retirement cash flows to assess the residual lifetime income. This method can be consistently applied to value financial and nonfinancial assets, and on the riskadjusted valuations, the workers will make optimal investments. To predict capital returns and economic variables, comparative statics will be numerically implemented via demographic structure, preference, and social security policy by aggregating workers’ decisions.

 작성일 20220901
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 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 for option prices of this type under the Black–Scholes model by deriving the multistep reflection principle, that is, the multistep boundarycrossing probability of Brownian motion. 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 applied to pricing barrier options under jumpdiffusion models.

 작성일 20220901
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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, 류두진, 손지훈 : RiskAdjusted Valuation in the Worker’s Economic Decision Making
 We suggest a riskadjusted valuation approach for workers at retirement risk to make decisions in an overlapping generations economy. The riskaverse workers use greater weights than expected, socalled riskadjusted probability, on their retirement cash flows to assess the residual lifetime income. This method can be consistently applied to value financial and nonfinancial assets, and on the riskadjusted valuations, the workers will make optimal investments. To predict capital returns and economic variables, comparative statics will be numerically implemented via demographic structure, preference, and social security policy by aggregating workers’ decisions.

 작성일 20220901
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 이항석, 류두진, 손지훈 (2021) : Heterogeneous Agents' Financial Decision and Asset Accumulation
 We develop a lifecycle OLG model with heterogeneous characteristics for population share, work probability, survival rate, labor productivity, subjective discount rate, and intertemporal elasticity of substitution. We can observe the deviations in financial decisions and asset accumulation among heterogeneous agents and obtain aggregate economic variables such as consumption and capital in equilibrium. In a numerical example, we find that workers and retirees in the group with high labor productivity, long working periods and long remaining life expectancy have the highest proportion of financial assets and the lowest propensity to consume out of wealth.

 작성일 20220901
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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

 작성일 20210804
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