• The Korean Journal of Financial Management
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• v.24 no.4
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• pp.1-43
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• 2007

The traditional Black-Scholes model for option pricing is based on the assumption that the log-return of the underlying asset follows a Brownian motion. But this assumption has been criticized for being unrealistic. Thus, for the last 20 years, many attempts have been made to adopt different stochastic processes to derive new option pricing models. The option pricing models based on L$\acute{e}$vy processes are being actively studied originating from the Gerber-Shiu model driven by H. U. Gerber and E. S. W. Shiu in 1994. In 2004, G. H. L. Cheang derived an option pricing model under multiple L$\acute{e}$vy processes, enabling us to adopt drift and jumps to the Gerber-Shiu model, while Gerber and Shiu derived their model under one L$\acute{e}$vy process. We derive the Gerber-Shiu model which includes drift and jumps under L$\acute{e}$vy processes. By adopting a Gamma distribution, we expand the Heston model which was driven in 1993 to include jumps. Then, using KOSPI200 index option data, we analyze the price-fitting performance of our model compared to that of the Black-Scholes model. It shows that our model shows a better price-fitting performance.

• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1463-1481
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• 2018

Let $X=\{X_t\}_{t{\geq}0}$ be a $L{\acute{e}}vy$ process in ${\mathbb{R}}^d$ and ${\Omega}$ be an open subset of ${\mathbb{R}}^d$ with finite Lebesgue measure. The quantity $H_{\Omega}(t)={\int_{\Omega}}{\mathbb{P}}^x(X_t{\in}{\Omega})$ dx is called the heat content. In this article we consider its generalized version $H^{\mu}_g(t)={\int_{\mathbb{R}^d}}{\mathbb{E}^xg(X_t){\mu}(dx)$, where g is a bounded function and ${\mu}$ a finite Borel measure. We study its asymptotic behaviour at zero for various classes of $L{\acute{e}}vy$ processes.

• Communications of the Korean Mathematical Society
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• v.9 no.4
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• pp.975-983
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• 1994

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.725-741
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• 1997

This paper considers the problem of deriving exponential probability inequalities for product symmetric Poisson processes. As an application they are used to show the existence of regular version of some product process derived from L$\acute{e}$vy process.

• Communications for Statistical Applications and Methods
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• v.17 no.4
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• pp.483-491
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• 2010

This study provides the explicit computation of the ruin probability of a Le￠vy process on finite time horizon in Theorem 1 with the help of a fluctuation identity. This paper also gives the numerical results of the ruin probability in Variance Gamma(VG) and Normal Inverse Gaussian(NIG) models as illustrations. Besides, the paths of VG and NIG processes are simulated using the same parameter values as in Madan et al. (1998).

• Journal of the Korean Mathematical Society
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• v.50 no.5
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• pp.1009-1031
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• 2013

In this paper, using elementary calculus only, we give a simple proof that Green function estimates imply the sharp two-sided pointwise estimates for Poisson kernels for subordinate Brownian motions. In particular, by combining the recent result of Kim and Mimica [5], our result provides the sharp two-sided estimates for Poisson kernels for a large class of subordinate Brownian motions including geometric stable processes.

• Management Science and Financial Engineering
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• v.19 no.2
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• pp.37-42
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• 2013

This paper suggests a numerical method for valuation of European and American options under the two L$\acute{e}$vy Processes, Normal Inverse Gaussian Model and the Variance Gamma model. The method is based on approximation of underlying asset price using a finite-state, time-homogeneous Markov chain. We examine the effectiveness of the proposed method with simulation results, which are compared with those from the existing numerical method, the lattice-based method.

• Communications for Statistical Applications and Methods
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• v.21 no.4
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• pp.327-334
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• 2014

Various modified GARCH(1, 1) models have been found adequate in many applications. We are interested in their continuous time versions and limiting properties. We first define a stochastic integral that includes useful continuous time versions of modified GARCH(1, 1) processes and give sufficient conditions under which the process is exponentially ergodic and ${\beta}$-mixing. The central limit theorem for the process is also obtained.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.863-875
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• 2010

In this paper we apply the martingale approach, which has been widely used in mathematical finance, to investigate the optimal investment problem for an insurer under the criterion of mean-variance. When the risk and security assets are described by the L$\acute{e}$vy processes, the closed form solutions to the maximization problem are obtained. The mean-variance efficient strategies and frontier are also given.

• Communications for Statistical Applications and Methods
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• v.18 no.2
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• pp.229-236
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• 2011

A continuous time stochastic volatility model for financial assets suggested by Barndorff-Nielsen and Shephard (2001) is considered, where the volatility process is modelled as an Ornstein-Uhlenbeck type process driven by a general L$\'{e}$vy process and the price process is then obtained by using an independent Brownian motion as the driving noise. The uniform ergodicity of the volatility process and exponential ${\alpha}$-mixing properties of the log price processes of given continuous time stochastic volatility models are obtained.