Earticle

Empirically we test the Merton-type model (1974) of credit risk in an emerging market such as the Korean corporate bond market. For that purpose, we assume two alternative firm value processes: Diffusion process for the Merton (1974) model and jump-diffusion process for our extended model in a general equilibrium setting. Our empirical results show that the diffusion model generally underpredicts spreads – which is referred to as “the credit spread under prediction puzzle” in the literature, while our jump-diffusion model somewhat raises the predicted spreads. We assert that jump raises the spreads on two grounds. Firstly, an extremely large (negative) change tends to increase the probability for a firm to default particularly over a short-time horizon. Secondly, jump requires the systematic risk premium for a positively correlated firm particularly when the market turns extremely volatile.

현실세계에서 증권시장의 상품들은 대체로 정규분포를 따르지 않으며, 실제로는 첨도가 높고 왜도의 꼬리부분이 두꺼운 fat-tail 분포를 따르는 것으로 알려져 있다. 따라서 본 연구의 목적은 확률변수가 안정적이고 독립적인 분포를 따르면서 실제분포의 속성을 잘 나타내는 분포를 이용한 옵션가격결정을 분석하는 데 있다. 이때 실증분석 데이터는 2019년 11월 15일부터 2020년 11월 20일까지 252개의 일별 KOSPI200 종가지수를 이용하였으며, 11월 20일 종가인 340.62를 등가격(ATM)으로 하여 335.62와 345.62 각각을 콜옵션(풋옵션)의 OTM(ITM)과 ITM(OTM)으로 설정한 후 분석하였다. 본 논문의 연구방법은 Furier 변환이 가능한 조절안정분포(Tempered Stable Distribution)를 가정한 CTS(또는 CGMY) 모형에 의한 옵션가격결정을 수행하였으며, 이와 더불어 정규분포 및 Fourier 변환을 가정한 Black-Scholes 모형, 그리고 Fourier 변환을 갖는 Merton Diffusion Process와 Variance-Gamma 프로세스에 의한 옵션가격결정모형을 비교·분석하였다. 본 논문의 연구결과는 다음과 같다. 만기 T=1 및 T=0.25 각각의 콜옵션 및 풋옵션 산출결과, (i) BS모형의 경우 기하 브라우니언 모션과 Fourier 역변환에 의한 콜옵션 및 풋옵션 모두 동일한 옵션가격을 산출하였으며, 이는 와 에 기인된 결과이다. (ii) Merton의 Jump-Diffusion Process는 Poisson 분포에 따른 jump 벡터(N=100)의 실증분석에 의해서 산출되었으며, 이때 jump가 0인 경우(와 의 값이 0인 경우)에는 BS모형과 동일한 옵션가격을 산출하나 여기서는 jump가 0이 아닌 경우를 고려하므로, BS모형의 옵션가격보다 약간 더 큰 것으로 나타났다. (iii) Fourier 변환을 이용한 Varinace-Gamma process의 옵션가격은 브라우니언 종속치(subordination)의 표현과 경로분석을 통해 산출되며, Fourier 역변환(FIC) 및 MCS 옵션성과가 BS 모형과 동일한 결과값을 산출하였다. 한편 (iv) CGMY방법에 의한 옵션성과에 대한 산출결과는 소수점 이하에서 미세차이는 있으나 무시할 수 있는 정도여서 CGMY와 BS 모형의 옵션가격이 점근적으로 동일한 것으로 나타났다. 따라서 기존 BS모형보다 Fourier 변환이나 tempered stable Lévy process 모형인 CTS(또는 CGMY)모형에 의한 옵션가격의 분석결과가 실제분포인 fat-tail 속성을 더 잘 설명하는 것으로 나타났다. 본 연구의 시사점은 확률변수가 Furier 변환이 가능한 안정분포인 Tempered Stable Lévy Process를 나타내는 6모수군의 CGMY 모형, Black-Scholes 모형, 그리고 Merton 및 Variance-Gamma 프로세스에 의한 옵션가격을 상호 비교함으로써 정규분포를 가정한 전통적인 Black-Scholes 모형의 분석한계를 극복한 데 있다.

In the real world, products in the stock market do not generally follow a normal distribution, but are known to follow a fat-tail distribution with high kurtosis and thick tails of skewness. Therefore, the purpose of this study is to analyze option pricing using a distribution in which random variables follow a stable and independent distribution and well represent the properties of the actual distribution. At this time, for the empirical analysis data, 252 daily KOSPI200 closing price indexes were used from November 15, 2019 to November 20, 2020. Call Options(Put options) were set as OTM(ITM) and ITM (OTM) and analyzed. In the research methodology of this paper, option pricing was performed by the CTS (or CGMY) model assuming a Tempered Stable Distribution that is capable of Furier transformation. In addition, the Black-Scholes model assuming a normal distribution and Fourier transformation, and the Fourier transformation were performed. The option pricing model using the Merton Diffusion Process and Variance-Gamma process was compared and analyzed. The research findings of this paper are as follows. Calculation results of call options and put options at expiration T=1 and T=0.25, respectively, (i) In the case of the BS model, the same option price was calculated for both call and put options by geometric Brownian motion and Fourier inverse transformation. which is the and . (ii) Merton's Jump-Diffusion Process was calculated by empirical analysis of jump vectors (N=100) according to Poisson distribution. The price is calculated, but since the case where the jump is not 0 is considered, it is slightly larger than the option price of the BS model. (iii) Option price of Varinace-Gamma process using Fourier transformation is calculated through expression of Brownian subordination and path analysis, and Fourier inverse transformation (FIC) and MCS option performance yield the same result as BS model did. On the other hand, (iv) the calculation results for option performance by the CGMY method showed that the option prices of the CGMY and BS models were asymptotically the same as there was a slight difference at the decimal point, but it was negligible. Therefore, the analysis result of the option price by the Fourier transform or the CTS (or CGMY) model, which is a tempered stable Levy process model, was found to better explain the fat-tail property of the actual distribution than the existing BS model. The implication of this study is that the random variable is normally distributed by comparing the CGMY model of the six-parameter group, the Black-Scholes model, and the option prices by the Merton and Variance-Gamma processes, which represent the Tempered Stable Levy Process, which is a stable distribution that can be Furier transform. It is in overcoming the analytical limitations of the traditional Black-Scholes model that assumes .

We consider counterparty risk in CDS rates. To do so, we use a multivariate jump diffusion process for obligors' default intensity, where jumps (i.e. magnitude of contribution of primary events to default intensities) occur simultaneously and their sizes are dependent. For these simultaneous jumps and their sizes, a homogeneous Poisson process. We apply copula-dependent default intensities of multivariate Cox process to derive the joint Laplace transform that provides us with joint survival/default probability and other relevant joint probabilities. For that purpose, the piecewise deterministic Markov process (PDMP) theory developed in [7] and the martingale methodology in [6] are used. We compute survival/default probability using three copulas, which are Farlie-Gumbel-Morgenstern (FGM), Gaussian and Student-t copulas, with exponential marginal distributions. We then apply the results to calculate CDS rates assuming deterministic rate of interest and recovery rate. We also conduct sensitivity analysis for the CDS rates by changing the relevant parameters and provide their figures.

본 연구는 마르코프체인 몬테카를로 시뮬레이션(MCMC)의 한 방법인 깁스샘플링을 이용하여 주요국 환율 변동의 점프확산과정을 추정하고 환율 변동에 내재된 점프 특성을 비교하였다. 비교의 초점은 점프확산과정의 intensity와 점프의 크기이다. 원/달러, 달러/파운드 등의 환율은 intensity는 작고 점프의 크기는 크게 나타난 반면에 스위스프랑/달러, 달러/유로 등의 환율은 intensity는 크지만 점프의 크기는 작은 편으로 나타났다. 원/달러 환율 변동의 뚜렷한 특징은 점프가 빈번하게 발생한 것은 아니지만 일단 점프가 발생하면 그 규모가 매우 컸다는 점이다. 이 사실은 위험관리에서 중요하게 받아들여져야 한다. 경제주체들은 빈번하게 발생하지 않는 위험에 대해서는 규모가 크다고 하더라도 소홀히 여기는 경향이 있기 때문이다.

The paper aims to estimate the jump-diffusion process of exchange rate movements, in using Gibbs sampling which is a popular method in MCMC. Focusing on intensity and jump size of jump-diffusion process of 15 exchange rates, also we compared jump-diffusion characteristics of major exchange rates. We obtained interesting results, such as that some exchange rates such as KRW/USD, USD/GBP showed small intensities and large jump sizes, but other exchange rates such as CHF/USD, USD/EUR large intensities and small jump sizes. These jump-diffusion characteristics are suggestive to comprehend different risk management behaviors.