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일반화극단치 내재확률분포를 이용한 Value at Risk

  • 김무성 부산대학교 경영학부 교수
  • 강태훈 경성대학교 경영학과 초빙외래교수
본 연구는 KOSPI 200 지수옵션시장에서 일반화극단치 내재확률분포(GEV)의 유용성을 검증하기 위하여, 확률분포의 꼬리영역과 관련된 Value at Risk(VaR)의 사후검정성과와 전체 확률분포에서의 예측성과를 분석하였다. 비교를 위한 내재모형으로는 Black and Scholes의 모형과 상태가격밀도의 형태를 two-lognormal mixture로 가정한 모형, 수익률생성과정을 분산-감마과정으로 가정한 모형을 사용하였다. 그리고 시계열모형으로 정규분포와 함께 비대칭과 초과첨도를 고려할 수 있는 GJR- GARCH 모형을 이용하였다. 또한 콜옵션시장과 풋옵션시장에 내재된 정보를 구분하여 이용함으로써, 의사결정의 유용성을 향상시킬 수 있는 가를 고찰하였다. 분석결과, GEV는 다른 내재모형에 비해 과적합될 수 있는 가능성을 낮추면서, 시장가격에 내재된 정보를 보다 정확하게 추론하였다. 그리고 추론된 내재정보를 VaR의 의사결정에 적용할 경우에도, GEV는 다른 내재모형이나 시계열모형보다 전반적으로 더 유용하였다. 특히 외가격 풋옵션가격에 내재된 정보만을 이용할 경우, GEV는 1%부터 5%까지의 VaR 유의수준에서 가장 우수한 성과를 보였다. 그러나 분포의 극단적인 꼬리부분과는 달리 전체영역에서의 실현분포에 대한 GEV의 예측성과는 상대적으로 열등하였다.
일반화극단치 내재확률분포; 불완성시장; 분산-감마과정; Two-Log- normal Mixture 분포; Value at Risk; Generalized Extreme Value Implied Distribution; Value at Risk; Non- Complete Market; Two-Lognormal Mixture Distribution; Variance-Gamma Process

Value at Risk Using Generalized Extreme Value Distribution Implied in the KOSPI 200 Index Options

  • Moo-Sung Kim
  • Tae-Hun Kang
Based on Harrison and Pliska(1981)’s no arbitrage equilibrium theory, Markose and Alentorn(2005) introduced an original analytical closed form solution for generalized extreme value distribution (GEV) in the European options. They showed that the three parameter GEV based risk-neutral density function for asset returns has great flexibility in defining the tail shape implied by traded option price data. They also found that the GEV option pricing model not only accurately captures the negative skewness and higher kurtosis of the implied risk neutral density but also delivers the market implied tail index that governs the tail shape. Hence, the model allows accurate estimation of the risk neutral density function by including extreme values and fat tails. Meanwhile, Ait-sahalia and Lo(2000) and Panigirtzoglou and Skiadopoulos(2004) have argued that Economic VaR, calculated under the option market risk neutral density, is a more relevant measure of risk than historically based VaR. Economic VaR can be seen as a forward looking measure to quantify market sentiment about the future course of financial asset prices whereas historical or statistically based VaR is backward looking based on the historical data. Thus, in this paper adjusting Markose and Alentorn (2005)’s closed form solution to improve the model stability for in-sample-fit and out-of-sampling pricing, we empirically investigate the usefulness of the adjusted GEV model implied in the KOSPI 200 index options prices in terms of effectiveness for value at risk(VaR). As benchmark models, we use two-lognormal mixture model(TLM) and variance-gamma model(VG) as well as Black-Scholes model(BS). Because TLM is the weighted sum of two Black-Scholes solutions, the model can relax the assumption of the lognormal distribution. And variance gamma process is obtained by evaluating the Brownian motion at a random time change given by a gamma process. So variance-gamma model can relax the assumption of the geometric Browian motion. As time-series model for comparative analysis, we use not only normal model but also GJR-GARCH model which can reflect the presence of excessively fatter tails and pronounced skewness, reflecting strong volatility clustering. To estimate parameters for each option pricing model, the non-linear least squares approach is applied to minimize the sum of squared errors between the model and the market prices. As for the back-testing of value at risk, we examine the number of violations, mean violation, and maximum violation. For the statistical testing of the performance, the frequency tests of Christoffersen’s conditional back-testing procedures are examined. The Christoffersen approach enables us to test both coverage and independence hypotheses at the same time. Moreover, if the model fails a test of both hypotheses combined, this approach enables us to test each hypothesis separately so as to establish where the model failure arises. The hypothesis of unconditional coverage means that the expected frequency of observed violations is precisely equal to observed violation. The hypothesis of independence means that if the model of VaR calculation is valid, then violations must be distributed independently. We also wonder whether it is helpful to eliminate the non-tail information, in which case we consider the whole return distribution equality tests by using the Berkowitz methodology. In this way, we aim to investigate whether the estimated densities are equal to the true densities not only in the tails of the density but also in the full range area of the density. Back-testing results of VaR show that, of all the models, GEV yields the least number of violations for the industry standard for 10 day VaR at the high confidence level of 99%. Because the risk-neutrality assumption was strongly rejected by the market data due to the second state variables such as the negative risk premium for the stochastic volatility and jump fear, the preference for moments of higher order than the variance, heterogeneous beliefs and risk preferences and market inefficiency and imperfections, call option prices are not perfectly correlated with put option prices and the information contents of the call option price are different from those of put options market price. So, under the non-complete market and the limited arbitrage, we examine whether the information implied in call(or put) option market price is more useful than that which is inferred from both call and put option market prices. The results show that for the non-complete put option market, the performance of GEV is improved much more incrementally than that of the other models. However, in the entire density forecast evaluation, GEV is not as useful as it is for the performance in the tail portions of the implied distribution.