초록 열기/닫기 버튼
Option markets are often characterized by the systematic and frequent deviations from the Black-Scholes assumption of constant volatility. In practice, however, Black-Scholes’ implied volatilities tend to depend on exercise prices and the time-to-maturity. Dumas et al. (1998), assuming that volatility is a deterministic factor of the asset price and time, tested a number of arbitrary models based upon a polynomial expansion across exercise prices and the time-to-maturity. Since then, numerous papers have proposed models in which implied volatilities play a quadratic function in the exercise price, the time-to-maturity, and the interaction term. In this article, we propose a model that forecasts the implied volatility for any given exercise price or maturity level in the KOSPI200 option market. We confirm that the variance of the intraday option implied volatility is larger than that of the underlying spot index return. We also find a sneer pattern in the KOSPI200 option market. Therefore, we believe it is important to examine the implied volatility functions on an intraday basis. We include an additional factor in the volatility function, expanding the moneyness term- the exercise price divided by the forward price-to the third degree to capture the sneer pattern among the exercise prices. The sample period is from January 2, 2005 to December 28, 2006. We first adjust an implied volatility model to each cross section of options available at three minute intervals in our sample by ordinary least squares (OLS). For each interval, we estimate the cross-sectional model, where iv is the Black-Scholes implied volatility, is the time-adjusted measure of moneyness, T is the time-to-maturity, ZT and is the interaction term of moneyness and the time-to- maturity. We then model the time variation of volatility function to capture the dynamics of the OLS coefficients in the cross-sectional model. For our analysis, we posit three models in terms of time series of OLS estimates. We consider the VAR (vector autoregressive) model, which completely reflects the autocorrelation of the OLS coefficient. In addition, we include the PBS (Practitioner Black-Scholes) model, where the fitted values for volatility at time t-1 are used as an estimate for time t. We also examine a simple AR(1) model. Our most important findings are summarized as follows. First, the Black-Scholes implied volatilities are different across exercise prices and the time-to-maturity while the sneer patterns are also found in the KOSPI200 option market. Our model, which includes the moneyness term to the third degree to capture the sneer pattern, gives the best goodness-of-fit result among the three models. These results are based on the adjusted and the Schwartz Bayesian Criterion (SBC). The coefficients of the OLS estimates also exhibit high autocorrelation. Second, among the three models, the PBS model show the best performance in terms of the prediction error and profitability of the trading strategy compared with the benchmark models. Third, the performance of the PBS model is the best for the in-the-money option group. The prediction error is lower for the medium-term maturity group than it is for the short-term maturity group. In conclusion, our own model of the intraday volatility function, which captures the nonconstant volatility across exercise prices and maturities, produces more accurate volatility forecasts than the previous models do. Also, we show that the model can be used to make actual trading profits in the KOSPI200 option market.
본 논문은 블랙-숄즈모형의 문제점으로 지적되어온 이분산성(nonconstant volatility)에 대한 연구로서 옵션시장에서 관찰되는 행사가격 및 만기별 내재변동성의 이분산성을 설명할 수 있는 변동성모형을 제시하였다. 결론적으로 일중 변동성이 큰 KOSPI200 옵션 내재변동성의 경우 일중 데이터를 이용하여 행사가격과 잔존만기에 따라 상이한 내재변동성의 변화를 설명할 수 있는 변동성모형의 도출이 가능하고 이를 통해 미래변동성을 추정하여 실제 옵션거래에 적용할 수도 있음을 확인할 수 있었으며 주요 분석결과는 다음과 같다. 첫째, KOSPI200 옵션시장에서도 행사가격과 잔존만기에 따라 변동성이 상이한 sneer 패턴에 가까운 현상이 관찰되었다. 그리고 sneer 패턴을 포착하고자 moneyness 3차항을 포함시킨 변동성모형이 기존연구의 모형보다 훨씬 적합도가 높은 것으로 나타났으며 추정된 변동성모형의 각 계수는 높은 시계열 자기상관 현상을 보여주었다. 둘째, 분석에서 설정된 3가지 검증모형, VAR 모형, AR(1), PBS 모형 들을 비교한 결과 전기에 추정된 모형상의 계수를 이용하여 추정한 내재변동성을 금기의 내재변동성으로 적용하는 PBS 모형이 미래변동성에 대한 예측오차와 거래전략의 수익성 등 부문에서 시계열 자기상관을 전부 모형에 반영한 VAR 모형과 AR(1)모형보다 우수함이 검증되었다. 세째, 옵션 moneyness 그룹별로는 예측력이나 수익성 면에서 그룹별로 차이가 있지만 전반적으로 ITM 옵션의 결과가 타 그룹에 비해 우수함이 확인되었다. 또한 옵션 잔여만기별로는 잔여만기 30일 초과 옵션이 만기가 짧은 최근월물보다 작은 예측오차를 보여주었다.
키워드열기/닫기 버튼
KOSPI200 옵션, 내재변동성, 변동성모형, 변동성 예측, 이분산성
