The Function of Intraday Implied Volatility in the KOSPI200 Options (The Function of Intraday Implied Volatility in the KOSPI200 Options)

서지정보

· 발행기관 : 한국증권학회
· 수록지 정보 : Asia-Pacific Journal of Financial Studies / 37권 / 5호 / 913 ~ 948페이지
· 저자명 : 이재하, 권상수

초록

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.

영어초록

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.

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