An empirical test of the variance gamma option pricing model
Introduction
Since Black and Scholes published their seminal article on option pricing in 1973, there have been lots of theoretical and empirical work on option pricing. One important direction along which the Black–Scholes formula can be modified is to generalize the geometric Brownian motion, which is used as a model for the dynamics of log stock prices. For example, Merton, 1976a, Merton, 1976b and Naik and Lee (1990) propose a jump-diffusion model. Hull and White (1987), Johnson and Shanno (1987), Scott (1987) and Wiggins (1987) suggest a stochastic volatility model. Naik (1993) considers a regime-switching model. Duan (1995) develops an option pricing framework based on the GARCH process. Recently, Madan et al. (1998) use a three-parameter stochastic process, termed the VG process, as an alternative model for capturing the dynamics of log stock prices.
The variance gamma (VG) approach proposed by Madan and Seneta (1990) and Madan and Milne (1991) has the advantages that the additional parameters in the variance gamma process provide control over the skewness and kurtosis of the return distribution, and there is a closed-form representation of the price of a European option. When the skewness parameter is set to 0, the VG model is called a symmetric VG model, abbreviated as SVG. When the skewness parameter allows for nonzero values, we then have an asymmetric VG model, abbreviated as AVG. Empirical tests have been carried out to evaluate the pricing performance of the VG option pricing model. Madan et al. (1998) show that while option pricing errors from the Black–Scholes (BS) model are observed to be correlated with the degree of moneyness and the maturity of the option, the VG model is relatively free of these biases.
In the empirical tests carried out by Madan et al. (1998), prices of the American style S&P 500 futures options were used. Since the VG option pricing model is developed for the pricing of European options, there is an obvious mismatch between the data used and the model that is to be tested. To effectively evaluate the pricing performance of the VG, it is desirable to test the model using prices of European options. This article intends to fill this gap and provide a rigorous test of the VG option pricing model using prices of European options. In this study, prices of Hang Seng Index options, which are European style options, are used to test the VG model.
The data used in this study have several other desirable characteristics that result in a more comprehensive evaluation of the model performance. First, unlike many existing studies, which are based on daily closing prices, model prices as well as observed prices in this study are extracted from tick-by-tick index futures and index options trading data. These data are closely examined and matched so that they are as synchronous as possible. Since the Hang Seng Index futures and option market is very active, using synchronized data still results in a large-sized sample. Our sample covers a 4-year time span and contains more than 100,000 pairs of observed prices and model prices.
Second, because of the enormous size of the data set, we can afford to use more robust methods for the comparisons of model prices and observed prices. A robust statistical test typically has the advantage of being insensitive to outliers, yet it may have the disadvantage of losing test power. However, the potential power loss can be compensated by the large sample size. Hence, the advantages outweigh the disadvantages. In this paper, we find that the use of nonparametric methods offers useful insights into the performance of the various price models.
Third, in comparing the out-of-sample pricing errors of the pricing models, we use both the historical approach and the implied approach to test for the validity of the model. In addition, we follow the approach in Bakshi et al. (1997) and compare the models using the following yardsticks:
- (i)
whether the implied structural parameters are consistent with those implicit in the relevant time-series data. See also Bates (1996);
- (ii)
whether out-of-sample pricing errors are small and whether systematic errors have been eliminated; and
- (iii)
whether hedging errors are small.
When historical data are used to estimate model parameters based on which model prices are produced, the model pricing error is found to be less for the VG models than for the BS model. Between the SVG and the AVG, the latter is found to have less error bias but have a larger standard deviation in error. Nonparametric test shows that the asymmetric VG fares better than the symmetric VG for option pricing. Thus, although the mean absolute error or mean-squared error is less for SVG than for AVG, the reverse is true if some outlying AVG errors are discounted.
As in Madan et al. (1998), we show that under the historical approach, the VG model can iron out some of the systematic biases inherent in the BS model. The results show that while the moneyness and maturity biases are prominent in the BS model, the VG models, either symmetric or asymmetric, are relatively free of these errors.
When model parameters are implied by a number of previously traded option prices, the VG model continues to exhibit systematic biases as in the BS model. Also, under the implied approach, there is no significant evidence that VG model is superior over the BS model in terms of hedging performance.
The outline of the paper is as follows. The VG option pricing model is reviewed in Section 2. The data used and methodology for analysis are described in Section 3. Empirical findings and its analysis are presented in Section 4. Results of regression analysis to identify variable for the systematic bias are presented in Section 5. Section 6 compares hedging performance of the pricing models. Section 7 concludes the paper.
Section snippets
The VG option pricing model
Since Black (1975) reported the systematic pricing error of the BS model, various efforts have exerted either to further investigate the pricing patterns or to propose alternative model specifications. One key factor that affects the pricing of an option is the choice of statistical distribution that governs the underlying asset's return. The BS model assumes that the change in the logarithm of the underlying asset price follows a normal distribution, which is found to be not accurate enough in
Study period
To perform an empirical test of the three-parameter VG option pricing model, data on Hang Seng Index options and futures are obtained from the Hong Kong Futures Exchange (HKFE). As noted by Duan and Zhang (2001), Hang Seng Index options have several unique features that potentially may facilitate a “cleaner” empirical examination of the performance of VG model than S&P index options. In particular, Hang Seng Index options are futures style European derivatives with the maturity date matching
Model comparison using the historical approach
In this paper, the pricing error of an option price is defined as the model price minus the market price, and the percentage pricing error is measured by the pricing error divided by the market price. Panel A in Table 2 reports the mean absolute error and the mean absolute percentage error, the mean squared error, and the mean squared percentage error using the BS model, the symmetric VG model and the asymmetric VG model. The comparison is based on a historical approach with a window size of
Volatility smile
It is quite well known that the BS model exhibits smile effects in that deep out-of-the-money or deep in-the-money options are relatively underpriced. In terms of model errors, deep out-of-the-money or deep in-the-money options tend to exhibit higher errors than the near-the-money options. A more suitable model is one that can be freed from these moneyness smiles. It is thus interesting to see whether the symmetric VG or the asymmetric VG model can iron out these systematic biases of the BS
Hedging performance
In this section, we investigate the dynamic hedging performance using the BS, symmetric VG or asymmetric VG models. Consider the dynamic hedging of a call option using a single-instrument hedge, i.e. the futures contract written on the underlying index. At time t, we shorten a call and undertake the usual dynamic hedging strategy in stocks and bonds to delta hedge it during its life. We then keep track of the tracking error as defined in Hutchinson et al. (1994), which is the worth (in index
Conclusion
In this paper, we test the three-parameter symmetric variance gamma option pricing model and the four-parameter asymmetric variance gamma option pricing model empirically. Prices of the Hang Seng Index call options, which are of European style, are used as the data for the empirical test. Since the VG option pricing model is developed for the pricing of European options, the empirical test gives a more conclusive answer to the applicability of the VG models.
We find that in terms of error
Acknowledgements
The authors would like to acknowledge the support provided by the Hong Kong University Grants Council (Grant number: RGC 2036/97H). The authors would also like to thank an anonymous referee for his helpful comments that has improved the presentation of this paper.
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