Empirical Analysis of GARCH models and their performance in pricing options in comparison to the Black Scholes Model

There are two dimensions to this paper. The first part aims at investigating two heteroscedastic models, namely the GARCH (1, 1) and its variant called the EGARCH (1, 1). The two models exhibit all the important features displayed by time series in the real-world, particularly satisfying the four most important properties namely lepto-kurtosis, volatility-clustering, mean-reversion, autocorrelation. The EGARCH (1, 1) successfully captures the leverage effect. This analysis is done on the time-series of daily closing price of S&P 500 index for the period 4th April 2005 to 4th April 2007. The more recent months are avoided due to the financial turmoil caused by the credit crunch 2007-08. The performance of the GARCH (1, 1) and EGARCH (1, 1) is further investigated and compared by applying the conditional variances obtained from them to option valuation. This is where the second dimension comes into picture. The second dimension focusing on GARCH(1,1), EGARCH(1,1) tends to diversify away to compare not only the two models but also the underlying option pricing models viz, Duan's(1995) GARCH option pricing model and the traditional Black Scholes. In order to accomplish this analysis, historical call options on S&P 500 index for the period 4th April 2005 to 4th April 2006 are considered. The two performance measures, RMSE and MAPE do not categorically declare any model to be the most superior. The pricing efficiency of models varies across maturity, strikes and error measurements. Overall RMSE value for EGARCH is the lowest whereas MAPE is the lowest for Black-scholes. Between GARCH and EGARCH, EGARCH outperforms GARCH for both the measures, though not significantly. RMSE shows EGARCH to outperform both Black-Scholes and GARCH (1, 1) model for short maturities and GARCH (1, 1) to perform the best in case of long maturities. Thus, the GARCH pricing out wins the Blacks-Scholes. But under MAPE, Black-Scholes outperforms both GARCH (1, 1) and EGARCH (1, 1) and thus the GARCH pricing model. Contrary to popular findings, black-scholes prices under constant volatility assumption do not differ significantly from market observed prices. But, the Black-Scholes model greatly overprices the ITM. Between GARCH (1, 1) and EGARCH (1, 1), it is difficult to point out categorically the superiority of one model as their efficiencies vary across moneyness (ATM, OTM, ITM) and maturity. Further from here, we delve into the analysis and investigate our aim empirically