Pricing Taiwan option market with GARCH and stochastic volatility

Abstract

This study compares the out-of-sample performances among Black–Scholes (B-S), Stochastic Volatility (SV) and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) models in the Taiwan option market. Using Absolute Relative Pricing Error (ARPE) as the performance criterion, the empirical result reveals that the performance for GARCH is the best, and SV slightly dominates B-S. Additionally, this study performs the regression of ARPE on time-to-maturity, moneyness and a binary variable that is set to unity, if the option is a call and to zero in the case of a put. For the three models, the regression result displays that the pricing error is consistently decreasing in time-to-maturity and moneyness, and the out-of-sample performance in puts are more accurate than those in calls. Since the corresponding R ² of the regression in GARCH is the smallest, the pricing error for the other two models is relatively severe with respect to the three explanatory variables.

Notes

¹ TXO daily average trading contracts were 6316, 87 229, 175 298, 324 277, 390 847, 374 841 and 372 519, respectively, from 2002 to 2008.

² For detailed introduction about B-S model, please see Hull (2006).

³ Since estimating the parameters of SV model is a nonlinear multivariate optimization problem, taking more terms of Taylor series would add the computational difficulty but need not improve the accuracy.

⁴ For detailed introduction about GARCH model, please see Maddala (2005).

⁵ In order to avoid calculating stock dividends, this study replaces stock index with stock futures index. Additionally, the close times for TXO and TX are the same 1:45 pm, so that the nonsynchronous trading problem can be avoided.

⁶ Because of the approaching of the maturity date, investors are under the pressure of the expiration of the options; this will easily lead to unreasonable market prices. In addition, investors in Taiwan prefer short-term investments so that the volumes are very low for options with long time to maturity. These types of options are excluded in order to avoid their impacts on the empirical results. In addition, TX and TXO have different maturity months. Maturity dates for TX are in two consecutive months plus three quarter months; maturity dates for TXO are in three consecutive months plus two quarter months. Hence, options whose maturity dates are in two consecutive months are selected so the corresponding index futures can also be obtained.

⁷ This study applies the MATLAB fmins command to estimate the implied volatility in the B-S model and the MATLAB fmincon command for parameters in the SV and GARCH models. SV and GARCH are nonlinear models with four unknown parameters so they are much harder to apply than the B-S model. For the purpose of accuracy, this study uses the parameters obtained by Lehar et al. (2002) as the initial values for the solutions of the SV and GARTH parameters.

⁸ The reason which causes this inconsistency, first of all, may be the selection of time-to-maturity. This study only selects options within 50 days to maturity while Lehar et al. (2002) had no limitation. Moreover, the inconsistency may be caused by the differences in transaction systems for the FTSE100 and Taiwan stock market.