Calibrated GARCH models and exotic options

Abstract

This article examines the differences between various Generalized Autoregressive Conditional Heteroscedastics (GARCH) models in pricing exotic options, given that the models have been calibrated on the same data sets using information on both returns and plain vanilla options. We focused on four widely recognized specifications: the Heston–Nandi (HN), Leverage, News and Power models, of which the first is an affine model, and the others represent the family of nonaffine models. First, we found that when the models were calibrated using option data, the previously reported superiority of nonaffine models over the HN in option pricing may not be generally true. On the other hand, the HN, Leverage and News models priced various exotic options quite similarly; but contrary to the others, the Power model yielded somewhat abnormal prices, especially for barrier options. Using the Maximum Likelihood Estimation (MLE) approach with return data, however, yielded different conclusions. Especially with long in-sample periods on stock returns, the nonaffine characterizations may outperform the HN model in terms of both the likelihood value and the European option pricing error, and in this case, the nonaffine models also price exotics differently than the HN model. We also demonstrated that different estimation approaches can affect exotic prices substantially.

Keywords:

• GARCH
• exotic options
• stochastic volatility
• calibration
• option pricing

JEL Classification::

• G13
• C00
• C51
• C52
• C58

Notes

¹Note that this estimation method assumes implicitly that option markets price plain vanilla options correctly.

²To speed up computation, we used distributed computing with empirical martingale methods (Duan and Simonato, 1998) and antithetic variables.

³Based on options data with the NLS, the RMSEs for the in-sample (Sample A) must, by definition, be lower than the RMSEs based on return data with the MLE for the same option sample.