Time-varying volatility models and indices : a GARCH option pricing approach

Abstract

In this thesis, the generalised autoregressive conditional heteroskedasticity (GARCH) option pricing model is applied to illiquid markets, volatility indices and in a modern derivative pricing framework. Chapter 2 provides empirical support for the use of a volatility index to obtain a more accurate GARCH option pricing model (applied to the South African equity market). In Chapter 3, the analysis (GARCH option pricing and volatility indices) is extended to FX markets. Empirical results show that asymmetry is an important factor to consider when modelling FX volatility indices.

The aim of Chapter 4 is to quantify the effect of asymmetry in the cryptocurrency market. Furthermore, the accuracy of the GARCH option pricing model applied to cryptocurrencies is also considered. Results indicate that the GARCH option pricing model produces reasonable price discovery, and that asymmetric effects are not significant when pricing cryptocurrency options. Chapter 5 focuses on the construction of a cryptocurrency volatility index, the models in Chapter 4 are used as a basis. The term structure of the GARCH generated volatility indices are consistent with expectations. Furthermore, short term volatility tends to increase when large jumps occur in the underlying asset.

In Chapter 6, the Heston–Nandi futures option pricing model is applied to Bitcoin (BTC) futures options. The model prices are compared to market prices to give an indication of the pricing performance. In addition, a multivariate Bitcoin futures option pricing methodology based on a multivatiate GARCH model is developed. The empirical results show that a symmetric model is a better fit when applied to Bitcoin futures returns, and also produces more accurate option prices compared to market prices for two out of three expiry dates considered.

Chapter 7 focuses on the pricing of volatility index options respectively. In Chapter 7, the GARCH option pricing model is applied to the Standard and Poor's 500 (S&P500) Volatility Index (VIX) option pricing. The different GARCH models are fitted to VIX futures returns. The results show that the symmetric GARCH model with skewed Student-t errors is the best performing model, and that the GARCH option pricing model provides reasonable price discovery when applied to the VIX.

In Chapter 8, the standard Black model and Heston-Nandi futures options pricing model are applied to the hedging of VIX futures options. The hedge performance is compared based on the stability of the profit and loss distribution (P&L) of the hedged portfolio. Empirical results show that the Heston-Nandi futures option pricing model is more reliable when applied to hedging of VIX futures options.

The focus of Chapter 9 is the application of the GARCH model to the pricing of collateralised options in the South African equity market. Symmetric GARCH and nonlinear asymmetric GARCH (AGARCH) models are considered. The models are used to price fully collateralised and zero collateral options (European, Asian, and lookback options). The effect of collateral is illustrated by the difference between zero collateral and fully collateralised option price surfaces. Finally, the effect of asymmetry is shown by the difference between the symmetric and asymmetric GARCH option price surfaces.

Finally, a closed-form expression for a collateralised European option in the presence of counterparty credit risk and stochastic volatility is derived in Chapter 10. The model is applied to S&P500 index options. The option prices obtained are consistent with expectations, default risky options are cheaper than options with no counterparty credit risk, and fully collateralised options are more expensive when compared to zero collateral options. The effect of correlation is tested by plotting the default risky at-the-money (ATM) option price for different levels of correlation. The results indicate that correlation has an insignificant impact when pricing using the calibrated parameters.