Comparison of Performances of XARCH and SVM family of Models

Student Name: K.P.Abraham

Volatility of returns of security prices has a direct impact and relevance on applications in derivative pricing, portfolio optimization, interest rate modeling etc. Thus accurate estimation of volatility is vital to most applications in the area of quantitative finance. Several approaches have been suggested in the literature for volatility estimation, such as historical volatility models, implied volatility models and stochastic volatility models. The existence of stylized empirical facts of the concerned financial time series makes this a particularly complex problem.

In this thesis, comparison would be made among performances of historical volatility models belonging to the XARCH family (here X is a place holder, which could be blank, indicating the ARCH or AutoRegressive Conditional Heteroscedasticity model, which is the expansion of ARCH for other XARCH’s as well; G for Generalized, EG for Exponential Generalized, T for Threshold etc.) and the Stochastic Volatility Models (SVM). Although volatility estimation itself is an important activity in terms of understanding “risk” of an underlying security, but as mentioned above, these volatility estimates have further use in financial management. In this thesis, apart from the usual comparison of out of sample performances, comparison would also be made in terms of two particular fields of application, namely derivative pricing and portfolio optimization.

Towards the derivative pricing comparison, first volatility of the S&P CNX Nifty index of the National Stock Exchange of India (NSE), has been modeled using eleven univariate models belonging to the XARCH family and the SVM, for the period from 2 Jan 2003 to 9 Jan 2008. Out of sample forecasts are used to compare the performance of the different models. Next, the implied volatility series, which is computed assuming the Black-Scholes formula of option prices, are compared with the twelve estimated volatility series, to comment on the optimality of option prices.

Keeping an eye towards the portfolio optimization problem, select indices of NSE representing different sectors are chosen to form a portfolio. Returns of these indices will then be modeled using eight multivariate XARCH models and three multivariate SVM. As before, first out of sample forecasts will be compared to judge the performance of the different models.

In the next step, the eleven different multivariate models built above will be used to estimate the mean vector and unconditional covariance matrix of the returns of the constituents of the above mentioned portfolio. The portfolio allocation will be done following the traditional Markowitz mean variance approach which uses quadratic programming. Thus there will be eleven different portfolio allocations, one each for the multivariate models. Out of sample performance of these portfolios will then be observed, in terms of which ones yield higher returns, to arrive at an optimal recommendation.