Are implied volatility levels suitable for forecasting? A study comparing the performance of volatility implied by options and covered warrants with the performance of ARCH forecasts. Hampus Granberg Hampus Granberg Spring 2016 Master Thesis, 15 ECTS Separate course in economics Abstract This study examines which of the implied volatilities from options and covered warrants with exactly the same terms and cash flows that deviates least from the subsequent week’s realized volatility levels. Their suitability as methods for forecasting is also examined by comparing their predictive abilities with the forecasts of ARCH models. The study was performed on options and covered warrants traded in Sweden between February and May 2016. The results indicate that volatility levels implied by covered warrants generally overestimates realized volatility and that neither instrument outperforms the forecasts of ARCH models.1 *I wish to express my gratitude towards Carl Lönnbark for much appreciated guidance during the work of this study. 1 Table of contents 1 Background and purpose ..................................................................................................................... 3 1.2 Earlier research.............................................................................................................................. 3 1.3 Characteristics of a covered warrant ............................................................................................ 5 1.4 Problem ......................................................................................................................................... 6 1.5 Purpose and limitations................................................................................................................. 8 2 Theory ................................................................................................................................................... 9 2.1 Black-Scholes-Merton model for pricing European options ......................................................... 9 2.1.1 Black-Scholes-Merton differential equation .......................................................................... 9 2.1.2 Dividends .............................................................................................................................. 13 2.1.2 Usefulness ............................................................................................................................ 14 2.1.3 Volatility smile and term structure ...................................................................................... 14 3 Data and Methodology ....................................................................................................................... 16 3.1 Data collection ............................................................................................................................. 16 3.1.1 Collection of the data of warrants ....................................................................................... 17 3.1.2 Collection of the data of option contracts ........................................................................... 18 3.1.3 Collection of the data of the underlying stocks ................................................................... 19 3.1.4 Risk-free interest rate and dividend yield ............................................................................ 19 3.2 Deriving the implied volatility ..................................................................................................... 19 3.3 ARCH-modelling and forecasting ................................................................................................. 20 3.3.1 ARMA modelling ................................................................................................................... 22 3.3.2 ARCH modelling .................................................................................................................... 23 3.3.3 Forecasting ............................................................................................................................... 26 3.4 Realized volatility ........................................................................................................................ 26 3.5 Hypotheses .................................................................................................................................. 28 4 Results ................................................................................................................................................ 29 5 Conclusions ......................................................................................................................................... 32 References ............................................................................................................................................. 34 Data sources .......................................................................................................................................... 36 Appendix................................................................................................................................................ 37 Econometric tests .............................................................................................................................. 37 ARCH modelling ................................................................................................................................. 38 2 1 Background and purpose Understanding the nature of the fluctuation that financial securities exhibit has become a large field of research for both academics and market participants. Fluctuation is usually measured as the degree of the variation of a financial time series for a specific time period and often expressed as its ‘volatility’. We start out this section with a short recap of the development of volatility models. 1.2 Earlier research Mandelbrot (1963) was the first to provide evidence of so-called volatility clustering, meaning that large (or small) moves, in a financial time series, tend to be followed by other large (or small) moves. A popular family of models that accounts for this is the autoregressive conditional heteroscedasticity, or ARCH models, originally developed by Engle (1982), in which the GARCH (1,1) model is one of the more widely used today. These models assume a conditional volatility level that is a function of the size of the volatility levels in previous time periods. Later, many researchers (Black 1976; Christie 1982; Nelson 1991, to name a few) found evidence of volatility levels being negatively correlated with stock returns, i.e. negative shocks tend to cause higher volatility than positive. One explanation behind this phenomenon is that the, generally risk averse, market participants’ demand for a particular asset falls when faced with news of increased volatility. The decline in the assets value then causes the volatility to additionally increase. Since the traditional ARCH models does not impose this asymmetry, models that do so were developed. One of the more popular models is the exponential GARCH, or EGARCH model, developed by Nelson (1991). The most common reason to model volatility is that the models can, with varying accuracy, be used to forecast volatility for future time periods. A great number of case studies have been performed in the last 30 years which test the forecasting ability of the different models belonging to the ARCH type family, which today have become quite copious in number. In 2005 a study was performed by Hansen and Lunde, in which 330 ARCH-type models was examined. The models predictive ability of the one-day ahead conditional variance of the daily DM–$ exchange rate and the daily IBM returns was tested in an out-of-sample procedure. The findings showed no evidence of the GARCH (1,1) model being outperformed in the analysis of the exchange rate data. It was however clearly outperformed in the analysis of the stock returns. Another way to acquire estimates of future volatility levels is to look at the implied volatility from various derivative pricing formulas. In the Black-Scholes option pricing model, which will be described thoroughly in the theory section, the only not directly observable factor 3 determining the price of the option is the volatility level of the underlying asset. Hence, the volatility level implied by the market can be derived from an option contract that has been priced. Plenty of case-studies have tested the implied volatility’s ability to forecast actual volatility. Latane and Rendleman (1976) were among the first to derive an implied volatility level from the Black-Scholes model. They performed their study on 39 weekly returns from 24 different companies. They found that generally the implied volatility forecast does correlate with actual volatility to a great extent (0.827). However, they also found that the predictive ability was superior for at-the-money options and that deep in-the-money options close to maturity were almost completely insensitive to volatility movements and hence priced in such a way that their implied volatility had very little ability to forecast correctly. This phenomenon is discussed further in the theory section below. To account for this the researchers suggested creating a weighted implied volatility by combining options with the same underlying asset but with different strikes and times to maturity. Beckers (1981) did perform such a study where different types of weights were used. All weights were based