Volatility in spot and derivatives market are measured and compared in two different ways. The first approach is standard deviation approach which is a time invariant measure of volatility. The second approach deals with time variant and conditional volatility models like ARCH class of models.
In the first stage, standard deviation along with other descriptive statistical measures of daily index and stock returns in spot and futures markets are calculated and compared among the two markets. The descriptive statistical measures for ten underlying blue chip stocks are calculated within a sample period starting from 1st January 2002 till December 2010. The stock futures returns for near month or one month contracts are only taken into consideration.
The second stage deals with the modeling of time variant conditional volatility of index and stock returns both in spot and futures markets. The modeling of conditional variance or volatility of different security returns in spot and futures markets are made through the utilization of ARCH (1) and GARCH (1,1) class of models.
Auto Regressive Moving Average (ARMA) class of stochastic models are very popularly used to describe time series data. ARMA models are used to model conditional expectation of current observation Yt of a process,. Given the past information such that
Yt = f (Yt-1, Yt-2,———-) + st, where ‘et’ is a white noise component and Var (et) = a2
Under standard assumptions, the conditional mean is considered to be non constant while conditional variance is a constant factor and the conditional distribution is also assumed to be normal. However, in some situations, the basic assumption of constant conditional variance may not be true. For example, consider the markets are experiencing high volatility, then tomorrow’s returns is also expected to exhibit high degree of volatility. If we model such stock return data using ARMA approach, we can not capture the behavior of time variant conditional variance in the model. This behavior is normally referred to as heteroskedasticity, i.e. unequal variance. A time series is said to be heteroskedastic if its variance changes over time. On the other hand, time series with time invariant variance is known as homeskedastic.
Engle in the year 1982 had developed a model called Auto Regressive Conditional Heteroskedasticity (ARCH) wherein today’s expected volatility is assumed to be dependent upon the squared forecast errors of past days. In other words, in a linear ARCH (p) model, the time varying conditional variance is postulated to be a linear function of the past ‘p’ squared innovations or residuals. The time series data relating to the return of a specific asset can be modeled as an Auto Regressive (AR) process where the forecast errors (et) could be assumed to be conditionally normally distributed with zero mean and variance ht2.
Therefore, the conditional mean equation, following an AR (p) process would be,
In the above equation, ‘et’ is conditional upon a set of lagged information and also assumed to follow normal distribution with zero mean and time variant variance ht2. Now, according to above process, the residuals ‘et’ following an ARCH (p) model, could be used to form the variance equation such that
depends upon the past squared residuals of the return series. But the question is what would be the accurate no. of parameters, i.e squared residuals. In empirical applications, it is often difficult to estimate models with large no. of parameters, say ARCH (p). Therefore, to circumvent this problem, Bollerslev (1986) proposed the Generalised Auto Regressive Conditional Heteroskedasticity or GARCH (p,q) model.
Wherein, ‘P’ and ‘y’ in the above mentioned equation represent Recent News Coefficient and Old News coefficient respectively. If the value of ‘у’ equals zero, then the GARCH (p,q) process will be converted into ARCH (p) process. Though there may be different specifications for ‘p’ and ‘q’, but it has been commonly observed that it can very nicely capture the heteroskedasticity in the asset return series. The summation of two types of coefficients, i.e. ‘P’ and ‘y’ can be used to get the overall conditional volatility and also can reveal volatility persistence for the next period. Apart from these, the coefficients of conditional variance equation in a GARCH (1,1) framework, can be used to calculate the value of unconditional variance as shown below.
Pq/(1- Pi- Yi) (4)
After estimating the asset return volatility by applying ARCH and GARCH class of conditional volatility models, the next task would be to test the forecasting power of different models. In other words, a good model should forecast or predict the future volatility up to a maximum accuracy level. The lesser the difference between the actual volatility and the forecast volatility, the stronger is the forecasting power of that model. The process of volatility forecasting can be done in two different ways; dynamic forecasting and static forecasting. In case of dynamic forecasting, forecasting is made by considering all the values in the series, whereas in static forecasting, estimation and forecasting is done though within a sample, but on different set of observations. Since dynamic forecasting considers all the values in the series, so we have tested the forecasting power using dynamic forecasting only.
In all the above statistics, ‘n’ represents the no. of observations used for forecasting, ‘o t’ and ‘ot’ respectively represent the forecasted volatility and the actual volatility. Therefore, any volatility model with the least value of any or all of the above errors is treated to possess the superior forecasting power.