# Realised volatility models

Another important class of models in volatility modeling is the realised volatility models. This class of models forms an assessment of variation in returns of a time-series by utilising its historical returns within a defined time horizon. The estimation of realised volatility is easy as it can be approximated by computing the sum of squared returns of the time-series of interest over the time horizon of interest:

Assuming that returns (r) are independent and with a zero mean, *RV* is an unbiased estimator of the true volatility (Andersen et al., 2003). For instance, the RV for a particular month can be approximated by the sum of squared daily returns over the trading days of the month (typically involving 21 to 22 trading days). In this way, RV forms a measure of price variation (a measure of volatility) over this particular month.

As discussed earlier in this chapter, it may be of interest to compute the annualised realised volatility to enable comparisons across different time horizons of interest. Then, the annualised realised volatility can be computed as the period-specific realised volatility (RV) multiplied by a suitable constant to yield the annualised volatility'. For example, if the realised variance is computed for one month (RV_{momh]y}) as the sum of squared daily returns over that particular month, then the annualised realised volatility can be obtained by^{; R}*Kn»u,i =' ^{2}*^{RV}„_{m}„i,iy*

Industry practitioners, for example financial analysts, often use high-frequency intraday data to determine measures of volatility at hourly, daily, weekly or monthly frequencies. For example, the daily realised variance (RV^ ) is computed as the sum of squared intraday returns for a particular day. The realised volatility is useful and widely used because it provides a relatively accurate and model-free measure of volatility which can be used as input for volatility forecasting and forecast evaluation.

*Figure 14.9* Realised volatility estimates for spot freight rates of different sizes of vessels in the tanker market

# The class of ARCH and GARCH models

The GARCH family of models consists of one of the most widely used classes of volatility models in the financial and shipping literature. This family of models and its extensions originate from Engles (1982) seminal paper where he introduced the concept of the conditional variance of a time-series by including in the variance equation autoregressive errors of the square of lagged error-terms (shocks) — the ARCH model. After the introduction of the original ARCH model, numerous extensions of it have been developed. For example, Bollerslev et ah, 1992 provide a review of extensions and applications of the GARCH-type model.

## Introduction to ARCH and GARCH models

One of the most challenged assumptions of the Classical Linear Regression Model (CLRM) is that the residuals of the model, e, need to be independent and identically distributed (i.i.d.) with mean zero and constant variance in order for the regression to yield efficient estimates. The CLRM can be defined as:

where, r stands for returns of a variable, *x _{u}* to

*x*are the explanatory variables used, £

_{kt}_{(}is the estimated residual series and ot

_{()}, oq, to Oq, are the estimated coefficients of the model. Following the OLS method we make a number of assumptions to make sure the model will yield Best Linear Unbiased Estimators (BLUE) of the true values of the regression equations coefficients. One of the most challenged assumptions under which the OLS estimator yield BLUE estimates is the assumption that the variance (a

^{2}) of the residuals (e

_{(}) is constant over time, the so-called homoskedasticity. However, in practice the variance of the residuals is time-varying giving rise to the phenomenon of heteroskedasticity, which results in non-BLUE coefficient estimates. Thus, instead of using

*o,.*The ARCH model introduced by Engle (1982) provides an autoregressive-type variance equation for

*a,*which includes lagged squared error-terms as explanatory variables as follows:

where, *(3 _{Q}* and /?.,

*i*=1 are parameters to be estimated. The statistical significance of the parameters (З

_{й}to

*/3.*would indicate significant autoregressive behavior in the volatility of the errors and in this way suggest the existence of heteroskedasticity in the error-term, i.e. ARCH effect would be present. In order for the conditional variance to be positive and stationary the following conditions should be met:

*(3*0 and 0 < £/3. < 1.

_{g}>The Generalised ARCH (GARCH) model was developed by Bollerslev (1986) and allows the conditional variance to be dependent also upon previous own lags. In this way, the conditional variance equation becomes:

*Figure 14.10* GARCH(l.l) conditional variance estimates for spot freight rates of different sizes of vessels in the tanker market

where, /3, * _{i}* and

*P*

_{2}.,

^{are}the parameters to be estimated. If /3

_{2}, is statistically significant then the current value of the conditional variance depends on its own lagged values. Nonnegativity constraints apply in the specification of the GARCH model, where /3 > 0 and /3,, ,/3

_{2}^ >0. Also, the summation of the estimated parameters must be less than 1 (A, + Aj < lj in order for the unconditional variance to be: defined, stationary and non-explosive.

The number of lagged error-terms *(p)* can be different to the number of lagged variance terms (*q*), therefore the models are typically denoted as ARCH(p) and GARCH(p,ij). It has been shown empirically that a GARCH(1,1) benchmark model performs very well in capturing the stylised facts of volatility in most financial and shipping time-series variables. The estimation of GARCH-type models typically relies on the Maximum Likelihood (ML) method due to its power and simplicity. Specifically, the ML method uses mathematical optimisation techniques to maximise the log-likelihood function with respect to the parameters of the model. In this way, when evaluating different numbers of lags to be included in a GARCH model or different specifications that capture properties of the data (e.g. asymmetric shocks) the ML log-likelihood function forms one strong statistical criterion for the choice of the optimal GARCH model.