## Estimation of volatility models

Although there is no consensus about the forecasting properties of GARCH-type models, there is ample empirical evidence that they can successfully explain the in-sample dynamics of the volatility of financial time series. Pooh and Granger (2003) provide an excellent literature review on different models for forecasting volatility. In this post, we briefly discuss parameter estimation of discrete-time and continuous-time volatility models.

The general approach for GARCH-type models is the method of maximum likelihood. Although the implementation may not be trivial, the idea is simple – express the error terms as a function of the observations and the model parameters. Since, by construction, the error terms are independent and identically distributed, having assumed a particular distribution, it is possible to calculate the log-likelihood function. The parameter estimates are then obtained as a solution to the problem of maximizing the log-likelihood. The standard distributional hypothesis is the Gaussian distribution, but GARCH-type models with other distributional hypotheses are also available (e.g. Student’s t distribution). GARCH toolboxes are available for many modeling environments such as MATLAB, S+ or R.

Parameter estimation of GARCH-type models is straightforward because the likelihood function can be derived in closed form. This is, generally, not possible for stochastic volatility models and their estimation is an involved affair. The main difficulty is due to the fact that the instantaneous volatility is not directly observable and is driven by stochastic dynamics of its own. In the following, we review the basic approaches applied to Heston’s stochastic volatility model which, as mentioned in this post, is interesting because it allows for analytically tractable option pricing framework. Although we consider Heston’s model, the discussion is not limited to it because most of the techniques are developed for more general stochastic processes of which Heston’s model is a special case.

Two general methods for fitting Heston’s model exist – (i) estimate the parameters using only the available historical price data of the risky asset and (ii) extend the historical data with the price series of an option written on the risky asset. The general advantage of (i) is that the data requirements are simpler. However, the instantaneous volatility of the log-return distribution of the risky asset is not observable and, therefore, the parameter estimation problem becomes a statistical filtering problem. In contrast, the advantage of (ii) is that volatility can be extracted from option prices. The usual strategy for the option is to choose an at-the-money European call-option with a one-month time to maturity in order to deal with potential liquidity problems. Another advantage of (ii) is that it provides the only way to estimate the market price of volatility risk which, although not a parameter of Heston’s model, is important in case the fitted model is to be applied subsequently for option pricing.

As far as (i) is concerned, there are several estimation methods discussed in the academic literature. Moment-based techniques are described by Chacko and Viceira (2003). The approach is based on the characteristic function of the logarithm of the price of the risky asset. The parameters of the model are estimated through the generalized methods of moments. A similar technique is discussed by Jiang and Knight (2002). Other methods include approximating the conditional moments of integrated volatility through high-frequency data (see Bollerslev and Zhou (2002)), filtering techniques based on the characteristic function (see Bates (2002)), and a Bayesian approach (see Eraker (2001)).

The approach in (ii) leads to a much more computationally demanding estimation problem. The option price is a function of the model parameter, the market price of volatility risk, and the current level of the instantaneous volatility. Since instantaneous volatility is not observable, we use the option price to gain inferences about it. Computational complexity arises because the option price has to be calculated for any intermediate approximation of the parameters within the estimation algorithm. Although more demanding, this approach allows full identification of all model parameters. Different statistical methods can be used in this context. Aït-Sahalia and Kimmel (2007) apply this technique with the method of maximum likelihood while Eraker (2001) uses a Bayesian approach. A MATLAB toolbox for maximum likelihood estimation of the parameters of different continuous-time processes can be found on Aït-Sahalia’s research page.