In Chapter 3, Time Series Modeling, we looked at various approaches to modeling time series. However, models such as ARIMA (Autoregressive Integrated Moving Average) cannot account for volatility that is not constant over time (heteroskedastic). We have already explained that some transformations (such as log or Box-Cox transformations) can be used to adjust for modest changes in volatility, but we would like to go a step further, and model it.
In this chapter, we focus on conditional heteroskedasticity, which is a phenomenon caused when an increase in volatility is correlated with a further increase in volatility. An example might help to understand this concept. Imagine the price of an asset going down significantly—due to some breaking news related to the company. Such a sudden price drop could trigger certain risk management tools of investment funds, which start selling the stocks as the result of the previous decrease in price. This could result in the price plummeting even further. Conditional heteroskedasticity was also clearly visible in the Investigating stylized facts of asset returns recipe, in which we showed that returns exhibit volatility clustering.
We would like to briefly explain the motivation for this chapter. Volatility is an incredibly important concept in finance. It is synonymous with risk and has many applications in quantitative finance. Firstly, it is used in options pricing, as the Black-Scholes model relies on the volatility of the underlying asset. Secondly, volatility has a significant impact on risk management, where it is used to calculate metrics such as the Value-at-Risk (VaR) of a portfolio, the Sharpe ratio, and many more. Thirdly, volatility is also present in trading, as it can be directly traded in the form of the CBOE Volatility Index (ticker symbol: VIX). The name comes from the Chicago Board Options Exchange, by which the index is calculated in real time.
By the end of the chapter, we will have covered a selection of GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models—both univariate and multivariate—which are one of the most popular ways of modeling volatility. Knowing the basics, it is quite simple to implement even more advanced models. We have already mentioned the importance of volatility in finance. By knowing how to model it, we can use such forecasts to replace the previously used naïve ones in many practical use cases in the fields of risk management or derivatives valuation.
In this chapter, we cover the following recipes:
- Explaining stock returns' volatility with ARCH models
- Explaining stock returns' volatility with GARCH models
- Implementing a CCC-GARCH model for multivariate volatility forecasting
- Forecasting a conditional covariance matrix using DCC-GARCH