Classic Econometric Models. Part 2: Volatility and Probability

A closer look at modelling the market uncertainty.

In this part of the series, I'll focus on models designed to capture volatility and probability distributions in financial markets. Our last article examined returns, but the strength of econometric models lies in their ability to model volatility (conditional standard deviation or conditional variance). This is proven by empirical evidence that suggests the heteroskedastic nature of volatility, showing its dependency on past observations.

Here, I will present some of the most commonly used models for volatility and probability forecasting, aiming for clarity and straightforward explanation.

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Volatility Models

GARCH (Generalized Autoregressive Conditional Heteroskedasticity) - Introduced by Bollerslev (1986) and Taylor (1986) as an extension of Engle's ARCH model (Engle, 1982), GARCH serves as a one of the most fundamental models in the field of financial econometrics. It captures the dynamic nature of financial markets by modeling conditional variance as a function of past returns and past conditional variances. Its various generalizations, such as EGARCH, GJR GARCH, and APARCH, offer more detailed ways to analyze volatility dynamics, taking into account factors such as leverage effects and asymmetry of returns.

SV (Stochastic Volatility) - In contrast to GARCH, which treats volatility as a deterministic function of past observations, SV models introduce an additional stochastic component to represent random volatility shocks (Clark, 1973; Taylor, 1986). This added complexity makes SV models more flexible in fitting the data but also introduces computational challenges in parameter estimation.

OLHC (Open, Low, High, Close Estimators) - These range-based estimators assess volatility by analyzing price movements during opening, closing, low, and high intervals (Tsay, 2010). Examples include Parkinson’s, Rogers-Satchel, Garman-Klass, and Yang-Zhang specifications, which account for the gap between the opening price of the current time period (day) and the closing price of the previous period (day), for both zero and non-zero drift values.

Extensions and Multivariate Models - Both GARCH and SV can be extended into multidimensional models, known as MGARCH and MSV, capable of handling multivariate time series. Examples such as VECH, BEKK, and DCC GARCH offer different ways to model the covariance structure among multiple financial instruments. Additionally, hybrid models such as MSV-MGARCH combine elements of both approaches. For comprehensive review of multivariate GARCH models see Silvennoinen & Teräsvirta (2009).

Implied volatility - An alternative method for estimating volatility involves calculating it from options prices, which provides the standard deviation of the underlying instrument. This approach is known as implied volatility. One of the most widely used models in this context is the Black-Scholes model (Black & Scholes, 1973), originally designed for European options. Other implied volatility models, such as the Heston (1993) or SABR (Hagan et al., 2002) models are also worth further exploration.

Markov Switching Models - this family of models employ state-dependent parameters to capture abrupt changes in volatility regimes (or states), such as shifts from periods of low volatility to high volatility or from bullish to bearish trends (Goldfeld & Quandt, 1973;  Hamilton 1989). Markov Switching Models offer a powerful way to capture the nonlinearity and time-varying nature of financial data, making them particularly valuable for risk management. They are commonly estimated using Bayesian techniques such as Markov Chain Monte Carlo (MCMC, see Tsay, 2010).

Probability models

As opposed to point forecasts described earlier, contemporary financial forecasting has seen the rise of models that allow for the prediction of entire probability distributions. Below are some examples in this category.

GARCH for Probability Distributions - GARCH models can be adapted to forecast the complete probability distribution of financial returns. For example, with the Student’s t-distribution, we not only forecast the mean and variance but also estimate the degrees of freedom, which gives a fuller picture of the distribution's tail behavior. This approach can also be adapted to accommodate other types of assumed distributions (see Wilhelmsson, 2006; Curto et al., 2009; Trottier & Ardia, 2016).

Copula Models - These models generate joint probability distributions by capturing the dependency structure between disparate financial assets. They are especially useful for risk assessment in portfolio management, as they can accurately describe the joint behavior of asset returns during periods of market stress.  The flexibility of copulas in selecting appropriate copula families, such as Gaussian, t-copula, Clayton, or Gumbel, makes them great tools for researchers trying to understand relationships in financial markets (see Patton, 2009, 2012; Dewick & Liu, 2022). 

Bayesian Models - Bayesian methods allow for the integration of prior knowledge and the quantification of uncertainty in a systematic manner. In the context of financial forecasting, Bayesian models provide a comprehensive framework for estimating and predicting probability distributions. They also allow for parameters to be modeled as probability distributions, which results in a more realistic representation of market dynamics. The application of Bayesian statistics in finance spans a wide range of models, from Bayesian regression to Bayesian neural networks (BNN) and structural time series (BSTS) models. For more about bayesian modeling of financial time series see (Tsay, 2010; Bauwens et al., 2000; Tsay, 2010; Osiewalski & Pajor, 2010 ) 

Summary

In summary, models designed to capture volatility and probability distributions provide essential tools for both traders and risk managers. The variety of models discussed here, each with its own set of advantages and limitations, provide a starting point and are worth further exploration for various financial applications. I'll go into more detail about some of these models in future editions of the newsletter, so stay tuned!

References

Bauwens, L., Lubrano, M., & Richard, J. F. (2000). Bayesian inference in dynamic econometric models. OUP Oxford. 

Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637–654.

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307–327.

Clark, P. K. (1973). A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices. Econometrica, 41(1), 135–155. 

Curto, J. D., Pinto, J. C., & Tavares, G. N. (2009). Modeling stock markets’ volatility using GARCH models with Normal, Student’s t and stable Paretian distributions. Statistical Papers, 50(2), 311–321. 

Engle, R. F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica, 50(4), 987–1007.

Dewick, P. R., & Liu, S. (2022). Copula Modelling to Analyse Financial Data. Journal of Risk and Financial Management, 15(3), 104. 

Goldfeld, S. M., & Quandt, R. E. (1973). A Markov model for switching regressions. Journal of Econometrics, 1(1), 3–16.

Hagan, P., Kumar, D., Lesniewski, A., & Woodward, D. (2002). Managing Smile Risk. Wilmott Magazine, 1, 84–108.

Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 57(2), 357–384.

Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies, 6(2), 327–343.

Patton, A. J. (2009). Copula–Based Models for Financial Time Series. In T. Mikosch, J.-P. Kreiß, R. A. Davis, & T. G. Andersen (Eds.), Handbook of Financial Time Series (pp. 767–785). Springer.

Osiewalski, J., & Pajor, A. (2010). Bayesian Value-at-Risk for a Portfolio: Multi- and Univariate Approaches Using MSF-SBEKK Models. Central European Journal of Economic Modelling and Econometrics, 2(4), 253–277. 

Patton, A. J. (2012). A review of copula models for economic time series. Journal of Multivariate Analysis, 110, 4–18.

Silvennoinen, A., & Teräsvirta, T. (2009). Multivariate GARCH Models. In T. Mikosch, J.-P. Kreiß, R. A. Davis, & T. G. Andersen (Eds.), Handbook of Financial Time Series (pp. 201–229). Springer.

Taylor, S. (1986). Modelling Financial Time Series. Wiley.

Teräsvirta, T. (2009). An Introduction to Univariate GARCH Models. In T. Mikosch, J.-P. Kreiß, R. A. Davis, & T. G. Andersen (Eds.), Handbook of Financial Time Series (pp. 17–42). Springer.

Trottier, D.-A., & Ardia, D. (2016). Moments of standardized Fernandez–Steel skewed distributions: Applications to the estimation of GARCH-type models. Finance Research Letters, 18, 311–316.

Tsay, R. S. (2010). Analysis of Financial Time Series. John Wiley & Sons.

Wilhelmsson, A. (2006). Garch forecasting performance under different distribution assumptions. Journal of Forecasting, 25(8), 561–578.