Volatility Forecasts (Part 1 - STES Model)
Updates
(Code snippets included in this post can be found in the Github repository.) I have updated the data used in this post to the end of 2023 to compare the results with the new model in the next post.
Table of Contents
Introduction
The exponential smoothing (ES) model is a popular yet simple volatility forecasting method in finance and economics. It is formulated as
\[\begin{equation}\label{eq:expsmooth} \hat{\sigma}_t^2 = \alpha r_{t-1}^2 + (1-\alpha)\hat{\sigma}_{t-1}^2 \end{equation}\]where \(\hat{\sigma}_t\) is the estimated volatility at time \(t\) and \(r_t\) is the asset log returns at time \(t\). It is an exponential-weighted moving average of the squared log returns \(r_t^2\). While technically it is the second non-central moment of the log-return distribution, in finance, it is often treated as a variance estimate, assuming either \(\mathbb{E}\left[r_t\right] = 0\) or is difficult to estimate precisely.
Exponential-weighted moving average (EWMA) is a smoothing and time-series forecasting technique used in many fields, and within finance, its application goes beyond volatility forecasting. Treated as smoother, the parameter \(\alpha\) controls the degree of smoothing. Treated as a moving average, it controls the weight of the most recent observation. If the most recent observation is deemed more important, it is given higher weight.
Smooth Transition Exponential Smoothing
I came across a paper on Smooth Transition Exponential Smoothing (STES) (Taylor, 2004) a few years ago. It started with the empirical observation that past return shocks exhibit an asymmetric relationship with future realized volatility. Therefore, one should assign more different weights to the recent observation depending on the characteristics of the recent shock. The STES uses past asset returns to help determine the value of \(\alpha\). Specifically, STES is formulated as
\[\begin{equation}\label{eq:stexpsmooth} \begin{aligned} \alpha_t &= \frac{1}{1+\exp\left(X_t \beta\right)} \\ \hat{\sigma}_t^2 &= \alpha_{t-1} r_{t-1}^2 + (1-\alpha_{t-1})\hat{\sigma}_{t-1}^2 \end{aligned} \end{equation}\]In this formulation, \(\alpha_t\) is no longer a constant as in the ES model but is determined by the set of transition variables \(X_t\), which can include a constant term and \(\lvert r_t \rvert\) or \(r_t^2\) that measures the magnitude of the most recent return, helping in accounting for volatility persistence and sudden price movements. It can also contain \(r_t\) which introduces leverage effects into the model. The author demonstrates that STES performed competitively against ES and other GARCH models in terms of 1-step forecast error for major equity indices. These variables collectively enhance the STES model’s ability to forecast volatility by considering different aspects of historical return data, including magnitude, nonlinearity, and direction, leading to improved predictive power.
Not being on a vol trade desk, my research interest was not forecasting realized volatility. I wanted to tackle a different but related problem: quantitatively selecting the \(\alpha\) parameter of an exponential smoothing model. The technique used in this paper not only addresses this problem in the context of vol forecasting but also resonates with some other work I have seen before at work. In the end, I chose another method more suited to my specific problems, but this paper left an impression on me. Years later, in the present day, I saw its follow-up paper (Liu, Taylor, Choo 2020) that extended the model to test if past trading volume helps forecast realized volatility and analyze the robustness of the STES model to outliers. I decided to replicate their results and extend their model with other variables.
Results
STES vs ES on Simulated Returns
First, let us replicate results from the (Liu et al. 2020) paper. They horserace a set of volatility forecast models that include the ES and STES models by fitting their parameters on a training sample and comparing their test-sample metrics that include RMSE, among others. They do this first on a simulated GARCH time series contaminated by extreme outliers. They found that the STES can already slightly outperform the ES model and better handle outliers on the simulated data. My result is listed in Table 1, along with their results. While I cannot reproduce their numbers, I actually do not think there is a reason why STES should outperform the ES model on the simulated data, as the ground truth is a constant parameter GARCH model with some added outliers. I follow the author’s model naming convention: STES-AE means only the absolute return (AE) is used as the transition variable. Similarly, STES-E&AE&SE contains both returns (E), absolute returns (AE) and squared returns (SE) as transition variables.
| Model | RMSE | (Liu et al 2020) |
|---|---|---|
| STES-AE | 2.85 | 2.43 |
| STES-SE | 2.88 | 2.44 |
| ES | 2.82 | 2.45 |
[Table 1: Comparison of the STES and ES models on simulated data (\(\eta = 4\)).]
STES vs ES on SPY Returns
When fitting the model to the SPY returns, STES has the potential to outperform the simple ES model. Table 2 shows the out-of-sample RMSE of the STES model on SPY’s realized variance and confirms my prior belief.
| Model | Test RMSE | Train RMSE |
|---|---|---|
| STES-E&AE&SE | 4.48e-04 | 4.98e-04 |
| STES-AE&SE | 4.49e-04 | 4.96e-04 |
| STES-E&SE | 4.52e-04 | 4.95e-04 |
| STES-E&AE | 4.54e-04 | 4.93e-04 |
| ES | 4.64e-04 | 4.99e-04 |
[Table 2: Comparison of the STES and ES models on SPY returns. Since my sample and data differ from the authors, their results are not listed. Train sample: 2000-01-01 - 2015-11-26, Test sample: 2015-11-27 - 2023-12-31]
While I cannot exactly replicate the results in Liu et al. (2020) on the simulated time series, STES performs better on the SPY’s data in terms of the out-of-sample RMSE, consistent with the author’s calculations on different data sets. However, the gains seem small, not an order of magnitude better, and its net-of-cost performance as a trading strategy needs to be tested. I will be curious to test this forecast as a volatile trading strategy.
Other Observations
The author observes that STES is not a statistical model, so they cannot conduct significance tests on the parameters. In both Taylor (2004) and Liu et al. (2020), they draw interesting observations about the parameters of the fitted STES model. In particular, the fitted parameters imply that the realized volatility responds to past shocks differently depending on the sign and magnitude of the shock. Moreover, in terms of robustness to outliers, the STES also responds better by downweighting the weights on outliers. In the next post, we’ll discuss an extension of the STES model to use tree-ensembles to improve its performance.