In support of today’s frequent and continuous decision makings in the financial stock market, we employ Functional Data Analysis (FDA) to study intra-month stock price volatility. FDA considers covariates as functional curves and gives response forecasting as functional trajectories. Based on a Dow Jones Industrial Average (DJIA) dataset, we train a functional autoregressive model and use it to do forecasting. We find that FDA is empirically effective in intra-month volatility trajectory forecasting. Further, the flexibility and information richness of the forecasted functional trajectories are highly valuable, especially for today’s fast-moving markets where high-frequency trading plays an important role.
Modelling and predicting stock price volatility is one of the key research fields for risk management in today’s financial stock market. In support of today’s frequent and continuous decision makings in the market, more sophisticated and informative forecasting techniques are strongly desired.
In our project, we employ Functional Data Analysis (FDA) to study intra-month stock price volatility. FDA considers covariates as functional curves and gives response forecasting as functional trajectories inheriting valuable functional natures including flexibility and information richness. Typically, we can literally obtain the volatility value at any given time point within the domain of the predicted functional trajectory. We can even take derivatives to have some idea about the changes of volatility over time or even the accelerations.
The key idea of FDA is curve smoothing. We observe volatility daily as the absolute logarithmic returns of the day. Then, we smooth daily volatility data points within a month into a smooth functional curve, and we consider this functional curve as the observed volatility function for that particular month. Finally, our target is to predict the volatility function of this month from that of last month. Specifically, we train a functional autoregressive model and use it to do forecasting.
We empirically examine the effectiveness of FDA on intra-month volatility trajectory forecasting. For the experiment, we consider the volatility of the Dow Jones Industrial Average (DJIA) from year 2018 to 2022. The DJIA is a price index representing the general stock price level of the U.S. stock market. We extract daily volatility data points from the DJIA data, and then perform some curve smoothing. We would use the smoothed intra-month volatility curves from the training set (i.e., 2018-2021, visualised in Figure 1) to train the functional autoregressive model, and then use the test set (i.e., 2022) for forecasting validation purposes. Note that the green curve with ultra-high volatility in Figure 1 is related to the stock market crash in 2020 March. A stock market crash is actually informative in a business cycle, so we decide to keep this meaningful piece of data in the training set.
After training, we do a 12-step forecasting based on the trained functional autoregressive model. The forecasting results are visualised in Figure 2. The 12 forecasted intra-month volatility curves for year 2022 are shown in red lines, in contrast to the blue lines obtained by direct curve smoothing with the daily volatility data points (shown in black circles). We can visually see that the forecasted curves and the directly smoothed curves inherit similar patterns and both give reasonable fits to the observed daily volatility data points. The effectiveness of our model is also supported by a functional F-test, which will not be presented in details here due to limited space.
In conclusion, Functional Data Analysis (FDA) provides an insightful view in stock market research and forecasting. The flexibility and information richness of functions are highly valuable, especially for today’s fast-moving markets where high-frequency trading plays an important role.
Runqiu Fei
The University of Melbourne
