Volatility Modeling and Forecasting in Financial Markets

Options market pricing depends on implied volatility and is priced based on how the volatility forecast would look in the future. This story focuses primarily on predicting the volatility with time series models in the S&P500 index based on historical data.

The Options market is an attractive source of investment for people across the globe and has been substantially growing at a double-digit pace every year from 2003 through 2008. In 2022, despite a brutal bear market in stocks, total U.S. options volume reached 10.32 billion contracts, up 4.6% vs. 2021.

Various option pricing models are being used in predicting the price of the options. One commonly used model is the Black Scholes model. Implied volatility (IV) is the expectation of future volatility in the options market. Higher implied volatility results in high option premiums and is also a forward-looking measure. IV differs across the options with different expiration dates.

The assumption about volatility does not hold in the real world as the volatility keeps on changing. Predicting volatility using historical data seems to be a good alternative as the financial market repeats itself over some time. ARIMA and GARCH time series forecasting models have been developed which predict the IV based on the historical data and have a better accuracy.

Black Scholes Model

The Black-Scholes model is the primary model used in predicting the price of the options. It is given by the equation,

C0=N(d1 )- X/e^(rf*T) N(d2)

Where N(d*) is the cumulative normal distribution function for (d*). C is the theoretical call premium, S is the Current stock price, t is the time until the option expiration, K is the option striking price, and rf is the risk-free interest rate. The assumptions in the Black Scholes model include the direction of the stock’s price following a random walk, constant volatility, normal distribution of returns, and risky assets not paying any dividends during the options life. Implied volatility is based on the pricing of the at-the-money call and put options. Forecasting the IV would help to determine the option premium accurately and can also be used by the investors to generate profits based on the straddle and strangle strategy.

ARIMA modeling

VIX levels were modelled in Figure 1 and we could see that the VIX level remained considerably stable from 2000 to 2007. However, in 2008 we could that the VIX index was very volatile because of the global financial crisis and there was a huge spike in 2020 because of Covid and turbulent times in financial markets.

The historical data from 2000 were plotted as residuals as shown below.

To model the residuals, ACF and PACF plots were plotted to determine the number of lags and the model to be used.

The above figures depict the autocorrelation and partial autocorrelation plot for the log of the VIX levels and its 1st difference levels. From the ACF plot we could see that 1st difference level with MA (2) and from PACF AR (2) seems to be a good fit. On the above information, initially, an ARIMA (2,1,2) was plotted and checked for the p-values. However, the p values were insignificant and hence an ARIMA (1,1,1) was incorporated to predict the residuals.

Below are the specifications for the ARIMA (1,1,1) model.

We can see that AR and MA coefficients are significant and the model has a low AIC and BIC which suggests ARIMA (1,1,1) better fits the VIX data.

ARIMA+GARCH modeling

To also consider the volatility shocks just like the events that happened in 2008 and 2020, we tried combining GARCH with the ARIMA model to see if we could get an improved forecast in the VIX index. On modeling GARCH(1,1) we got the regression results

The coefficients of the GARCH model were significant at 99% level. From the residual plot below, it could be inferred that the model picked up the overall shocks in the volatility as it is far less fluctuating but in 2020 the market was so volatile that the model could not pick up the variation.

Volatility Forecasting

After fitting the ARIMA+GARCH model with exogenous regressors, this paper analyzes on the prediction efficiency of the model. We have split the input datasets into training and test samples. The training dataset contains data from 04/01/2000 to 12/31/2020 and the test dataset contains data from 01/04/2021 to 11/30/2023.

After training the model with the fixed training regression window, the below result was obtained.

The above result is the VIX observed and the predicted first differenced levels. We could see that the model picked up the variations in 2021 and till the middle of 2022. As inflation was starting to pick up in mid to late 2022, the predicted level remained higher and the was quite a difference in those time frames.

Conclusion

VIX forecasting involves a combination of qualitative and quantitative metrics, incorporating factors from various domains. Traders and investors should be aware of the inherent uncertainties in predicting volatility and approach forecasting with a well-informed and adaptive mindset. Forecasting VIX helps us to gauge the market sentiments to some extent even if not accurately. The main drawback of the Black-Scholes model is that the volatility is assumed to be constant. If we use the historical forecasting method, it can be accurately priced to avoid mispricing. Based on the results we obtained VIX prediction is highly likely with the historical IVs using ARIMA and GARCH model with improved accuracy and efficiency.

References

1. https://www.investors.com/research/options/options-trading-today-after-50-years-of-growth/
2. https://finance.yahoo.com/
3. The Bond & Money Markets, Strategy, Trading, Analysis, A volume in Securities Institution Professional Reference Series, Book,2001
4. Engle, Robert. 2001. "GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics." Journal of Economic Perspectives, 15 (4): 157-168.DOI: 10.1257/jep.15.4.157