Volatility Models
Question 3: Evaluate the forecasting capabilities of three alternative volatility models following guidelines provided in Brownlees et al. (2011) and Hansen and Lunde (2005).
A forecast comparison of volatility models: does anything beat a GARCH (1,1)?
(Peter Hansen and Asger Lunde, 2005)
- Arch (1) model
The Autoregressive Conditional Heteroscedasticity (ARCH) model describes the variance of the error term in a financial time series data. From the study by Hansen and Lunde (2015), the simple ARCH (1) model when used as a benchmark to estimate the volatility of exchange rates and IBM data was outperformed by other competing models such as the GARCH (1,1), as evident from p-values in tables 1, 2, 3 and 4. From the analysis of data on exchange rates and IMB, there is a much significant advantage of forecasting volatility using other models, because the ARCH (1) model’s limited flexibility is insufficient to capture volatility persistence. Also, the ARCH (1) model is only suited to situations where there are short durations of volatility, implying that the model’s capability is suited to limited volatility periods.
- GARCH (1,1) model
In addition to the ARCH (1) model, Hansen and Lunde (2015) used the GARCH (1,1) model to forecast volatility of exchange rates and IBM data. The model outstandingly outperformed the ARCH (1) model when estimating volatility of exchange rate data and IBM data, as evident from the p-values in tables 1, 2, 3, and 4. There was not enough evidence to show that other models outperformed the GARCH (1,1) model, and further analysis shows that the two models reveal that GARCH (1,1) has better sample performances. Results from both exchange rate data (DM/USD) and IBM data all saw GARCH (1,1) model exhibit better suitability in volatility forecasting as compared to other models. The model equally corresponds to impact curves and does not generate a leveraging effect. Also, the model is flexible enough to capture persistence in volatility, as well as its suitability in forecasting volatility in situations where there are longer durations in volatility.
A practical guide to volatility forecasting through calm and storm (Brownlees et al., 2011)
From the analysis, Brownlees et all (2011) find that asymmetric models, in particular, the TARCH model can perform very well across different methods, assets, and samples. As more financial time series data is availed, such models have the capability of performing even better. Though the model suffers from effects of drifting of parameters, when the parameters are updated weekly, the move counteracts the drifting impact, thereby increasing the performance of the TARCH model. The authors also find that despite the student t distribution model having a potentially more realistic description of tail returns, there was no evidence that the model improves the performance of the TARCH model.
Question 4: Following the Shu and Zhang (2013) paper, where authors used daily S&P 500 index option prices over the period between January 1995 and December 1999, report and commented on a novel empirical application, evaluating the relationship between alternative measures of volatility.
The Relationship Between Implied and Realized Volatility of the S&P 500 Index (Shu and Zhang, 2013)
The paper analyzed the relationship between implied and realized volatility using daily S&P 500 index option prices over five years by testing how different measurement errors affect the stability of the relationship. Two sources of measurement errors were considered, error in realized volatility and error from the model specification. To compute errors in realized volatility, estimators like the standard deviation of daily return, extreme value volatility estimator, range estimator, and the square root of intraday return squares are used.
The implied volatility computed fromthe Black-Scholes model is compared with that from the calibrated Heston stochastic volatility option pricing model. One of the relationships between the models is that an improvement in the measurement of realized volatility significantly improves the forecasting ability of implied volatility, with the realized volatility estimated from intraday return data the most predictable. However, there was no significant difference in forecasting realized volatility using implied volatility from either the Black-Scholes model or from the Heston model.
Another relationship between the two volatility models is that when both implied volatility and historical volatility are used to forecast realized volatility, the implied volatility outperforms historical volatility as well as subsuming information from the historical volatility. The relationship was observed to hold for measurements from both realized and implied alternatives of measuring volatility.
From the study, the measurement error in both the realized volatility and implied volatility affects the stability of the relationship between the two alternative measures of volatility. Implied volatility contains information that can forecastrealized volatility, and the correlation appears to be stable under different measurements of realized volatility and implied volatility. Notably, the forecasting ability can be improved by constructing more accurate measurements of realized volatility. For example, creating the realized volatility from 5-minute returns is more predictable.
The implied volatility that was calculated using the Black-Scholes model had higher explanatory power than the one computed using the Heston model, because the Heston stochastic volatility model is too restrictive, and the actual true data generating process can deviate from Heston model significantly. As a result, historical volatility has less explanatory power than the implied volatility in predicting realized volatility.
Though the univariate regression of historical volatility to the realized volatility shows that historical volatility also has information in predicting realized volatility, the regression of the historical volatility and implied volatility simultaneously shows that the slope coefficient for the historical volatility is not statistically different from zero. This result indicates that implied volatility dominates historical volatility in forecasting realized volatility, or that the implied volatility has reflected all the information contained in historical volatility, and the historical volatility has no incremental forecast ability. Our study shows that the options market process information efficiently.
Question 5: Elaborate an essay on systemic risk measures based on your reading of the Benoit et al. (2017) paper and references therein. In particular, highlight pros and cons of the MES of Acharya et al. (2017), of the SRISK of Brownlees and Engle (2016), and the CoVar of Adrian & Brunnermeier (2016) with respect to the alternative source-specific and global approaches in Benoit et al. (2017)
Where the Risks Lie: A Survey on Systemic Risk (Benoit et al., 2017)
Systemic risk refers to the probability of an event at the company level to trigger acute uncertainty or even collapse the entire industry and economy and had a significant contribution to the global financial crisis in 2008. It is caused by a combination of different factorslike the economy, rates of interest, political issues, corporate health, among others. Some systemic risk measures are CoVaR, as proposed by Adrian and Brunnermeier (2016) and a closely related SRISK measure by Brownlees and Engle (2016), while Acharya et al. (2017) proposed using Marginal Expected Shortfall to measure systemic risk.
One of the advantages of the marginal expected shortfall model by Acharya et al. (2017) is that the model can be back-tested by the due shortfall methods that are non-parametric, independent of the underlying distribution and makes no assumption of any asymptotic convergence (Banulescu-Radu et al., 2020). Such tests are easy to implement, and in general, they display superior power compared to the back-test of Value at Risk. On the other hand, a disadvantage of the model is the complexity it brings. There is a necessity to record the forecasted cumulative distribution functions daily, which undermines the remarkable stability in the important levels across several shapes of the tail (Hansen, 2013).
A significant advantage of the systemic risk (SRISK) measures by Brownlees and Engle (2016) is that the measure provides more information about the firms’ systemic risk when compared with the capital asset pricing model (CAPM). On the other hand, the authors find that it is challenging to measure tail dependence when estimating systemic risk because of the limited systemic data, while also obtaining estimates will require various extrapolations from historical time series of returns. The extrapolation is because financial data usually have limited extreme value numbers. Also, another disadvantage in systemic risk measurement is that there are pitfalls in dissemination and collection of data, as some of the required data may be confidential, making it a challenge for the same to be shared with researchers.
Adrian and Brunnermeier (2016) propose the use of covariance of risk (CoVaR) as a measure of systemic risk, which can be predicted using commonly available characteristics like the leverage, size, mismatch, maturity, and asset boom. Another advantage of CoVaR is that it can straightforwardly be adapted from similar measures of risk like the expected shortfall. One of the drawbacks of using covariance in measuring systemic risk is that measurement is subject to skewness because of the presence of outliers that are in the historical data being used. As a result, a sizeable single-period movement could skew the overall volatility, and the resulting outcome will not represent the actual systemic risk a company may be facing.
References
Acharya, V. V., Pedersen, L. H., Philippon, T., & Richardson, M. (2017). Measuring systemic risk. The Review of Financial Studies, 30(1), 2-47.
Adrian, T., & Brunnermeier, M. K. (2016). CoVaR (No. w17454). National Bureau of Economic Research.
Banulescu-Radu, D., Hurlin, C., Leymarie, J., &Scaillet, O. (2020). Backtesting marginal expected shortfalland related systemic risk measures.
Benoit, S., Colliard, J. E., Hurlin, C., &Pérignon, C. (2017). Where the risks lie: A survey on systemic risk. Review of Finance, 21(1), 109-152.
Brownlees, C. T., Engle, R. F., & Kelly, B. T. (2011). A practical guide to volatility forecasting through calm and storm. Available at SSRN 1502915.
Hansen, L. P. (2013). Challenges in identifying and measuring systemic risk. In Risk topography: Systemic risk and macro modeling (pp. 15-30). University of Chicago Press.
Hansen, P. R., & Lunde, A. (2005). A forecast comparison of volatility models: does anything beat a GARCH (1, 1)?. Journal of applied econometrics, 20(7), 873-889.
Idier, J., Lamé, G., &Mésonnier, J. S. (2014). How useful is the marginal expected shortfall for the measurement of systemic exposure? A practical assessment. Journal of Banking & Finance, 47, 134-146.
Shu, J., & Zhang, J. E. (2003). The relationship between implied and realized volatility of S&P 500 index. Wilmott magazine, 4, 83-91.