By: Andrey Fradkin Date: 7/16/07

1. A Study of the Informational Content of Implied Volatility and the Effect of Jumps on Forecasting Volatility By: AndreyFradkin Date: 7/16/07

2. Introduction • Examine relative informational content of implied volatility forecasts against historical volatility forecasts using high frequency data for 10 large cap stocks trading on NYSE and for the SPY. • Important prior studies on this subject are Andersen et al(2007) and Jiang and Tian(2006). Find contradictory evidence as to which measure is better. • This study is first to use individual stocks, HAR-RV CJ model, RAV, and robust regression in examining this question. • HAR-RV-CJ model allows for the separation of the jump component of volatility from continuous component of volatility. • Robust regressions Andrey Fradkin: Informational Content of Volatility

3. Outline • Jump Statistics • HAR-RV-CJ Models • Data Preparation • Summary Statistics • Summary Plots • Significance of Jump Statistics in HAR-RV-CJ Models • Robust Regression • Comparison of Model Performance In-Sample Andrey Fradkin: Informational Content of Volatility

4. Background Mathematics Realized Variation: Realized Bi-Power Variation: Andrey Fradkin: Informational Content of Volatility

5. Background Mathematics Part 2 • Tri-Power Quarticity • Quad-Power Quarticity Andrey Fradkin: Informational Content of Volatility

6. Background Mathematics Part 3 • Z-statistics (max version) – Paper uses .999 significance level Andrey Fradkin: Informational Content of Volatility

7. Original HAR-RV-J Model(Developed in Andersen, Bollerslev, Diebold 2006, Extended for use with RAV measure in Forsberg and Ghysels(2007)) Andrey Fradkin: Informational Content of Volatility

8. The HAR-RV-CJ Model Andrey Fradkin: Informational Content of Volatility

9. Log and Square Root Transformations HAR-RV Model HAR-RV-CJ Model Andrey Fradkin: Informational Content of Volatility

10. Data Management • Uses TAQ high-frequency data that was cleaned by Tzuo Law. Prices are sampled at five minute intervals. • 10 equities and the SPY are used in the study. Equities chosen are based on the high open interest in their options. • Implied volatilities are obtained from the OptionMetrics database. One implied volatility is used per day. This implied volatility is taken from an at-the-money call option expiring about a month ahead. • In-Sample data ranges from 2001 through 2004 • The Out-of-Sample data encompasses all of 2005 Andrey Fradkin: Informational Content of Volatility

11. 10 Stocks Referred to from now on by Ticker • BMY - Bristol-Meyers • C - Citigroup • GE - General Electric • GS -Goldman Sachs • HD - Home Depot • KO – Coca Cola • MDT - Medtronic • MOT – Motorola • NOK – Nokia • TXN – Texas Instruments • SPY – SPY RV’s with Vix Implied Volatility • SPX – SPY RV’s with SPX Option Implied Volatility Andrey Fradkin: Informational Content of Volatility

12. Summary Statistics Andrey Fradkin: Informational Content of Volatility

13. Summary Statistics (Cont) Andrey Fradkin: Informational Content of Volatility

14. Some Facts • Average Log Difference between the mean of implied variance and the mean of realized variance is .56, with the minimum difference equal to .33 and the maximum difference equal to 1.01 • Average RV over the timeperiod in all stocks is 3.75 and the average RV in the SPY is 1.03 • Average number of jumps in the stocks is 35 and the number of jumps in the spy is 37 Andrey Fradkin: Informational Content of Volatility

15. HAR-RV-CJ – Level Month Ahead – Newey-West Standard Errors Lag(60) * p<0.05, ** p<0.01, *** p<0.001 Andrey Fradkin: Informational Content of Volatility

16. HAR-RV-CJ – 1 week Newey-West Standard Errors Lag(60) Andrey Fradkin: Informational Content of Volatility

17. HAR-RV-CJ – Level Day Ahead – Newey-West Standard Errors Lag(60) Andrey Fradkin: Informational Content of Volatility

18. Realized Variance and IV Andrey Fradkin: Informational Content of Volatility

19. Realized Variance and RAV Andrey Fradkin: Informational Content of Volatility

20. Continuous and Jump Andrey Fradkin: Informational Content of Volatility

21. Vix vs SPX options Andrey Fradkin: Informational Content of Volatility

22. SPY RV and RAV Andrey Fradkin: Informational Content of Volatility

23. SPY Continuous vs Jumps AndreyFradkin: Informational Content of Volatility

24. Adj. R-Squared For In-Sample Data - Levels Andrey Fradkin: Informational Content of Volatility

25. Adj. R-Squared For In-Sample Data – Levels (Continued) Andrey Fradkin: Informational Content of Volatility

26. Adj. R-Squared For In-Sample Data – Square Root Transform Andrey Fradkin: Informational Content of Volatility

27. Adj. R-Squared For In-Sample Data – Square Root Transform Andrey Fradkin: Informational Content of Volatility

28. Adj. R-Squared For In-Sample Data – Log Transform Andrey Fradkin: Informational Content of Volatility

29. Adj. R-Squared For In-Sample Data – Log Transform (Cont) Andrey Fradkin: Informational Content of Volatility

30. Conclusions from In-Sample R^2 • HAR-RAV performs as well or slightly better than HAR-RV-CJ Model • HAR-RV-CJ performs as well or slightly better than HAR-RV Model • Combining IV and HAR-RV-CJ yields a significant increase in R^2 • In Square Root or Log form IV diminishes in importance to historical measures. • IV performs best when forecasting a month in advance. (This is to be expected ) Andrey Fradkin: Informational Content of Volatility

31. Robust Regression • Regressions that are meant to correct for problems with standard OLS regressions, specifically non-robustness to outliers. • There are several types of robust regressions including least absolute deviation, least median of squares, trimmed least squares, M-estimators. • For ease of calculation, this paper uses a form of robust regression easily implementable in Stata. Andrey Fradkin: Informational Content of Volatility

32. Robust Regression Continued • The robust regression used is a type of Weighted Least Squares Rregression. • It first runs an OLS regression, calculates Cook’s Distance values which approximate the influence of an observation on the estimation. • Cook’s D Value that are higher than 1 are dropped. • Then the remaining observation get weighed based on the size of their absolute residuals. This is an iterative process where Huber weighing and Bi-weighing are used. The result is a regression that is about 95% efficient. Andrey Fradkin: Informational Content of Volatility

33. Robust Regression Visualization Minimize by solving Andrey Fradkin: Informational Content of Volatility

34. Robust Regression Pseudo Adj. R-Squared Andrey Fradkin: Informational Content of Volatility

35. Robust Regression Pseudo Adj. R-Squared (Cont) Andrey Fradkin: Informational Content of Volatility

36. Conclusions from Robust Regressions • In robust regressions the IV becomes a comparatively worse than historical measures. • This might imply that IV is best at forecasting leverage points or points with high residuals which are down weighed in the robust regression. Andrey Fradkin: Informational Content of Volatility

37. Conclusions • Jumps do not have persistent effects on volatility. • The use of the HAR-RV-CJ yields little if any benefit as measured by R^2 • Robust regressions mitigate the predictive power of IV • The HAR-RAV model has slightly more explanatory power than that HAR-RV model • Both implied and historical measures have unique information. Andrey Fradkin: Informational Content of Volatility

38. Future Work • Compute RMSE for all of the models and compare. • Create graphs showing forecast against actual behavior • Try to implement conditional forecast model used in Forsberg and Ghysels (2007) • Explore biasness of Implied Volatility • Incorporate Overnight Returns Andrey Fradkin: Informational Content of Volatility