1. Predictive Models of Realized Variation and the Effects of Realized Semi-Variance on Implied Volatility Haoming Wang 3/19/2008

2. Introduction Want to examine predictive regressions for realized variance and realized semi-variance (variance caused by negative/postive returns). Sector realized variance and realized semi- variance is introduced as a regressor. Regressions are of the form of the HAR-RV regression talked about in Anderson, Bollerslev, and Diebold (2005). Semi-variance taken from Barndorff-Nielsen, Kinnebrock, Shephard (2008)

3. Introduction Also want to examine the effects of various variance measures on changes in implied volatility and the level of implied volatility. All data has been narrowed to 2006. Datasets for Pfizer, S&P 500, and Pfizer options data was examined.

4. Introduction Will first address HAR-RV regressions featuring sector RV. Then will examine HAR-RAV regressions also featuring sector RV. Finally, will examine the various effects of realized variation, realized semi-variance, sector variance, and market variance on implied volatility.

5. Implied Volatility Implied volatility is considered the “price” of an option since at any given time, all other inputs are predetermined. Used implied volatility data from OptionMetrics. Implied volatility is quoted from at the money calls which are about a month away from maturity. ATM implied volatility was used as the best proxy for expected future volatility.

6. Implied Volatility Volatility skew: Equity options exhibit volatility, in other words, implied volatility is often a decreasing function of the strike price.

7. Implied Volatility Intuition: Could be a reflection of equity investors fears of market crashes. Implied volatility on lower strikes would be bid up as investors purchase protection. The volatility skew for equities has only existed since the stock market crash of October 1987, before that, implied volatility was much less dependent on strike price. Motivation for research: Increases in downside semi-variance should cause increases in implied volatility. Increases in upside semi-variance should cause decreases in implied volatility.

8. Implied Volatility

9. Background Mathematics Realized variation (where r t,j is the log-return): Realized sector variation: An average of daily realized variation for ABT, BMY, JNJ, MRK, and PFE, excluding whichever stock is being regressed. Realized sector portfolio variance: average of daily stock prices is used as the price of the portfolio.

10. Background Mathematics Realized variation (where r t,j is the log-return): Realized semi-variance (where 1 is the indicator function that the return is negative) :

12. Background Mathematics Further, according to BNKS, the realized semi- variance converges to half the bipower variation plus negative squared jumps, or:

13. HAR-RV Model The multi-period normalized realized variation is defined as the average of one-period measures, or: The daily hetereogeneous autoregressive realized variance (HAR-RV) model of Corsi (2003) is used with daily, weekly, and monthly periods:

14. Extended HAR-RV model Added sector variance for daily, weekly, and monthly periods. For example, the RV of PFE would be regressed on the average RV of ABT, BMY, JNJ, and MRK. Examined regressions for PFE.

15. RAV RAV (realized absolute value) is of the form: HAR models using RAV as a regressor instead of realized variance have been shown to be superior. Important to note that RAV is in different units than realized variance (not squared).

16. Why RAV? “Sampling error is shown to reduce persistence of volatility measures based on squared returns more than the persistence of measures based on absolute returns. Hence, the latter have higher persistence based on population moments, and the downward bias due to estimation error is much less severe than it is for volatility measures based on squared returns.” Forsberg and Ghysels (2007) Using RAV amplifies the effects of the lagged measures? Lower bias due to estimation error.

17. OLS vs. Robust Regression The pricing data most likely has leverage points (data points that have an extremely large effect on the coefficient estimates) as well as sampling errors (trading days that were removed). This could create small disturbances for the data which are hard to isolate. Thus, standard OLS regression might not be the best way to estimate these coefficients since OLS is sensitive to leverage points.

18. Leverage Point Example

19. OLS vs. Robust Regression Thanks to Fradkin (2007) we can mitigate these problems by using robust regressions instead of OLS regressions. The “rreg” command in STATA is used, which iteratively reweighs least squared based on M- estimators. Robust regressions will be compared with OLS regressions in the following slides.

20. Robust Regression Background Standard regressions of the form: Coefficients are found by minimizing squared errors: This leaves the regression coefficients sensitive to leverage points. To mitigate the effect of leverage points, robust regressions can be used. This is down with the rreg command in STATA

21. Robust Regression Background STATA uses a recursive algorithm to calculate the coefficients. Roughly, what happens is that points with large residuals are down-weighted. This means that points with high leverage play a smaller role in the regression. Weights are then recalculated until a certain threshold.

22. Robust Regression Example:

23. Newey West Standard Errors Newey West standard errors are assumed to be heteroskedastic and possibly auto-correlated up to some lag. Heteroskedasticity: means that the variance of the error terms are a function of the regressors. Makes sense to assume this with financial data since we’ve seen that volatility and the volatility of our variance measures change a lot over time. Assuming homoskedasticity means that the standard errors of coefficients tend to be underestimated, which might make insignificant variables appear significant.

24. Newey West Standard Errors Use lag(60) as the parameter in STATA for the amount of time to be considered for possible autocorrelation. Makes sense that standard errors would be autocorrelated since we’ve seen in the data periods of high and low volatility.

25. Results HAR-class of regressions

26. Sector as a Portfolio? Professor Bollerslev suggested looking at the sector as a portfolio instead of simply averaging the relevant statistics. Previously, sector RV was just the average of RVs for ABT, BMY, JNJ, and MRK. Now, create a new stock that is the average of the prices of the four companies.

27. Sector Results (Non-Portfolio)

31. The previous two regressions were all done with each company in the sector treated individually. As before, we see very strong sector effects. With Pfizer, company specific realized variance is only statistically significant at the 5% in one day lags. In both regressions we see sector-specific one day lagged realized variance as having a greater effect. In the OLS regression with Newey-West standard errors, we see that sector realized variance is significant at both the weekly and monthly levels, results not seen in the company realized variances.

32. Sector Results (Porfolio)

33. Sector Results (Portfolio)

36. Across the board there are higher standard errors when we treat create a sector portfolio. Qualitatively, treating the sector as a portfolio seems to lessen the ability to predict spikes. Many sector effects become statistically insignificant, thus treating each company individually and then average the results seems to be the best method.

37. HAR-RAV-Sector

39. HAR-RAV-Sector (Robust)

40. HAR-RAV-Sector We can see the effect of robust regressions, downshifting the predictions. We can see from the OLS regression with Newey-West standard errors that RAV regression has greater persistence since the monthly lag is statistically significant. HAR-RAV model had the highest adjusted r-squared, although still low at around 5%. Next step: Take a look at sector RAV.

41. Results Implied Volatility

42. Implied Volatility Results Implied volatility is quoted as annualized volatility. As such, all relevant variance data has also been converted to annualized volatility. Again, from the structure of the volatility skew, we would expect downward semi- variance to have a positive effect on implied volatility and upward semi-variance to have an opposite effect on implied volatility.

43. Implied Volatility Results What should we expect to impact how much implied volatility changes? Candidates: previous change in implied volatility, the level of implied volatility, and realized semi-variances for the company, the sector, and the market. Separate the three and see which level of the market has the greatest impact on implied vol. First, we examine a regression of the change in implied volatility. Regressions are of the form: IV(t)-IV(t-1) = a + regressors+ epsilon(t)

44. Implied Volatility Results

47. Implied Volatility

48. Implied Volatility Results Surprisingly, the realized semi-variances of the individual company are not statistically significant (although the semi-variance is close). On the other hand, four factors are statistically significant: yesterday’s change in implied volatility, today’s implied volatility level, and the realized semi- variances of the S&P 500. We see that yesterday’s change in implied volatility has a negative effect on today’s change in implied volatility. Intuitively, this seems to mean that we see some mean reversion. On average, a 1% increase in implied volatility yesterday leads to ~0.3% decrease in implied volatility.

49. Implied Volatility Results For the level of implied volatility, I decided to use 4 days lag, since more recent lags already enter the regression in the lagged delta IV. Seems to imply that high levels of implied volatility are unsustainable. For example, an implied volatility of 40% four days ago implies a ~4.2% decrease in implied volatility. Both sector and company specific realized semi- variances are statistically insignificant at and beyond the 5% level.

50. Implied Volatility Results The results for market semi-variance are the most interesting. It’s interesting that stock-specific and sector-specific realized semi-variances wouldn’t have a statistically significant effect, but market wide semi-variances would. S&P 500 upside semi-variance has an economically and statistically significant negative effect on implied volatility. Downside semi-variance also has an economically and statistically significant positive effect on implied volatility. The results seem consistent with our hypothesis, that implied volatility becomes more expensive as the market goes down since options buyers seek to purchase protection.

51. Implied Volatility Results It seems intuitive to expect that increased volatility of any type would increase implied volatility, it’s interesting to note that in this case, that’s not true. On the other hand, options are not a pure volatility play, there is also a directional aspect to the payoff of an option. It’s also interesting to see that larger spikes in the change in implied volatility are better explained by the regression than smaller changes that are around average.

52. Implied Volatility Results Next, I looked at a predictive model of the level of implied volatility. Included regressors are: implied volatility, upside and downside semi-variances, upside and downside semi-variances for the sector, and upside and downside semi-variances for the market. Regressors were all at the lagged daily, weekly, and monthly level.

53. Implied Volatility Results Again, we would expect downward realized semi-variance to positively effect implied volatility and upward semi-variance to have the opposite effect.

58. Implied Volatility Results One surprising thing was the high adjusted r- squared of around 60%. Also surprising was the fact that outside of lagged values of implied volatility, no other regressors appeared to be statistically significant. Seems that we can simplify our regression by limiting it to simply include implied volatility.

60. Extensions I plan on looking at 5 and 22 day ahead predictions: intuitively it should be easier to forecast averaged realized variances and implied volatilities than it is to predict the next day’s. Also, perhaps include measures of sector RAV and see if that provides better forecasting. Do HAR style regressions for semi-variances.