Asset #1 – S&P 500 monthly returns



Forecasting the VIX

I. Executive Summary

The primary objective of this project was to develop both a model that correctly forecasts the CBOE Volatility Index (VIX) and a complementary option-based trading strategy that would generate profits as a result of accurate VIX predictions. Although the VIX measures the implied volatility of the S&P 100 index (OEX), it is a good proxy for volatility of the entire U.S. equity market. Forecasting movements in the VIX is an extremely constructive exercise because the VIX is a measure of implied volatility, and as such, can be considered a good substitute for conditional volatility. Since market volatility changes through time, it is well documented that conditional volatility provides much more insight, and its use can lead to much more accurate pricing models than an unconditional, historical volatility measure. For this reason, we believe that an accurate forecasting model for the VIX could lead to profitable OEX options trades that capitalize on changes in volatility.

The VIX, introduced by the CBOE in 1993, provides investors with up-to-the-minute market estimates of expected volatility by using real-time OEX index option bid/ask quotes. This index is calculated by taking a weighted average of the implied volatility of eight OEX calls and puts with an average time to maturity of 30 days. Consequently, the VIX is intended to indicate the implied volatility of 30-day index options. As seen in our data, implied volatility levels in index options change frequently and substantially.

Our hypothesis is that one can create a profitable option trading strategy based on forecasting the VIX. This hypothesis is based largely upon our belief that volatility is mean-reverting, and that a well-constructed model will be able to predict these reversions. Although our model correctly predicted the direction of the VIX a majority of the time, we feel that additional research should be performed in order to perfect the model and the trading strategy. In addition, we recognize that one cannot “trade the VIX” as we have done in our trading strategy; however, buying and selling straddles—or engaging in other volatility-based transactions—on the OEX might be one way to profit from correctly forecasting the VIX.

II. Methodology

In order to test our hypothesis that one can construct an options trading strategy based on forecasting the VIX, it was necessary for us to determine the sampling frequency with the most economic justification. Our intuition was that market volatility is very volatile, and therefore, a high-frequency trading strategy would be superior. However, the data collection issues for a daily strategy—along with the extremely high transactions costs that would most likely be incurred—caused us to look beyond a daily strategy. Because we still believed in the high volatility of the VIX, we did not want to base our forecast on monthly data, and as a result, we settled on a weekly sampling period (Wednesday to Wednesday to mitigate the effect of holidays).

Next, we developed multiple variables that we felt could have robust predictive power. (A summary of these variables is included as Appendix 1.) From that broad list, we focused our attention on several key variables and constructed numerous modifications of them—through parsing, the creation of interaction variables, or developing different short- and long-term moving averages. Despite our best attempts to create modified variables that provided greater insight than the base variables, only one of these was strong enough to be included in our final model (change in S&P 100, if positive).

After collecting weekly data from 1995 to the present (which gave us a statistically-sufficient number of weekly data points), we utilized a two-holdout sample methodology to reach our final conclusions. Initially, we used a data set of 263 weekly observations (October 1995 to November 2001) and ran in-sample regressions on numerous potential models using a data set of 213 weekly observations. After observing these in-sample results, we validated the models over the 50-week holdout period (weeks 214-263). Based upon our preliminary regressions, we felt that the top candidate was a three-factor model using the lagged level of VIX, the lagged change in the S&P 100 (if positive), and the lagged change in 10-year U.S. Treasury yield. After deciding upon this as our final model because of its high in-sample R2 and its 64% hit rate out-of-sample, we performed a second validation using data from 2001 to the present.

III. Regression Model Analysis

Dependent Variable:

Weekly % change in CBOE Volatility Index (VIX)

Independent variables utilized in final model:

VIX, Lag 1

Change in US 10yr Treasury Yield, Lag 2

Change in S&P100 (if positive), Lag 1

Discussion of variables utilized:

VIX, Lag 1:

This variable is the level of the VIX from the previous week. We expect a mean reversion of the VIX; i.e. the VIX is expected to rise when at low levels and fall when high. This is supported intuitively by the concept that there is a loosely fixed range in which the VIX trades, and therefore, there is more room to increase when volatility is low and more room to fall when volatility is high. Likewise, it is our intuition that extremely high levels of volatility are usually caused by transitory events that fade over time, and extremely low levels of volatility are generally unsustainable due to the inevitable onset of volatility-generating events. The regression output below supports our belief that mean reversion exists and that percentage change in VIX is negatively correlated withVIXt-1.

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This regression yields an R-squared of 5.15%, and an adjusted R-squared of 4.84%. The variable’s T-statistic and P-value both show a statistically significant relationship between the dependent and independent variables at the 99% confidence level.

Change in 10-yr U.S. Treasury Yield, Lag 2:

The change in the 10-year U.S. Treasury yield is determined as the weekly basis point change in Treasury yieldt-2. We posit that the Treasury market is more adept than the equity market at pricing in a multitude of risk factors—largely because the market is more quantitative in nature and because the equity market is often burdened by unsophisticated, retail investors. Therefore, the Treasury market reacts earlier and more fully than the equity market to new information. In our opinion, the link between volatility and Treasury yields is fairly straightforward; as yields fall, investors are moving into bonds (and, presumably, out of equities). This shift into bonds can often be associated with a so-called “flight to quality” during times of increased uncertainty. Because we feel that the Treasury market’s reaction to signals of potentially heightened risk occur more rapidly and more fully than the equity market’s, it is logical to assume that an increase in Treasury yields can say something about future market volatility. By contrast, as Treasury yields rise and money flows out of the government bond market, it is a signal that the risk of future volatility is lessened. The regression output below shows a negative correlation between percentage change in VIX and the change in Treasury yields, and supports our premise.

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This regression yields an R-squared of 3.74%, and an adjusted R-squared of 3.43%. The variable’s T-statistic and P-value show a statistically significant relationship between the dependent and independent variables at the 99% confidence level.

Change in S&P 100 (if positive), Lag 1:

This variable measures the weekly price change in the OEX and attempts to capture the impact of movements in the index from which the VIX is derived. We expected large movements in the S&P 100—either up or down—to contribute to increased volatility. This implies that as the market jumps (falls), investors enter (leave) the market rapidly, contributing to increased trading volumes and increased volatility. We believed that on a weekly basis, there is a momentum effect, but that it might affect volatility differently depending upon the direction in which the S&P 100 moved. Therefore, in our initial analysis, we parsed S&P 100 movement into positive and negative changes. Although anecdotal evidence supports the theory that decreases in the S&P 100 increase volatility, our regression showed this variable to have no explanatory power. Perhaps, volatility increase much more quickly and fully in response to negative market moves and the one-week lag is not appropriate. Regardless, our regression results show that percentage change in the VIX is positively correlated to the lagged change in the S&P 100 index, supporting the concept that a positive move in the market contributes to increased future volatility.

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This regression yields an R-squared of 1.726%, and an adjusted R-squared of 1.41%. The variable’s T-statistic and P-value show a statistically significant relationship between the dependent and independent variables at the 95% confidence level.

First Out-of-Sample Analysis:

Of 313 total data points, our first out-of-sample model was based on 213 and validated on 50. The regression formula for this model is as follows:

Pct Change VIX = 10.927 - 0.503*lag(VIX,1) + 0.221*lag(Chg SP100 if pos,1) - 11.169*lag(Yld Chg US 10yr,2)

Of the 50 out-of-sample forecasts, this model correctly predicted the direction of change in weekly VIX in 32 periods, or 64% of the time. The model correctly predicted positive direction 50% of the time and negative direction 76.92% of the time.

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Second Out-of-Sample Analysis:

Of 313 total data points, our second out-of-sample model was based on 263 and validated on the final 50. The regression formula for this model is as follows:

Pct Chg VIX = 12.526 - 0.529*lag(VIX,1) + 0.165*lag(Chg SP100 if pos,1) - 14.967*lag(Yld Chg US 10yr,2)

Of the 50 out-of-sample forecasts, this model correctly predicted the direction of change in weekly VIX in 30 periods or 60% of the time. The model correctly predicted positive direction 40% of the time and negative direction 80% of the time.

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IV. Final Model

Our final regression includes weekly data from September 1995 to January 2003 for a total of 315 observations. Below are regression results for our final VIX forecasting model:

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The final model regression yields an R-squared of 10.7%, and an adjusted R-squared of 9.84%. Each variable’s T-statistic and P-value show statistically significant relationships between the dependent variable and independent variables VIX, Lag 1 and DIFF US 10yr, Lag1 at the 99% confidence level. Independent variable DIFF SP pos, Lag 1 is significant at the 95% level. The Durbin-Watson (DW) statistic = 2.08 with a P-value = 0.239, indicating no serial autocorrelation in the residuals.

All coefficient signs are as expected from previous variable explanations. The DIFF SP pos variable has the only positive coefficient.

The correlation matrix of all independent variables shows no sign of multicollinearity.

| |VIX, Lag 1 |Diff 10 yr, Lag 2 |Diff SP pos, lag 1 |

|VIX, Lag 1 |1 |-0.052206241 |0.025347389 |

|Diff 10 yr, Lag 2 |-0.052206241 |1 |-0.068772587 |

|Diff SP pos, lag 1 |0.025347389 |-0.068772587 |1 |

Final Model In-Sample Analysis:

Please reference the Excel file titled A1AAM.xls, Tab: Full Model for corresponding spreadsheet data.

Direction count and percentage:

Our model correctly predicted the direction of the weekly percentage change in the VIX in 180 out of 313 periods, or 57.51% of the time. We predicted positive moves in the VIX 111 out of 154 actual positive periods, or 72.1% of the time. Our model was less accurate with negative moves, predicting only 69 out of 159, or 43.4% of the occurrences.

Trading Strategy #1: Assuming the VIX is tradable, this strategy explicitly follows our basic model and puts us in the market, long or short, every week. Average weekly return using this strategy is 3.89% with a standard deviation of 11.62%. Cumulative return since September 1995 is over 2,300,000%, yes, 2,300,000%. Not only does this strategy yield a 57% hit rate, but also the winning weeks outperform losing weeks 11.26% to -6.14%, respectively.

Trading Strategy #2: This strategy involves placing a trade, long or short, when the absolute value of our VIX prediction is greater than 5%. Under this strategy, we placed 73 trades in 313 periods. We correctly predicted direction in 46 of these periods, or 63.01% of the time. While this strategy does not produce a significantly greater percentage of hits, it is far less risky. Cumulative return for the strategy is 4,591%, yes, only 4,591%, but standard deviation drops to 5.73%. Of the weeks we place trades, average return is 5.95%. However, given the infrequency of this strategy, average weekly return over all 313 periods is 1.39%. As with strategy #1, winning weeks outperform losers by a significant margin, 12.52% to -5.24%, respectively.

Of these two strategies, we recommend going for the 2.3 million %; we leave for Bermuda tomorrow! On a serious note, it is interesting to consider the merits of these two crude trading strategies. While we recommend trading strategy #1 because of its ability to deliver spectacular returns and its higher Sharpe ratio (assuming weekly risk-free rate is negligible), we realize that this return is not really relevant since we cannot trade the VIX and there are implications to options trading that will significantly impact our results. In fact, some of the performance measures for trading strategy #2 give us reason to believe that it may perform better in the “real world”. The slightly higher hit rate is appealing as is the greater spread between winning weeks and losing weeks—5.12% for strategy #1 vs. 7.28 for strategy #2. Because we will have to consider transaction costs when implementing this model, a higher hit rate and higher spread between winners and losers is very desirable.

V. Issues/Future Recommendations

The primary issue detracting from the significance of this project is that while the model does well at predicting future implied volatility, the VIX is not a tradable security, and therefore, the returns examined in the in-sample/out-of-sample portion of this paper are not necessarily relevant. Because the VIX is not tradable, a trading strategy needs to be devised using securities that are tradable—most likely, options on the OEX. More specifically, a forecast of an increase in VIX could lead to the purchase of short-term, at-the-money OEX straddles. Even after a strategy is devised, examining the results of the strategy proves difficult, as weekly OEX option prices are not readily available. Additionally, such a strategy would introduce potentially high transaction costs. As a recommendation for further research, we would suggest examining returns based on such a strategy that incorporate both actual option pricing and transactions costs.

Additional recommendations for further research include examining trading strategies outside of the “long VIX”/“short VIX” option observed in this paper. This would include examining double moving average crossovers and various thresholds for forecasted VIX changes to trigger market entry or exit. It is likely that following a rule other than predicted change in VIX being greater than zero will yield superior returns or lower risk than the results presented in this paper. Furthermore, we would recommend looking at various holding periods.

VI. Conclusion

Our model demonstrates the ability to predict the direction of the VIX with some accuracy. By following either of the trading strategies described above, the portfolio return would have been significant. Although forecasting the VIX is difficult, the results of our model give us reason to believe that with additional research, a model similar to ours could be used to trade profitably.

APPENDIX 1 – Variables Considered

Change in VIX, lag 1 and 2

Percentage change in VIX, lag 1 and 2

Considered because of belief that volatility has autocorrelation and is persistent.

S&P100

Percentage change in S&P100

Change in S&P100

Change in S&P100, if negative

Change in S&P100, positive dummy

Change in S&P100, negative dummy

Change in S&P100 >3%, positive dummy

Change in S&P100 ................
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