Lehman, Forecasting, and the Vix

Jeff Kearns and Michael Tsang of Bloomberg have an article out today (VIX Fails to Forecast S&P 500 Drop, Loses Followers) in which the authors contend that largely because the VIX failed to predict the October losses in the S&P 500 index (SPX), the VIX is no longer considered to be an accurate gauge of future market activity.

One of the central claims made by Kearns and Tsang regarding the lack of effectiveness of the VIX is stated as follows:

On Sept. 11, less than a week before New York-based Lehman Brothers Holdings Inc. went bankrupt and four days after the government takeovers of Washington-based Fannie Mae and McLean, Virginia-based Freddie Mac, the VIX closed at 24.39. That meant traders bet the S&P 500 wouldn’t fluctuate more than 24.39 percent on an annualized basis, or about 7 percent in the next 30 days, and implied a range for the index of 1,161.11 to 1,336.99.

One month later, on Oct. 10, the S&P 500 closed at 899.22, or a record 23 percent lower than what the VIX predicted.

As fellow blogger Don Fishback was quick to point out, the VIX calculation actually estimates one standard deviation of annualized 30 day volatility expectations in SPX options. That standard deviation, of course, is meant to capture 68.3% of the Gaussian or normal distribution of prices. In fact, the distribution of VIX prices does not follow a normal distribution, but even if it did, the movements from September 11 to October 10 would not be that statistically improbable.

Some quick comments on the math involved here. Using a VIX of 24.39, the conversion of an annualized volatility of 24.39% to 30 day volatility yields a 30 day volatility of 7.04%. Two standard deviations translate into a 30 day volatility of +/- 14.08% and should cover about 95.5% of the normal distribution (I am assuming a normal distribution for the sake of mathematical simplicity). Three standard deviations increase the range of expected volatility to +/- 21.12% and should capture about 99.7% of the normal distribution. In fact, the 99.9% boundary for the normal distribution is 3.29 standard deviations and translates into expectations of an SPX move of 23.16%, about on par with what transpired. So, given that an event which falls outside of 99.9% probability distribution happens once every 1000 instances, and using 250 trading days per year as an approximation, the VIX calculation says the fall from September 11 to October 10 was on the order of a once every four year event.

Of course there are many ways in which to utilize the VIX to aid in market timing. Consider that the nature of the VIX calculation is such that the VIX acts as an unbounded oscillator whose values are derived from prices paid for options on the SPX. Like most oscillators, traders tend to used extreme values as opportunities to bet on a reversion to the mean.

Part of the problem with understanding the movements of the VIX is that some of the dominant patterns are different over the course of different time horizons. For instance, in the short-term, the VIX commonly spikes and mean reverts. Looking at the intermediate term, the VIX frequently establishes strong trends; and in the long term, the VIX has a tendency to move in cycles of 2-4 years. For the month of September and most of the month of October, the VIX was in an uncharacteristically strong sustained uptrend.

Part of the reason for the sharp move in the VIX during September and October is that it is highly dependent upon macroeconomic and fundamental events that help to shape investor perceptions of uncertainty, risk and fear. In the week leading up to the Lehman Brothers bankruptcy, for instance, very few investors believed that the government was prepared to let Lehman fail. Additionally, in retrospect it seems as if those who did believe failure was an option did not comprehend the nature of the systemic reverberations that a Lehman bankruptcy would trigger.

The bottom line is that during the second week in September, the VIX was pricing in a very low probability of a Lehman Brothers bankruptcy. Perhaps more importantly, investors were also assigning a significantly lower systemic threat potential as a result of the dominoes associated with a Lehman bankruptcy.

Ultimately, it took a full six weeks of a steadily trending VIX for the market to fully price in the global systemic risks associated with the sequence of events that began with the Lehman Brothers bankruptcy.

Consider that the prices and implied volatilities of SPX options have to account not just for the probabilities associated with various future scenarios, but also the magnitude of the impact of those scenarios on the stock market. For this reason, even while some of the probabilities may not have varied significantly from day to day during September and October, as the magnitude of the financial crisis was slowly revealed, the VIX continued to ratchet higher – and investors reacted to a rising VIX with increasing alarm.

Getting back to the question posed by Kearns and Tsang, yes the VIX underestimated future volatility back on September 1. At that time, the prediction of a four-year flood was consistent with mainstream thinking. Very few observers anticipated the SPX falling below 800 by Thanksgiving.

As the year winds down, I will have more about what we learned about volatility in 2008 and what some of the implications are for 2009 and beyond.

Comments

[Evan Grey Owl:]

The VIX isn't set up to predict large scale sudden selling of stocks by big entities with inside information about what Congress is going to do vis-a-vis a "bailout" of those same big players. I'm sure the insiders could imagine the public reaction to learning suddenly about ugly internals they had kept from sight so successfully.... until they couldn't. They cashed out before the lines got too long. Basically, taking from all the small investors (The un-annointed) with one hand, while holding a tin cup to the Congress with the other. I believe all small investors should be re-imbursed by the SIPC up to $100K for losses from mid-September to the present, or whenever this precarious ship rights itself.

2008 Dec 22 08:46 PM

[Moon Kil Woong:]

Just because 99.5% of the time something isn't expected to happen doesn't mean it won't. I suggest people think about probability before they decry statistics dead. In general, I think people have a very hard time considering big numbers or small odds. Even .001% occurrences happens all the time. Just ask a nuclear physicist or chemist. Just because they happen doesn't mean the statistics are wrong or the half life of uranium is off.

The author does a good job pointing out the fact the VIX number people are used to only covers 1 standard deviation. However, the ending saying VIX underestimated volatility is probably not an accurate statement. People might have not estimated the probability correctly but it's very easy that this event just happened to end up as a .3% probability event. That's the fact.

If you roll dice you sometimes roll 4 sixes. That's life.

2008 Dec 23 02:51 AM

[a. palmer jr.:]

That sounds illegal.

2008 Dec 23 10:40 AM

[OldLimey:]

How could the VIX have predicted what mood Paulson would be in when he got out of bed on the day he decided to let Lehman go? This comment, of course, doesn't so much defend the efficacy of the VIX as underline the new uncertainties now extant in our politicised financial markets.

2008 Dec 23 03:00 PM

[Lilguy:]

This article sounds like a long apologia for why the VIX was useless in anticipating September's market drop.

2008 Dec 23 04:04 PM