The Story of “New Normal” and the Fat Tail

Prior to 2007, few people had heard of Nassim Taleb and his Black Swan. Some people had heard of fat tails, but most were quant modelers and academics with seemingly boundless mathematical knowledge. Additionally, the predictability of the distribution curves was an easy sell when three standard deviations encompasses nearly all outcomes, four, five, and six standard deviations were hardly worth considering.

But the question that continues to arise is whether or not there is a "new normal" and, if so, is it more dangerous than the "old normal"? Will the "new normal" carry greater volatility and risk? The short answer is probably.

As we know, volatility can come in many forms but it is always descriptive of a range of outcomes. Whether it is implied, historical or expected, volatility is described by a percent annualized standard deviation. In other words, if over the last year a $100 stock traded with a 25% volatility, we can be reasonable sure that 68% of that time it traded between $75 and $125. Similarly, if we expect the stock to have a 25% range over the next year, we are expecting 68% of outcomes to fall between $75 and $125.

With the flash crash a recent memory and the near collapse of the financial system (the verdict is still out on that one, in my opinion) having occurred in the not so distant past, I asked myself, "Has there really a broad increase in volatility over the years or is it just a passing fancy?"

Obviously, I needed a lot of data to answer this question; fortunately, we have this thing called the internet. So I pulled the historical prices for the S&P 500 from Yahoo! from Jan 1962 to present day, put them on a spread sheet and went to work.

The first thing I thought was that if volatility is increasing we should see an increase in the standard deviation of daily percentage returns, meaning that overall the range of outcomes probable on a daily basis should expand. With over 12,000 data points, I decided some summary was in order so I broke the information down into annual groups.

Along the bottom of the chart there are the average daily percentage returns for each year. This is fairly constant with average returns in the +/- 1.5% range for each year.

The spiky reddish orange line is the standard deviation of the close to close changes for each year. Not unexpectedly, the largest increases in range occur in down years (2008, 2002, 1987, etc.). Additionally, there are higher highs and higher lows which are suggestive of a long term up trend. That trend can be seen easily with the overlay of the linear regression line (royal blue). Since 1962, the standard deviation of daily returns has more than doubled on an annual basis from .0575% to 1.25%. That's a fair sized increase and certainly evidence of more volatile markets. Remember, this is only close to close and does not take into account any increases in intraday volatility.

Now knowing the mean (average) daily return and standard deviation for each year gives us all the information we need to understand the annual distributions of returns for any given year. Unfortunately, that only applies to normal distributions and the stock market has never been so kind as to keep things simple. To really reinforce my thesis of increasing volatility I wanted to know more about the distributions.

This is about skew and kurtosis. These are descriptive statistics that provide you with additional information about the distribution in question. Briefly, skew can be positive or negative with the direction reflecting which side the fatter tail is on. Positive skew reflects a plethora of outcomes just below or to the left of the mean, with outliers, when the occur tending to be far right of the mean. Negative skew reflects the majority of outcomes just above or to the right of the mean, with outliers tending to occur at the extreme of negativity or far to the left of the mean. The broad stock markets tend to trade with a slight negative skew, with the majority of outcomes resulting in a small percentage gains, but the large moves, when they occur produce large daily losses.

Our other statistic, kurtosis, can also be positive or negative. Positive excess kurtosis indicates that the majority of outcomes are clustered around the mean with a number of both positive and negative outliers fattening the tails of the distribution. As a result, there are fewer median outcomes than would be found in a normal distribution or bell curve, a condition known as leptokurtosis. Negative excess kurtosis, or platykurtosis, is characterized by few outliers (thin tails) and a roughly even probability of outcomes around the mean. The broad markets tend to exhibit leptokurtosis.

My intention here was to determine whether over time the tails, and in particular the downside tail was increasing; were getting fatter by means of increasing kurtosis and decreasing (more negative) skew. However rather than examine each year individually, I grouped them together 6 years at a time. This was somewhat arbitrary, but it allowed for an easy to deal with 8 data sets.

Before we get into the results, take a look at the above chart for a little more understanding on skew and kurtosis. The period from May 1998 through May 2004 (red line) most resembles a normal distribution with the lowest skew and second lowest kurtosis of the 8 periods. It is compared to the April 1986 to April 1992 which had the highest kurtosis and most negative skew.

Hopefully you can see that the red hand drawn mean line virtually bisects the red distribution, which would be typical of a normal distribution. Whereas the blue median line shows the largest number of outcomes just to the right or the median with more bumps on the far left (negative skew). Additionally, the blue distribution is much narrower while also having more outliers (leptokurtosis). This chart isn't indicative of anything, but will hopefully help you better visualize the numbers in the table below.

Although I would not consider this conclusive, the four most recent periods (April 1986 to present) have a higher average excess kurtosis and a more negative skew compared to the prior 24 years. Obviously this is heavily influenced by the inclusion of the April 1986 to April 1992 data, which was unquestionably extreme.

PERIOD	MEAN	STD. DEV.	KUTOSIS	SKEW
MAY04TOJUN 10	0.00009	0.01436	11.0284	0.0208
MAY98TOMAY04	0.00008	0.01319	1.7328	0.1022
APR92TOMAY98	0.00069	0.00734	7.7143	-0.4487
APR86TOAPR92	0.00043	0.01189	63.1308	-3.5728
APR80TOAPR86	0.00059	0.00889	1.4935	0.3280
MAR74TOAPR80	0.00007	0.00897	2.0750	0.2814
JAN68TOMAR74	0.00006	0.00764	2.4569	0.2278
JAN62TOJAN68	0.00020	0.00646	12.8579	-0.5348

Never the less, the table does add additional weight to the theory that increasing volatility with fatter tails is the new normal. I would feel remiss however if I didn't propose some fundamental explanation as to why this might occur.

First, globalization has increased correlations between countries, businesses, and other entities. As we saw with our own financial crisis and now the crisis in Europe, when things start going wrong they go wrong for more people. It is like a pebble falling off a cliff into a pool of water. It used to be that when something failed it would fall by itself, maybe dragging a few smaller rocks along as well causing some nominal ripples. Now, the little pebbles are chained to rocks which are chained to boulders, and the ripples are more like the waves caused when a chuck of glacier falls into the ocean. No matter the level of regulation, if you feel that helps, this interconnectedness is not going to change.

Second, electronic trading and penny increments have diminished liquidity, in my opinion. While I admit this is an arguable point, back in the day, when stocks were quoted in fractions, size would build up on the bid and the offer because they were relatively wide spreads. Even if there is a thousand bid and offered on a stock that is only a penny wide, it is much scarier to execute a large order of 10,000 shares or more. Before, you might find 10-15,000 up on a market that was $50 - $50.25, now with 1000 at each penny you are not sure where you will get the order filled and once the buying or selling begins, algorithms are likely to jump on board in the same direction putting additional pressure on the stock. Furthermore the same computers that are following the momentum are also canceling liquidity providing orders on the other side. This also is not going to change.

This is not a doom and gloom piece, this is about knowing and understanding that the markets are evolving. So keep those puts pumped, don't take relative calm for granted, and enjoy the glacial waves.

Implied Volatility and Earnings

Earnings season is upon us and is harkened by this evening's release of Alcoa, Inc.'s (AA) numbers. The question traders are asking is: What impact will the earnings release have on the price of the stock? Well there is a quick and dirty method for estimating this and it works particularly well on expiration week (which this week is for those who weren't paying attention). Take the front month implied volatility of the at-the-money options and divide by the square root of time. What?

Let's use AA as our example. AA closed on Friday at $14.39, which isn't conveniently right at a strike, but since AA options have strike in $1 increments we can use an estimated average of implied volatility for the 14 and 15 strikes on the April options. Obviously, stocks are seldom right at a strike so the simple averaging of the two surrounding strikes is a good way to go. If you want to get really fancy you can calculate a weighted average, but that kind of defeats the quick and dirty part. So the average implied volatility for these strikes as of Friday's close was approximately 65%. It's important to remember that that is an annualized number. To convert that into the a daily expectation we need to divide that by the square root of time, which in this case is the number of trading days in a year. Generally there are about 250 trading days per year and √250 = 15.81, but 16 is good for quick and dirty and is also much easier to remember. 65%/16 = 4.06% or roughly $.58. Note that there is no directional implication.

Now there are three important points to keep in mind. First, the calculated value is based on Friday's close, a better estimate can be made when as we approach today's 4 PM close. In other words do the math again around 3:30 PM, since there is likely to be some active trading in these options ahead of the earnings report which will obviously impact the implied volatility value. Second, know what the number means. Implied volatility is the boundary of the first standard deviation, meaning that if you could repeat this particular earnings release 100 times, 67 of those releases would impact the stock by no more than + or – 4% and 33 of those releases would result in a gain or loss in value of more than 4%. Finally and probably most importantly, is the trend in volatility. It's a good idea to know whether both actual and implied volatility has been rising or falling over the last several days to get a feel for what expectations are out there. Unless you have been following the options for a few days it is hard to know this, however you can get a pretty good estimate for an individual stock at www.ivolatilty.com . But it is really about the last minute action ahead of the numbers, so looking at the 3:30 PM calculation compared to Friday's calculation may be sufficient.

In summary, the average implied volatility of the at-the-money strikes divided by 16 will give you a range of potential impact of the earnings release. While we are using this method for estimating the impact of an earnings event, you can use it whenever there is an impending event that could have an immediate impact on a stock.

Steady as she goes

A quick follow up on the Predictive VIX post:

I couldn't help noticing that on Monday, Tuesday, and Wednesday of last week that the VIX increased marginally (less than the 5% level that was examined in the original post) while the SPY also had marginally positive moves from close to close. So, since I have the data, I thought I might take a look at the occurrence of multi-day positive VIX and positive SPY over the March 1995 to present range.

Generally speaking, it proved to result in positive returns over the next 120 day period (there was no look back condition). On average, the SPY was approximately 4.2% higher 120 days from the event. However, results on a shorter 60 day forward looking period were much more mixed yielding an average return of 1.1% with a number of 1-3% down periods. In essence, there is the suggestion of a range bound market over the next 6 months.

So although the VIX is trading near pre-crash lows there is certainly the potential for it to continue lower since recent realized volatility in the SPY is considerably lower than the expectations reflected in the VIX. This would be consistent with a range bound market over the next 6 months. Additionally, as of last week, the market is still pricing in expectations of increasing volatility during 2010 (see Implied Volatility - Part 1), allowing the market a "wall of worry" on which to climb.

It should be noted that there were 3 multiple up/up events that produced significantly negative returns over 120 day periods: 1) late August 2000, while the in the throes of pricing in a bursting internet bubble, producing a -16% return, 2)December 2007, as the market began falling from its peak, yielded a -10% return, and 3) May 2008, when the crisis was really beginning to take its toll, resulting in a -36% return. Those were clearly unusual periods and current market conditions do not really resemble those times. Basically, it seems likely that the broad market is likely to find itself up or down 3-5% over the next 6 months with decreasing volatility. That is hard for me to say because I personally believe that there are a number of factors that could negatively impact world markets during 2010, but the numbers suggest that if there is an impending disaster on the horizon, it may take some time to surface.


Oh and HAPPY ST. PADDY'S DAY!

Implied Volatility – Part 1

While I intended to discuss the impact of implied volatility and time on the strategies we have talked about so far, I felt that some general information on implied volatility might be useful to cover first. I have included a chart that depicts estimated implied volatilities for near-the-money strikes on the SPY. These are the average of the call and put implied vols for the available 2010 months (LEAPS are not shown and there are many more strikes available in each month). What stands out is that the March values exhibit the classic "volatility smile" indicating that the nearest-the-money strikes are "cheapest" while the out-of-the money strikes are significantly more expensive. The April values are beginning to "smile" but still have significant skew, like the further out months. Skew in this case is characterized by higher volatilities in the downside options with the implied volatilities diminishing as the strike prices increase.

To understand this it is important to grasp what implied volatility represents. In theory, implied volatility reflects future expectation of the underlying's actual volatility. In an aggregate sense this is true in that if implied volatility is rising across all strikes then the market is pricing in expectations of increasing actual vol; if it is lower across the board then actual vol is expected to be lower over time. Implied volatility is really, however, a "fitting" of the volatility component of a theoretical options pricing model. In other words, based on supply and demand factors as well as time to expiration, market makers adjust their implied vols to reflect fair value or the approximate mid-point of a given option's bid/ask spread.

You might ask, "What's the difference?" An aggregate metric like the VIX can be used as a general measure of expected volatility (at least over the next 30 days), but the "fitting" of the volatility component is what causes smiles and skews to exist. Longer dated option trading (beyond the front two months) tends to be dominated by hedging, selling covered calls or buying protective puts, thus a downward sloping skew exists. As a given expiration approaches, however, a smile begins to develop since the vega of the out-of-the-money options diminishes causing changes in an option's price to have a greater impact on implied volatility. Natenberg** defines vega as: "the sensitivity of an options theoretical value to a change in volatility", but the caluclation goes both ways. As an example, let's say that a near-the-money option is fairly valued at $1.00 and has a vega of .05 and an out-of-the-money option is valued at $.20 with a vega of .01. Assuming no change in the underlying, if both options increase in value by $.05 then the implied volatility of the first option increases by 1 point, whereas the IV of the second option increases by 5 points. So as time passes and out-of-the-money options approach parity or zero, small price changes can dramatically impact the implied volatility of those options, thus the skew develops into a smile.

In addition to the time factor, order flows begin to change as expiration approaches. More speculators show up in the front two months looking to profit from a quick move in either direction, causing out-of-the-money demand and thus implied vol at those strikes to rise. Also, hedgers who were selling further out calls are now buying those calls back and selling new longer dated calls. For some reason, people are less likely to sell out their put protection, but that is another story.

In sum, over time it is reasonable to expect skew to become a smile. Is this a hard and fast rule? No, because option markets, like the underlyings they are derived from, are fluid. None the less, these points are good to keep in mind when considering the appropriate time to expiration and strike of a potential options position.

Hope I didn't cause too much confusion by getting overly technical today and I will start discussing the impact on strategies very soon.

**Sheldon Natenberg, Option Volatility& Pricing.