Most of my thoughts about bottoms recently have focused more on the beach-bathing variety I’m starting to see as spring creeps back to the beach. But since everyone is wondering about the market bottom, I’ll bounce the proverbial quarter off of it and see how high it goes.

I’ve talked before about market “gravity” and price clusters that attract future bids. It’s based on basic auction theory: the price that attracts the most bidding represents the best guess at the value of an item even if people who really want the item badly (or are ill-informed or excited) will pay more (or in reverse auctions, less).

I’ve advanced my work on the idea by taking to the computer and working with the R statistical platform to analyze markets from an auction theory perspective.

So how does the S&P look in this context?Well, I thought it made the most sense to take a snapshot as of the most recent major high, which in this case is 10-17-2007. I went back to an arbitrary 01-01-2005, just enough to make sure that I had prices included well below the recent low. Below is the output after the close on 10-17-2007. The right-hand panel is a display price on the x-axis and density on the y. Think of density as proportional to the popularity of a price. The dotted lines mark the peaks of detected clusters. These are the prices that stand out as most important. The left-hand panel attempts to delineate the clusters. Think of it as the current sphere of influence on price. The same dotted lines appear here. Each black bar corresponds to the range of prices around each dotted line price. When price is in a certain bar, it is most likely testing the value of the associated dotted line. Obviously, there was an attempt to make them mutually exclusive for clarity, but that in no way means they are. Just because price may fall in one cluster doesn’t mean it has no probability of belonging to another nearby cluster.

Now look at the red line. That is the recent low. Let me reiterate here that these charts were produced with data only up through the market’s high in October. This fingerprint was already in place as price retraced. While that sinks in, look at the solid vertical line. That is the Tuesday rally’s close. Notice that while price move into another sphere of influence, it is still contained in the lower mass of price density. I’ll leave the forecasts to others but a pull back to the read line would be like warm apple pie to the market. It’s familiar and “safe”. Maybe what the market wants is some time to move sideways in this general area for a while.

I’m facinated that when the fear hit, everyone ran right to the most popular price of the last three years. In some way that makes me feel comforted. Could fear lead to a rational conclusion? Was it mere chance? Maybe. This is just one example pulled out of thin air. But the fact that I didn’t know what I would find but that I knew I’d likely find exactly what I did leaves me a bit breathless…in a good way.

While most would agree that the stock market has certainly been more volatile this year, putting it in perspective with the long term trend shows that by at least one measure, the S&P 500 was less volatile this year than its long term average.

The chart below summarizes the average absolute daily price change in the S&P 500 by year. In 2007, the average worked out to 72 basis points, which means that, on average, the S&P 500 had a daily move (up or down) of 0.72% versus an average of 0.75% since 1928. While this year was more volatile than the last three years, prior to those years, the last time the market was this ‘placid’ was in 1996.

While this is a very good point and good analysis, I decided to do a little of my own analysis and found… I don’t have as much data as Bespoke does. Going back to 1950 (the furthest back I can go with free and easily available data from Yahoo Finance), the average daily change is 0.62%. That makes sense, considering that I couldn’t include the 1930s when volatility was so high.

Despite that, I forged on to find a few interesting things… if I look at the 30 day moving average of daily price change, we get a slightly different picture. The average volatility for 2007 has been skewed lower from the low-volatility of the first 6 months of the year. Outside of the last three years, the last time volatility was this high/low was early 2002 when the markets were mid-patriot rally.

Similarly, it is worth looking at volatility on multi-day time frames, such as weekly price change or monthly price change. As we go to longer time frames, the changes look more similar to the first chart that Bespoke published.

There are several parameters that need to be optimized and, since it’s non-linear, it requires some major computation power. All those parameters make it more difficult because, during fitting, you happen upon local minima that aren’t the real best minimum. So you have to run the optimization several times with different starting seed values and hopefully converge on the answer.

So enter Java. I wrote a program that would read in market data (S&P 500 since the ‘03 bottom) and try to perform a fit to the model. If no good fit is found, you can assume that perhaps we aren’t in a singularity pathway at the moment. If you do get a decent fit, then it tells you  a bubble might be forming and gives you the estimated time for the singularity. It should be noted that by singularity, I don’t mean the big one that is talked about in technology circa 2050. I just mean the end of any greater-than-exponential growth path. In other words, mini-singularities happen, like in 1987 and 1929.

My code is very crude and by no means am I an expert in curve-fitting but I figured it would be fun to see if I could get any good fits and if the various minima would agree on a general time that a major bubble and crash might be likely. Of course, crashes can occur any time but if the precursors of this model are present, you can get a bead on it. Sornette points out that the model gets more accurate as the real time approaches. Modeling too early will give no good results. Also, all those local minima might disagree wildly, leading to inconclusive results. What you are watching for is when all the answer are starting to flash red lights around the same time.

 So is yesterday’s big drop a precursor to a major crash? Has another 1987 started? Well, the answer is: not by Sornette’s model at least. Nothing about the rise in price from the 2003 bottom would lead one to believe that a exponential, log-periodic bubble is in place. The critical times of all the decent fits I could create were years in the future and also all over the map. Here is a chart showing the market data (denoised and coverted to log values) overlaid with the best model:

 I’ve extrapolated the model out to the estimated singularity time. As you can see, the best the model could determine was that we are in an early stage of a bubble at best. Please understand that the extrapolation is not saying that that is the likely path. Only that if the market does play out as a bubble, the behavior since 2003 doesn’t reflect it yet. Interestingly, this is in agreement with most value models that show no significant overpricing in the stock market like we saw in 2000.

Crashes don’t need these precursors to happen nor are they guaranteed when the precursors are there (though it is highly likely to a significant degree). Lack of evidence doesn’t equal evidence of lack. But it does mean that a bubble in the mathematical sense that Sornette lays out isn’t reflected in today’s market as far as my crude code can tell. Take it for what it’s worth.

Now take a look at the US stock markets. The S&P 500 has a long-term Sharpe Ratio of less that 0.5. To put this in perspective, it’s generally considered “good” if an investment strategy has a Sharpe Ratio near 1 and hopefully greater. What this means is that relative the return one can expect from stocks, the volatility is way to high. Now this sounded like an oversimplification to me so I decided to perform a little test. I took the full set of SPDR ETFs, representing various sectors and build a sort of UP-like portfolio out of them. It was nearly impossible to create a portfolio that could manage a decent Sharpe Ratio and consequenly a resonable yield/drawdown ratio. This was based purely on dividend yields mind you and so doesn’t factor in market timing. But I think the point remains valid because any investment can be made good with brilliant timing. I wanted to compare market-neutral strategies so that I was truly taking what the market is willing to give me no questions asked: yield and volatility. Yield in currencies are interest rates and yield in stocks are dividends. So the conclusion seems to be that unless you are a good timer, stocks are an uphill battle with risk. Certain investments have volatilities closure to their yield than others. I just think it’s amazing that something as popular as stocks have so steep a profile. Stock volatilities are almost always double digits and often over 20% long-term. Most currencies top out at 10-12% historically. Yet yields are similar. All else being equal, the choice seems obvious…for now. Things can change tomorrow and I’m not trying to tout currencies above all else. I’m just trying to point how what’s important when thinking about your portfolio of whatever.

Of course, one might feel they have a better sense of timing in one versus the other and that shouldn’t be discounted. But if anything, stocks are far from being the more “conservative” everyman investment they are made out to be, while more exotic things like forex seem to be much more suited to the risk-averse if handled with some basic care. It’s the overuse of leverage (see Jason’s posts on hedge funds), which you don’t have to use, that deserves the blame for this rep. This is just one facet of any investment decision so I don’t mean to overgeneralize. In fact, I’m reminded of the old adage that the greater the risk the greater the reward. I’m just going out on a limb to see what fruit I might find.

alpha = returns of investment  –  (volatility of investment / volatility of benchmark) * returns of benchmark

What this is is a measure of relative performance or advantage.  To stand higher than a benchmark, like an index for example, an investment must not only provide higher returns but do so with proportionally lower volatility.  If an investment doubles the returns of the index but also doubles the volatility (i.e. risk), then the alpha is zero and the investment provides no real advantage.  So the conclusion is seek alpha at all times.  Sounds easy enough, right?

Finding alpha is more important than ever, but it hasn’t become any easier.  And the cost is rising.

So investing is hard work.  But alpha is what you should work hard to find.

To begin, grab yourself some data.  Good ol’ OHLC data.  Any timeframe will do as we have been discussing in the “You Are My Density” thread.  I realize that daily data is by far and away the most commonly availble free data so don’t be afraid to use it even though I recommend 30 minute data or hourly if you can get it.  Now, throw out the open and close.  Well, don’t literally throw it out but you won’t need it to make the chart.  What you are left with are a bunch of high-low ranges for each time period.  How many time periods?  I usually don’t like to work with less that 250 periods and I avoid after-market periods and weekends data because they aren’t active enough to be given the same weight (or any at all).  In addition to the 250 period chart, I like to make a 500 and a 1000 period chart for reference.

So if you have a vertical column of highs and lows, create a horizontal group of buckets for your chart price axis.  If you have enough space, make a bucket for every possible price tick from the highest high to the lowest low of your data set.  Or otherwise make some bigger buckets, like every 5 or 10 ticks.  Now put a ”1″ in every bucket that the high-low range for that period covers.  For example, if the range of some market is 70-65 during one time period, and you have buckets every 0.5 then you’ve put a 1 in the 65.00, the 65.5, the 66.00 etc. all the way up to 70.00.  Do this for each time period then add up all your columns into a histogram.  That’s it.

Here is an example worksheet. I used daily SPX prices from Yahoo!  I figured using this data would be appreciated since it’s the most readily available.  Across the top are the buckets I created, one at each whole number.  The rest of the worksheet is the 1s and 0s used to flag the buckets at each time period.  The very last row is the sum of each column and is the final histogram that I charted.

Now you should be able to create these for anything you wish.

P.S. I included an exponentially-weighted version of the chart on the second tab if you want to play with that concept.  I tend to do this to my data often.  You can play with the weights. If you want, say a 200 day exponential weighting then the weight is 2/(200+1) or ~0.01.  This would go in the formula in place of the 0.008 I used (~250 days).  You could also just right the formula to accept some input from some cell but I was in a hurry.

Big Mac Index

Now, do you want fries with that?

First, some supplies: 

MCSim Excel Add-In: Just drop this in Application Data\Microsoft\AddIns under your Documents and Settings directory.  Then, in Excel, Tools/Add-Ins… to install.

Resample Demo: A demo worksheet to show how to create a random sample from your raw data.

To begin, look at the column labeled Resampled in the Excel demo.  Here you can see a formula that you can use to create a random data set with replacement from your actual raw data.  Note that the data in the first column are meant to represent PL numbers of some system or daily stock price changes.  After resampling, I can create a series that represents the chart for my phantom stock or whatever this is.  Now every time you press F9, you see a new random sample based on your real data.

Now for MCSim.  Select the cell for the max drawdown.  Go to Tools/MCSim…(assuming you installed it), change the reps to something low like 100 and click Proceed.  After a while, you’ll have your results.  You’ll want to use more reps when you want to do a serious analysis but it will take much longer.  Your results will show you the average, SD, Max and Min of the variable you measured, in this case the Maximum Drawdown.  The answer I got (your result might be slightly different with so few reps) was about 26% with a SD of 12.6%.  The max was 88%.  So quite a volatile set of data but you can begin to see how this is useful.  This would tell you that if you wanted to limit drawdowns to no more than 10% you’d probably want to trade this system or stock at about 40% of the size that was used to create this data set and probably less since you should consider the SD too.  Say that you wanted to base it off of +2 SDs.  You’d then need to limit things to 20% of the size (10/(26+12.6*2)) to control your risk appropriately.  In fact, I included a cell called Scale that you can use to adjust the leverage of the sample series and then rerun your MC simulation.  Think of it as the percent of equity you’d use to purchase this stock.

Hopefully, you can imagine many uses of the MCSim.  Basically it works for any formula result you want to test as long as somewhere on the page, the formula result is linked to a RAND() function so its value changes on recalculation.  I think this tool is invaluable and should be used by investors of all types to help shed light on the all important question of risk and position sizing.  It’s not a perfect tool and doesn’t erase risk but it certainly helps you understand it.

P.S.  There is another Excel Add-In that contains both a resample function and and Monte Carlo simulator among hundreds of other obscure things called PopTools that seems very good.  Give it a try as an alternative to the above tools.  I thought it would be easier to share MCSim but PopTools is probably a more powerful tool and having a Resample function that doesn’t require writing that complicated formula I gave you is nice.  It’s also light years faster, completing the 100 reps in less than 1 second.  Also, Jason, the PopTools simulator will allow you to put in a test criteria like “How many times did it exceed 10%” which should be perfect for your drawdown analysis techniques.

IV = 40 x p / SQRT(t)

where p is the price of the option (as a % of the underlying) and t is trading days until expiration.

That’s all.  Forget all the complications of the million dollar formula as this gives you all you need.  However, it only works at-the-money, but then again so does Black-Scholes.

So what if the price is between strikes so there is no at-the-money?  Well here is another spiffy formula:

CM = 1.04 – 0.04 * R

where R is the ratio of the more expensive to least expensive options that are nearest to the money.  Just multiply your option price p above by CM and there you have it: the price converted to the at-the-money equivalent.