Dec

12

An interesting thing to speculate about over recent days in the S&P is the abysmally small ranges:

date   high    low        range                                                          

12/11 1109  1101       8                                                             

12/10 1106  1099        7                                                             

12/09 1097  1086        11                                                             

12/08 1097  1088        09                                                             

12/09 1110 1100        10    

To put it in perspective, note that a range of less than 8 has occurred about 1/14 of all days, a mere 60 times or so in last three years. But it has occurred back to back 15 times. Thus, given that a range of 8 or less occurred yesterday, it's one in four to occur the next day. But on any day, the chances of a range of 8 or less is a mere one in 14. Low ranges tend to beget low ranges. Indeed the stand deviation of the move from close to close the next day is a shockingly low 3.5 on the day following a range of 8 or less. This compares to a standard deviation of 20 for all days during the sample.

Also note that five days have gone by without a range exceeding 11. This has occurred only 28 times in last several years. There appears to be no simple formula for the relation between the range of a normal distribution and its standard deviation. The range is important for the feeding relations between the higher species in the chain and the lower species. Without a big range, there is little incentive for the public to do the wrong thing and be stopped out of their positions. A small range therefore is generally antithetical to the market because it doesn't allow the proper amount of losses by the public to cover the big overhead. Indeed, the expectations following runs of small days during the last 10 years or so have been considerably lower than the normal tendency to declines of 1 a day that has afflicted us during these years.

The ideas should be generalized to individual stocks. What is the relation between ranges and subsequent return for a individual stock? The matter should be considered relative to the individual stock's history, taking into account the market's range during that period. The relation between range and the usual size of a move, nay swing, must also be considered. Holbrook Working used to show that commodities were non-random by considering the daily range relative to the transaction size. Such numbers should be updated with simulation for actual changes in price in current days with reference to all of the above.

Michael Cohn writes:

Is it worth looking at the samples with and without SPX and equity-related options expiration, which is 12 weeks per year, or perhaps the smaller set of the SPX calendar expirations which is four weeks out of 52? I have often felt most misled by option expiration and perhaps it is worth quantifying how misled I can become.

Victor Niederhoffer adds:

There appears to be no simple formula for the relation between the range of a normal distribution and its standard deviation.

Alex Castaldo replies:

For a Brownian Motion starting at zero, with no drift and with variance sigma^2 per unit time,

the expected value of the range (high minus low) during time [0,T] is given by

E[R] = 2 SQRT( 2 sigma^2 T / pi ]

this is equation (48) of the paper below, which credits Feller, 1951, and gives a more general formula for non-zero drift.

For example for 1 day (T=1) the formula reduces to E[R] = 1.6 sigma, if we further assume a daily standard deviation of 1% we get an expected daily range of 1.6%.

==Reference==

Magdon Ismail et al.: The Sharpe Ratio, Range and Maximal Drawdown of a Brownian Motion (2002)

How well does this approximation fit the data?

For the last 100 days the standard deviation of the open-to-close move is 0.825%

The average range (low to high) is 1.36%

These are in the ratio 1.65 to 1, close to the theoretical 1.6.

[the OHLC used was the pit-traded S&P adjusted contract]

Alex Castaldo adds:

For the last 100 days, the daily returns on S&P futures have a standard deviation of 1.05 %

the daily ranges have been on average 1.36% of the previous day's close, substantially below the 1.68% that would have been predicted by the formula.

[the OHLC used was the pit-traded S&P adjusted contract]

Alston Mabry writes:

Here's is a comparison using SPY, of the most recent 100 tday period with a previous period:

23-Jul-09 thru 11-Dec-09
sd of daily C-C log%:  1.253%
expected range C-C (sd*1.6):  2.01%
actual O-C range:  1.39%

9-Oct-07 thru 17-Mar-08
sd of daily C-C log%:  1.302%
expected range C-C (sd*1.6):  2.08%
actual O-C range:  1.84%

Bill Rafter comments:

Sorry for the width of this picture, but it shows the relationship over relevant history. The picture of course does not *prove* anything, but is food for future thought.

William Weaver Jr replies:

What's interesting about that is the regime shift. Note how the blue line has greater amplitude than the red prior to MAR09. It would be interesting to take the absolute value of each and then smooth the result.

Maybe:

252 ex pave abv data

One could decrease n in the original standardization process (21?) as well as change n in the above.

My hypothesis is when realized volatility is gt implied volatility markets are falling. The problem then becomes lag.

Craig Mee comments:

I think what is interesting here on a vanilla bases, is there is "no fight in the market" the cowboys have gone home for the winter. Market has had good extension, we see reasonable employment results (at least on the surface), for the first time in x, why would people want to duke it out?. With the market seasonally bid at this time of year , a general upward bias holds, however without any real pullback in Sept/Oct, and the market already having squeezed say a quarter off the lows, then found new buyers , in the next quarter…is sitting precisely where its showing its content ie 50% off the lows. Like a happy boxer back from the brink, where all fight has been taken out after a few heavy rounds and the latest rounds have mostly been done on the ropes and hes now taking a breather in his corner. (Its looks like Vic's uncanny tight ranges are directly a result of the unprecedented flogging which has proceeded it).What next? well without any volatility at these highs, the expectations for a major reversal, even though an expansion is on the cards, one would expect would be somewhat limited.

Jeff Sasmor replies:

Or maybe anyone who has tried to short has been so badly burned over and over again this year by the incessant bid on any big dip, that, like the dead parrot of the eponymous Python sketch, they're all shagged out after a long squawk.

Lately most days after the morning session the action turns into barbed wire. The talking heads can't even figure it out as one rep after another from one oracular brokerage firm or another appears on CNBC or BBG TV.

In many different areas (CRE deterioration, manipulation of accounting standards & timing, govt supplied macro figures, bailouts, one time subsidies, warping of contract law as it relates to creditors, whistling past the graveyard w.r.t. FASB 166/167 and the failure of highly touted mortgage modification efforts), the year could be labeled "The emperor has no clothes but who cares?" Particularly when there are bigger supra-national fish to fry.

I was always under the erroneous impression, that in prior recessionary periods when the likes of Citigroup were considered distressed, they were but not terminally on a mark-to-market basis. With new evidence of their extremely dire conditions during those less challenging times, it becomes obvious how badly underwater so many fin. institutions are currently and therefore structures in place during normalized conditions will be suspended. Witness FASB's recent attempts to shake themselves of the demand that they align their valuation rules to governmental regulatory actions & guidance. Some are just increasingly unwilling to toe the line anymore.

Vince Fulco writes:

In many different areas (CRE deterioration, manipulation of accounting standards & timing, govt supplied macro figures, bailouts, one time subsidies, warping of contract law as it relates to creditors, whistling past the graveyard w.r.t. FASB 166/167 and the failure of highly touted mortgage modification efforts), the year could be labeled "The emperor has no clothes but who cares?" Particularly when there are bigger supra-national fish to fry.

I was always under the erroneous impression, that in prior recessionary periods when the likes of Citigroup were considered distressed, they were but not terminally on a mark-to-market basis. With new evidence of their extremely dire conditions during those less challenging times, it becomes obvious how badly underwater so many fin. institutions are currently and therefore structures in place during normalized conditions will be suspended. Witness FASB's recent attempts to shake themselves of the demand that they align their valuation rules to governmental regulatory actions & guidance. Some are just increasingly unwilling to toe the line anymore.

Jim Sogi comments:

Talk about low range. How 'bout today's 6.25. Lowest in last couple years. Reminds me of water trying to boil. Takes quite a bit of added energy to to change states at the boundary.


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