Fractal Market Analysis (long book review for math types).
Wiley by Edgar E. Peters
Take the day over day percentage return (or week over week or
etc.) of the Dow or just about any other market. Make a histogram
plot of them. You expect to get a gaussian (normal curve)
distribution. This expectation is a consequence of the law of large
numbers: If you take the sum of n independent, identically
distributed random variables, the sum gets closer to a normal curve
as you increase n. But people's decisions to buy and sell are not
independent. Instead they are correlated. The correlation makes
the big gains bigger and the big losses bigger. When plotted on top
of a normal distribution with the same standard deviation, the market
return plot has a higher peak at the average return, and is higher for
the far outliers. On the other hand, the market return has a dearth of
returns with values around a standard deviation away from the
average return.
With a plot this would be easier, but I don't have a scanner handy.
I'll try to describe with words:
Returns above or below 2.5 standard deviations of normal are more
common than expected. These are the extremely rare (BK) returns.
Returns between 2.5 and 0.75 s.d's above or below normal are more
rare than expected. These are the second most likely returns.
Returns between .75 below and .75 above normal are about 1.5X
more common than expected. These are "average" returns and are
the most likely.
When I first looked at the graph of the two distributions, I thought
that they had simply calculated the wrong standard deviation for the
market return graph. If they replaced their calculated standard
deviation with a smaller value, then this would have the effect of
broadening the market return graph. This broadening would lower
the peak at zero standard deviations, while increasing the frequency
of returns in the range of 2.5 to .75 above or below normal. Its a
shame his book doesn't include an example of such a scaling of the
standard deviation. But such a scaling, while it would better match
the values within 2.5 standard deviations, would leave a worse
difference between the normal return and market returns at the
outliers outside 2.5 standard deviations (more or less).
Now some consequences of this regarding fair options pricing. The
usual model for options pricing is the Black-Scholes equation which
assumes normal distribution of returns. When you calculate the
implied volatility of an option, it is probably using those equations.
The guys who make the markets in options are not stupid, (One of
them, Steven? Thorpe, a professor of mathematics and economics
at U. Cal Irvine, made $93M in options back when I was a grad
student studying partical physics there. He also wrote a book on
counting cards in black-jack called, Beat the Dealer, if I remember.)
so they charge more for the time premium in the outlying options than
you would expect from the implied volatility of the options struck near
the security price. That is, the outlying options have a higher implied
volatility than the inner ones. I'll verify this with numbers if I ever get
the CBOE options calculator to work decently.
But market makers for those options have to provide both a selling
and buying market. On most far out of the money index options on
indexes (indices) you will see relatively low spreads (for options) but
zero open interest. Thus no market exists and the MM is determining
their prices completely theoretically, presumably hedging using deltas
off other options or the underlying future market. Those close
spreads are a big hazard for him. If the price is too high, then the
rest of us can make a profit writing them to him. If they are too low,
we can make a profit buying them. I think he has to be tying their
prices to the volatility that the market chooses for the more heavily
bought options. So the MM has to estimate volatility. Shouldn't be
too much of a problem., right?
But it gets interesting when you examine long term trends in market
volatility. Historical market volatility is where the MM gets an
estimate of current volatility and thus what he needs to charge for
implied volatility. Volatility is known to be "mean reverting." That is,
rises in volatility tend to be followed by declines and vice-versa. But
that doesn't mean that volatility is tame. Take a look at the chart on
page 144. S@P 500 volatility, annualized 1945-1990. Usually 10
percent, it occasionally gets to 30, and at a spot looking to be
perhaps some time in 1987 (guess) hits 95. That is 95 percent
standard deviation per year (brutal).
The author continues giving some evidence that market returns
actually have no long term average standard deviation. That is, they
bounce around enough that no matter how long you watch, you
never get a good estimate. This should come as a mild shock to
those of you familiar with probability and statistics. To check for this
we could use a set of daily dow jones closing values from 1880 to
say 2880. But unless medical treatment improves a whole lot during
the next 100 years, I won't be around to benefit from analyzing that
long a data set, so I'll leave the question to the theoreticians.
In any case, (not in the book) we can make some estimates of the
probability of sufficient market moves to the downside to bring those
outlying puts into the money. Over a period of 77 years, count the
number of quarters that the DJIA made a move 25 percent below
the previous quarter's low. I've just got the (infamous) Value Line
Long Term Perspective chart in front of me, and the ones I find are:
4Q29, 4Q30, 2Q31, 3Q31, 2Q32, 3Q37, 2Q40, 3Q62, 3Q74, 4Q87.
That's at total of 10 occurrences out of 304, or 1 out of 30. You can
fudge the numbers whatever way you want to. Try ignoring drops
that occured during bear markets, or only dividing by the years where
the dow was more than average over book. This will put that
probability higher or lower. I personally think its a lot higher. We just
don't have enough data to figure it out. In any case, choose your
own probability, we just don't have the millenia of market data
required to put an estimate on it. And anyway, when 2880 finally
rolls around, they'll be saying "But this time its different!". So choose
your probability (my guess is 15 percent) and pick your index, and
buy puts way out of the money. But don't come complaining to me
when the crowd runs the market higher, you knew you were
gambling big time.
As for me, I am quite certain that this market will end with a bang.
The market of the 60s ended with a whimper, but it never got to the
multiples of book we are at now. I swear to keep throwing money at
puts until this market is undervalued, or certain other events happen.
Like I lose my source of income : (.
But on to the fractal market hypothesis, the subject of the book:
Hypothesis: There are two types of investors who trade in stocks.
The day traders work on the day's price, while the long-term investors
work on longer trends. Day traders move the market prices up when
long term people start loading up on stocks, down when they start
dumping them. Markets remain stable only when all sorts of
investors participate with many different time horizons. "When a
five-minute trader experiences a six-sigma event, an investor with a
longer investment horizon must step in and stabilize the market. The
investor will do so because, within his or her investment horizon, the
five-minute trader's six-sigma event is not unusual. As long as
another investor has a longer trading horizon than the investor in
crisis, the market will stabilize itself." Now the fractal market
hypothesis says that the risk structure of investors with different time
horizons is the same. That is, it is scaled appropriately to their time
horizons, rather than with the simple square root of time that standard
deviations of normal random variables do. That is, when you plot
histograms of market returns for different time periods (i.e. 1-day,
5-day, 30-day dow jones returns,) you get graphs that look similar.
Always with those high peaks and fat tails.
Before I go on, this makes some sense to me, though I would
suggest reading the original if my explanation is unclear. In fact, as
a bottom feeder, stabilizing the market is what I do. When I see a
stock drop 90 percent, I start salivating, and looking through Edgar
files. I'll buy before the stock has made an obvious bottom. For
me, a stock panic is no big deal. Every day I hope for one. And I
don't care what industry the stock is in. But to the guy who's bot a
stock at 50 times earnings, and then has it collapse to a PE of 5,
this is a major disaster. He is very glad when I show up and keep
the stock from dropping further, though he usually seems to resent
it. Oh well, try to lend a hand, and look what they call you!. This is
dangerous habit, but I guess I've been lucky so far....
Anyway, his take on the crash in '87: "Prior to the crash of October
19, 1987, long-term investors had begun focusing on the long-term
prospects of the market, based on high valuations and a tightening
monetary policy of the Fed. As a result, they began selling their
equity holdings. The crash was dominated entirely by traders with
extremely short investment horizons. Either long-term investors did
nothing (which meant that they needed lower prices to justify action),
or they themselves became short-term traders, as they did on the
day of the Kennedy assassination. Both behaviors probably
occurred. Short-term information (or technical information)
dominated in the crash of October 19, 1987. As a result, the market
reached new heights of instability and did not stabilize until long-term
investors stepped in to buy during the following days."
If this is the correct way of looking at things, then a market top
would be correlated with an increase in volatility. This is associated
with the long-term investors being more aware of the lack of
fundamental valuation support for their positions. I get the impression
that fundamental investors are being crowded into fewer and fewer
stocks. We are buying the same things. The herd is taking over a
larger and larger percentage of the market, and every experienced
investor I know thinks the market is over valued. Most of them are
at least partly invested in it, but they give off a feeling of anxiously
looking into the faces of their fellow dancers, wondering if they will
be able to get out the only unlocked door after the herd notices that
the ballroom is on fire. Maybe the herd will only notice after the first
year goes by with the dow increasing less than the checking
account interest rate. But it is inevitable that the herd will eventually
notice. If someone has data for long term volatility in stocks, please
share it. People always talk about volatility at market tops, but I
suspect that most of it disappears when you graph on a log scale.
But back to the book.
Another set of fascinating plots is that of standard deviation versus
time period between measurements. That is, the horizontal axis is
the time-horizon of the investor, the vertical axis is the standard
deviation (risk). He plots these on log-log paper so that powers
become straight lines. For a gaussian process, the scaling is to the
square root of time, as in the Black Scholes options estimate. But
volatility actually grows a little faster than normal for time periods
below 4 years. For time periods longer than four years, volatility
decreases again. The implication is that the buy and hold a long
time types do achieve reduced volatility. In other words, over the
long-term, stock returns are somewhat deterministic. That's exactly
what all those mutual fund buyers would like to believe. Close your
eyes and buy a stock.
But mutual funds are an interesting beast. The person who supplies
the money is not the same as the one who chooses the investments.
Consequently, even though the time horizon of the man who
purchases shares in the mutual fund may be quite long, to
determine the fractal market hypothesis consequences of the mutual
fund, you must instead analyze the activity of the man who runs the
mutual fund. His time horizon is definitely longer than a day trader's,
but I don't think it is as long as 4 years. The reason for this is that
the turn-over on most mutual funds is a lot faster than 25 percent
per year. (Is this true?) The mutual fund's stock picker isn't really
a long-term investor. He is instead trying to maximize return on a
quarter by quarter basis, and thereby maximize his own current
income. He can turn some percentage of his stock holdings to
cash, and he can always choose which stocks to invest in. This
means that he can sell when he likes. Most people are inclined to
be doers rather than watchers in that job, and I think that they slosh
money around a little too much. That's at least partly why the index
funds regularly clean their clocks. At least the indexers are
continually driving up the same shares.
So buy/read the book already. You will, no doubt, glean more
insights. Other book suggestions?
-- Carl |
