So what's Bollinger?

Following Mr. Bollinger, we do this:

Plot the price of a stock, each day, for the past umpteen months.
Pick a number like N = 20, then, for each day, compute the Standard Deviation and Average for the previous N days.
Pick a small number like k = 2, then, for each day, compute

Upper Bollinger = Average + k (Standard Deviation) ^* and
Lower Bollinger = Average - k (Standard Deviation).
Plot Upper Bollinger and Lower Bollinger along with the stock Price.

^* Standard Deviation² = SD² = (1/N)Σ (P_k - A)²
where there are N stock prices, P_k (k=1 to N), and A = (1/N)ΣP_k is their average and SD is the Mean Square Deviation between the prices and their average and it can also be computed like so: SD² = (1/N)Σ P_k² - {(1/N)ΣP_k}² namely the difference between the average of the squares and the square of the average ... nice, eh?

You'll get something like this:

where the Price of the stock seems to bounce between the Bollinger bands.
(I should mention that Mr. Bollinger picks N=20 and k=2, but we can pick any numbers ... right?)
Anyway, (if you have a great imagination) you might think that when the stock Price crashes thru' the Upper Bolli-band, we should SELL, and when it drops below the Lower Bolli-band we should BUY.
However, when it crashes thru' the Upper-B it may keep going and who'd want to sell then? So maybe we wait for it go above then drop back below the Upper-B ... then we SELL.

Now some folk wait for another piece of data to indicate a SELL (besides going above then below the Upper-B). That's the (are you ready for this) Relative Strength Index (or RSI) which measures the percentage of times when the stock Price increased over the past N days.
Because it's a percentage it goes from 0 to 100 and when the stock Price goes above then below the Upper-B (that's part of our SELL signal) and, in addition, the RSI is at least 70% (meaning the stock increased at least 70% of the time over the past N days), then we conclude that the stock has been overbought and we should SELL.
Uh ... it's not really we who conclude but those who play with Bolli-bands.

Anyway, we can also consider a BUY signal to be when the stock Price drops below the Lower-B then rises above again ... AND the RSI is less than 30% (meaning the number of increases over the past N days is no more than 30%).

Of course, the 70% and 30% are arbitrary.
Here's a graph of what the RSI might look like ... and the levels 60% and 40%

For example, things might look like so:

If Bolli-bands don't improve your financial health you might want to try band-aids ... and/or Tylenol.

P.S. I have it on good authority that Bollinger himself does NOT think that using Bollinger bands and RSI as described above is a good idea. Indeed, John Bollinger's words are:

"Perhaps you would be so kind as to mention that John Bollinger, the father of the eponymous bands, thinks that using Bollinger Bands and RSI in the manner described on the gummy-stuff site is a pretty poor idea."
John Bollinger, CFA, CMT

Then there's MOVING AVERAGES

Each day we compute the average stock price over the past M days ... and plot it.
We also compute the average price over the past N days ... and plot it.
When they cross we BUY ... or maybe we SELL
depending upon whether the crossing is from above or below
It might look like so:

So you play with the numbers M and N until you're happy.

Of course, a simple 200-day Moving Average, like MA = (P₁ + P₂ + P₃ + ... + P₂₀₀)/200, gives equal weight to all Prices, even the one that occurred 200 days ago! So, if we want to give more weight to more recent Prices, we could use a Weighted Moving Average:

WMA = (P₁ + 2P₂ + 3P₃ + ... + 200P₂₀₀)/K

where K is a magic number (which we'll explain in a minute).

Note that we are assuming that P₁ is the price 200 days ago and P₂₀₀ is the most recent Price ... and this most recent price is multiplied by 200 so it's 200 times more significant than the 200-day old Price, right?

Okay, what's K?
If all Prices are equal to, say P, we want the weighted average to equal P as well. That means:

WMA = (P + 2P + 3P + ... + 200P)/K = P(1 + 2 + 3 + ... + 200)/K
must equal P, so K must equal (1 + 2 + 3 + ... + 200).

As you might imagine, there's a magic formula for this sum, namely:

1 + 2 + 3 + ... + 200 = 200*201/2 so that's the value for K.

In general, for an N-day moving average, we have
K = 1 + 2 + 3 + ... + N = N(N+1)/2
(I think Euler discovered this when he was six years old. How does that make you feel?)

Our Weighted Moving Average is now:

Weighted Moving Average = WMA= 2(P₁ + 2P₂ + 3P₃ + ... + N P_N)/{N(N+1)}

Of course, nobody sez that the relative weights 1, 2, 3, ... are etched in stone. We could choose any weights w₁, w₂, w₃, ... w_N, and get:

Weighted Moving Average = WMA = (w₁P₁ + w₂P₂ + w₃P₃ + ... + w_N P_N)/K
where K = w₁ + w₂ + w₃ + ... + w_N = Σ w_k

Note that the number K is chosen so that, in the case where all Prices are equal, the Moving Average is equal to that Price as well.

It's time for a picture (where we use the weights 1, 2, 3, ...):

Note that the Weighted Moving Average (which emphasizes more recent Prices) follows the stock price more closely than the Simple 20-day average.

Then there's
Exponential Moving Averages
and
Moving Average Convergence/Divergence (MACD)
and ... well, you get the idea.

See also Moving Averages

Another thing ...

In order to simplify the calculation of the Weighted Moving Average WMA (with relative weights 1, 2, 3, ... N) we do this:

We have, at the current time period (which we'll call "Now"):

WMA(Now)= (P₁ + 2P₂ + 3P₃ + ... + NP_N)/K

and, at the next time period (where P_N+1 is the "Next" stock Price):

WMA(Next) = (P₂ + 2P₃ + 3P₄ + ... + NP_N+1)/K

and if we subtract, we get:

WMA(Next) - WMA(Now) = (-P₁ - P₂ - P₃ - ... - P_N + N P_N+1)/K

and here we recognize P₁ + P₂ + P₃ + ... + P_N as N times the Simple Moving Average at time period "Now" ... which we'll call MA(Now), what else?

That gives a prescription for computing our "Next" Weighted Moving Average:

WMA(Next) = WMA(Now) + {- N MA(Now) + N P_N+1}/K

and it's time to stick in K = 2/{N(N+1)} and get:

WMA(Next) = WMA(Now) + 2/(N+1){P_N+1 - MA(Now)}

So, assuming you're at time period 123 (that's "Now" and it could be 123 days or 123 weeks or ...) and you're workin' on 26-day moving averages (so N = 26) and you have the values of WMA(Now) and MA(Now) and the "Next" stock Price, P₁₂₄, then you get the "Next" Weighted Average like so:

WMA(Next) = WMA(Now) + 2/27 {P₁₂₄ - MA(Now)}.

So now, having the "Next" stock Price you also compute the Simple Moving Average of the last 26 prices, namely MA(Next), then "Next" becomes "Now" and you start again, to compute a new "Next" ... now, ain't that right?

Okay, now we're ready for:

Exponential Moving Average(Next) =
EMA(Next) = EMA(Now) + 2/(N+1){P_N+1 - EMA(Now)}

Okay, so why does this give an "Exponentially Weighted Average"?
Let's pick some weight, say α < 1 (like, mebbe α = .9, meaning 90%) and do the following:
EMA(Now) = K { P_N + α P_N-1 + α² P_N-2 + ... }
where the series goes on forever ... or, at least back to the first day you start to compute the sum(!) ... and, as usual, K is chosen so that the weighted average of equal stock prices, say P, is just P itself. That means that
EMA(Now) = K { P + α P + α² P + ... } must equal P, and, since the infinite series 1 + α + α² + ... adds up to 1/(1-α) (believe me, it does), we get:
K P/(1-α) = P, hence K = (1-α) and (finally!):

EMA(Now) = (1 -α) { P_N + α P_N-1 + α² P_N-2 + ... }
Now we see that this is definitely a weighted average, with yesterday's price weighing in at 90% (if α = .9) and the day before having a weight of α², or 90% of 90% (namely 81%), and the day before that having a weight of α³ or 90% of 90% of 90% (or 72.9%) and one hundred days ago, that price has a weight of just (0.9)¹⁰⁰ = 0.00002 (or 0.002%) ... well, you get the idea, right?

Now, if we also write
EMA(Next) = (1 -α) { P_N+1 + α P_N + α² P_N-1 + ... }
and compute EMA(Next) - α EMA(Now), we get
EMA(Next) - α EMA(Now) = (1 - α)P_N+1 (since most everything cancels out), and we get:

EMA(Next) = α EMA(Now) + (1 - α)P_N+1

See? It's the EMA equation we got before 'cept, now, we do recognize it as an exponentially weighted average!
(Did I mention that the weights α, α², α³, etc. are the reason for calling it exponentially weighted"?)

One curious thing. For a 12-day EMA, one chooses α = 1 - 2/(N+1) with N = 12 and for a 26-day EMA, one chooses α = 1 - 2/(N+1) with N = 26, etc. even tho' the average is certainly not an average over just 12 or 26 days, but over all previous stock prices!

Now, we're ready for:

Moving Average Convergence/Divergence(Next) =
MACD(Now) = EMA_12-day(Now) - EMA_26-day(Now)

where EMA_12-day means the 12-day (N = 12) Exponential Moving Average.

Here's MACD for a fast-moving exponential average (12-day) minus a slow-moving exponential average (26-day):

Of course, y'all don't hafta use the numbers 12 and 26 used by all them thar investment gurus ...

When MACD goes positive (meaning the fast average moves above the slow), that's a BULLish signal. When it goes negative, that's BEARish.

VOLUME-WEIGHTED MOVING AVERAGE
There are jillions of Weighted Moving Averages; pick your weights as the periods of oscillation of a butterfly wing and you've got yerself another WMA. Most play with just the stock price and ignore the volume of stock traded at that price.

If today's closing price is $16 and last month it closed at $12, then is today's price more significant ... because it's more recent (hence more relevant)? I don't think so, not if only ten shares traded at $16 whereas ten million traded last month at $12. (Okay, I exaggerate, but you get the idea, no?) That brings us to my favourite (which we'll call VMA), not necessarily because it gives better BUY/SELL signals, but because it makes some sense (to me, cuz VMA_N-day is approximately the average price paid for each share of stock over the past N days).

It's simple.
We take, as weights w₁, w₂, w₃, etc. the volumes of stock traded at Prices P₁, P₂, P₃, etc. and get:

Volume-Weighted Average = VMA = (V₁P₁ + V₂P₂ + V₃P₃ + ... + V_N P_N)/K
where K = V₁ + V₂ + V₃ + ... + V_N = Σ V_k

Here's a stock and the volume of trades:

The trading price in May was really important ... look at the volume! Anyway, we plot the weighted moving average over, say N = 100 days (why not?) and get:

I'm getting ahead of myself.
You see the stock price and the 100-day VMA in a lovely blue and also the 5-day Moving Average. (I hate to look for places where the stock price crosses some moving average; the stock price is too finicky, too nervous, too apt to spike-then-fall, too volatile ... so we use a fast moving average, like 5-day, 'cause it follows the stock price pretty closely and it's smooother, right?)

So what're the BUY/SELL signals?
Stare at the plot and decide when you'd like to BUY or SELL.
At those points, note that the VMA is a long way from the fast (5-day) average. That provides our signals:

When VMA - (5-day) > A then we BUY.
When VMA - (5-day) < -B then we SELL.

For example, if we plot the difference VMA - (5-day) we get:

OOPs! I've used the label VA for the Volume Weighted Average, rather than VMA; sorry 'bout that.

And what're optimal choices for A and B?
(Above, I chose A = B = 1.5 at my wife's suggestion.)
And is 5 the best number of days for the fast average?
And why is 100 chosen in the VMA?
(Because there are 100 cents/dollar or 100 asparagus in my garden?)
You Decide!

One other thing:
Some people caculate the Volume Weighted Average Price after each trade, throughout the day! If they can buy a stock at less than this VWAP, that's good. If they can sell stock at a price higher than the VWAP, that's good, too.

VOLUME-WEIGHTED EMA
As you might expect (since I think the volume of stock traded shouldn't be ignored), I have for you a Volume-weighted Exponential Moving Average which I'll call V-EMA. It's obtained as follows:

V-EMA(Next) = Num(Next)/Den(Next)

where

Num(Next) = α Num(Now) + (1 - α) V_N+1 P_N+1

and

Den(Next) = α Den(Now) + (1 - α) V_N+1

and "Num" and "Den" stand for Numerator and Denominator, respectively
and the numbers V_N are the volumes of stock traded each day
and, as before, α = 1 - 2/(TimePeriod + 1) so, for a 12-day V-EMA, we'd have α = 1 - 2/13 = .846

You can, of course, take V-EMA_12-day - V-EMA_26day to get a Volume-weighted MACD which I call V-MACD (and my son calls VD).

Does it differ much from the garden-variety MACD? Yes, if the volume of stock trades changes a bunch, like so:

Here, the price dropped but the volume increased thereby increasing the significance of these lower prices; when the garden-variety MACD dropped ('cause the price dropped so the 12-day EMA dropped) the increased volume kept the V-EMA from dropping so dramatically.

Note: If the volume doesn't change from day to day, then V-MACD and MACD are identical.

I think that's enuff, for now.

Except ... uh ... I should mention the Signal Line. You see, some technical gurus use yet another line (curve?) called the Signal Line and take as BUY/SELL signals the times when MACD crosses this Signal Line.
You get it by (are you ready for this?) taking the 9-day Exponential Moving Average of the MACD. In case y'all done forgot, that's got from:

EMA_9-day(Next) = α EMA_9-day(Now) + (1 - α) MACD(Next)

where α = 1 - 2/(9+1) = .80 and it looks like so:

Just one last thingy: formulas which look like:
A(N+1) = α A(N) + (1 - α) P(N+1)
compute a sequence of exponentially-weighted averages of the sequence of numbers P(1), P(2), P(3), ...
Aah, but how to start?
Y'all kin start with A(1) = (1 - α)P(1) and go from there, like so:
A(2) = α A(1) + (1 - α) P(2)      which is really (1 - α)[ P(2) + αP(1)]
A(3) = α A(2) + (1 - α) P(3)      which is really (1 - α)[ P(3) + αP(2) + α²P(1) ]
A(4) = α A(3) + (1 - α) P(4)      which is really (1 - α)[ P(4) + αP(3) + α²P(2) + α³P(1)]
etc. etc.
Eventually, how you started becomes lost in the distant (hence irrelevant) past.

TREND LINES
Okay, now we talk about stock price trends ... is it generally heading UP or DOWN? Some like to draw a trendline connecting a series of highs (or lows) going maybe UP or going maybe DOWN:

Trending UP: the Bulls are winning

Trending DOWN: the Bears are winning

As you can see, it's sort of a personal thing; you draws 'em as you see 'em. The stock price keeps bouncing off the red line as though it's being supported by that line ... so it's called the line of support. Prices can't seem to break through the green line; it's called resistance. One (presumably) waits for some sign that the stock price has changed trends and bravely crosses either line, leaving the channel. (The channel between the red and the green lines is called the ... uh ... channel.)

Here's a beauty; it's trending UP ... uh ... or is it trending DOWN?

It's called trendless

I invented the above charts to illustrate trends, but here's a real live example:

Me? I find it difficult to identify any trend, just by eyeballing the chart ... especially the beginning of a trend. (After it's over, it's too late!)

Maybe there's a more analytical/technical/sophisticated method to identify trends, which brings us to:

Directional Movement Indicator
or Directional Motion Indicator
or Directional Moving Index
or ... just plain DMI
We consider a competition between the bulls and bears. The bulls gobble up stock and drive the stock price to the high for the day. The bears dump their stock and drive the price to the low for the day. Who wins?

Each day we compute BULL points if today's high is greater than yesterday's high.
BULL points are (today's high) - (yesterday's high) ^* ... or zero if the high goes DOWN.

Each day we compute BEAR points if today's low is smaller than yesterday's low.
BEAR points are (yesterday's low) - (today's low) ^* ... or zero if the low goes UP.

^* In order to be somewhat more meaningfull, we'll divide these differences by "today's" closing price so they become percentages.
I'm not sure if Mr. Wilder (the author of DMI and RSI) did this, but we will.

Then, each day we see which points are bigger: the BULL points or the BEAR points (recognizing that, on some days, they may both be zero).

If the BULL points are bigger, they get awarded to the bulls.
If the BEAR points are bigger, they get awarded to the bears.
(Get it?)

Here's the BULL and BEAR points, for a sequence of days:

Uh ... the horizontal axis includes weekends when there were no points ... like Sep 25/26 and Oct 2/3

Note that, on Sep 24, there were both BULL and BEAR points, but only the BEAR points were awarded - to the bears (cuz their points wuz bigger).

Okay, here are the points that were actually awarded (to either the bulls or the bears) for the six month period covered by the stock price chart (above):

and, for reference, the stock chart itself:

The bulls were winning in November, eh what? (I owned CBR then!)

Okay, here's what we do ... actually, what Mr. Wilder does, 'cause DMI is his baby:

We take the sequence of awarded BULL points, say B(1), B(2), B(3) ... and compute the 14-day Exponential Moving Average of this sequence (Remember the EMA? Use: EMA(n+1) = α EMA(n)+ (1-α)B(n+1) with α = 1 - 2/(14+1))
The resultant curve is called +DI.

Then we take the sequence of awarded BEAR points and compute the 14-day Exponential Moving Average of this sequence (remember the EMA?).
The resultant curve is called -DI.

Then we plot 'em both, like so:

and, for reference, the stock chart itself:

Notice something interesting? When the +DI gets bigger than the -DI, there's a UP trend (the bulls are winning) and when the -DI gets bigger than the +DI, there's a DOWN trend (the bears are winning) and these events usually happen near the beginning of the trend!!

Of course, if we take the difference between +DI and -DI we'd get a chart that goes positive when the former exceeds the latter (the more the bulls are ahead, the bigger this difference would be) so we compute:

(+DI) - (-DI)

or, better still (to "normalize" things) we divide this difference by their sum, giving (finally!):

ADX = 100 { (+DI) - (-DI) }/ { (+DI) + (-DI) }

(assuming the denominator doesn't equal zero ... else just use (+DI) - (-DI) and forget my attempt to normalize)

That'd give this guy:

and, for reference, the stock chart itself:

Uh ... did I mention it was called ADX (Average Directional Indicator)?
Oh yeah ... that ritual of dividing each of the high and low differences by the closing price, each day (in order to get a percentage), makes absolutely no difference to the ADX. I think that most techies just look to see when +DI crosses -DI (regardless of their values, or the "scale" one uses) and stare raptly at ADX (whose value isn't influenced by that "divide by the closing price" ritual).

It's in the nature of the sport that bulls get very excited when the ADX is increasing ... especially when it exceeds 50.
(I guess bears get excited when it's decreasing.)

I should mention that, if the spikes in ADX are bothersome,
then a moving average of the ADX values will provide some smoothing.
Here's the 10-day moving average:

and, for reference (have I said that before?), the stock chart itself:

Check out DI+/- (sometimes called DMI+/-) and ADX on your stock at bigcharts.com. It'll look like so:

PS#1

So, did DMI predict the 1987 C^RA_SH ?

You decide:

Note that I've called them DMI+ and DMI- for a change of pace

PS#1.5

... and what does DMI say of the recent past?
You decide:

PS#2

It's been said (not by me!) that one can't use DMI on single-value sequences of numbers, like Mutual Funds (that don't have daily Highs and Lows). Take a peek at these charts (where I just made the High & Low stock prices equal to the Close ... hence a single daily value):

PS#3

Another thing: the use of the Exponential Moving Average is not the only choice one has (to incorporate the recent stock price history). We could also use (of course!) the Volume-weighted Moving Average. Here it is (for the CBR stock), where we're changing the names of +DI and -DI to VDI+ and VDI- and the ADX we'll call VDX:

15-day VMA of Bull & Bear Points	VDX = 100 [(VDI+) - (VDI-)] / [(VDI+) + (VDI-)]
Stock Prices	the Volume (with the Average shown)

Note: Volume-weighting is important if there's a large variation in the volume of trades. Large volume in early October affects the VMA hence the VDI hence the VDX, causing it to stay low ('cause the bears were winning) whereas the ADX rose (then fell back to where it belonged, rejoining the VDX). On the other hand, the high volume in mid-November (when the bulls were winning) allowed the VDX to stay high (while the ADX fell more rapidly and even go negative for a day or two, just after Nov 26).

Conclusion: with Volume-weighting, there's a certain amount of inertia associated with stock prices accompanied by high volume; the VDX tends to be more sluggish after high volume days. The effect hangs around for a while and the VDX is less temperamental, less likely to frivolously change direction with every spike in stock price - especially prices with low volume. That may - or may not - be a good thing.

Another note: in the Volume chart, there are blanks along the horizontal time axis. Them's weekends!

For more on VDI and Volume-weighted EMA, see EMA.

STOCHASTIC OSCILLATOR

Here we compare the current stock price, P, with the smallest and largest stock prices over the past N days: its smallest daily Low and its largest daily High. If these are L and H respectively, then we determine how far up the range from L to H that the current price lies. That percentage is:
%K = 100 (P - L) / (H - L)

(You get %R = 100% when the Price equals the High over the past N days.)

As you might imagine, we plot the values of %K (which lie between 0% and 100%) ... either with or without smoothing. (Smoothing involves taking a 2- or 3-day average of the %K values). No smoothing? It's a fast stochastic.

In addition, we calculate the M day moving average of %K ... and call it %D. This might be a weighted average, such as described above.

In any case, we watch to see when %K falls below or above some magic number (like below 20% or above 80%) ... or when it crosses %D. The chart below shows a few months of %K (with N=10 days) and a 2-day (smoothed) version and an M=5-day, simple moving average (that's %D) and some red and green arrows at the 20% and 80% values ... meaning SELL ... or maybe BUY ... or maybe ...

WILLIAMS %R

Whereas the Stochastic Oscillator compares the current closing price with the Lowest price over the previous N days, the Williams %R determines how far down the range from H to L that the current price lies. %R = 100 (H - P) / (H - L)

(You get %R = 100% when the Price equals the Low over the past N days.)

Note that %K + %R = 100%