In part II we looked at the magic formula: (1)
Here: **T**is some kind of period (like**T**= 5 days)**P**_{0}is some parameter (maybe a 10 day Moving Average)**P**(n) is today's stock price and**P**(n-1) is the price yesterday**P**(n+1) is the*next*stock price in the sequence ... namely*tomorrow's*price.
Pay attention. I want you to watch very closely ... Figure 4a, b, c >Yeah, so?
>Yes, I can see Figure 5, but shouldn't the curve drop like a rock? We're talking Crash, right?
Uh ... it seems that the singularity when t = T _{c} identifies the crash. One looks at recent market gyrations and tries to
fit a curve lby picking appropriate values for A, B, C etc. and, in particular, T_{c}.
If successful, the value of T_{c} indicates a time, in the future, when a crash will occur.
>Or an earthquake or the mylar breaks or the concrete fractures or ...
>Wait! What about the log-periodic equation?
In the meantime, I think we can get singularities with our Equation (1) if we ... >What's this >And you get that with Equation (1)?
> Figure 6a, b >But decreasing that - As the price moves above some "benchmark" (that's P
_{0}), investors sell to lock in their gains. - Selling drives the price down.
- When the price moves below P
_{0}, investors buy, thinking the stock is cheeep. - For some reason (a war? bad news? coffee prices increase?), stock changes occur more quickly ... higher volatility.
- Investors, seeing this, react more quickly.
- Faster investor buying and selling increases the volatility, changes occur more rapidly.
- Investors, seeing this, react more quickly.
- Faster investor buying and selling increases ...
>Yeah, I get it. Doesn't that have a name ... like feedlot?
>And you get this terrible squeal. Yeah, I did that once and ...
>Doesn't that have a name ... like cattle herd?
(3a)
Now we divide each side by (3b)
Now we set (3c)
decreasing.
Masochists may be interested in the following:
>zzzZZZ
We can solve Equation (3a):
- Put p(n) = u
^{n} - Get u
^{n+1}= A u^{n}- u^{n-1} - Divide by u
^{n}and get u = A - 1/u or u^{2}- A u + 1 = 0 - Solving this quadratic equation gives:
u = u_{1}= B + SQRT(B^{2}- 1) or u = u_{2}= B - SQRT(B^{2}- 1) where B = A/2 - The general solution to Equation (3a) is then:
p(n) = L u_{1}^{n}+ K u_{2}^{n}where K and L are constants (as yet unknown). - Suppose we know the values at n = 0 and n = 1 (namely p(0) and p(1)) hence we get:
L u_{1}^{0}+ K u_{2}^{0}= K + L = p(0) L u_{1}^{1}+ K u_{2}^{1}= Ku_{1}+ Lu_{2}= p(1) - We can then solve these two equation for K and L to get:
K = {p(1) - p(0) u_{2}}/(u_{1}- u_{2}) L = {p(0) u_{1}- p(1)}/(u_{1}- u_{2})
u _{1} = 1.5+SQRT(1.5^{2}-1) = 2.618
u _{2} = 1.5-+SQRT(1.5^{2}-1) = 0.382
If p(0) = 4 and p(1) = 3 (meaning the first two prices, P(0) and P(1), differ from P Hence p(n) = 0.658 * 2.618 Note that 0.382 Now comes the interesting part:
^{iλn} and e^{-iλn}
are each linear combinations of
cos(λn) and sin(λn).
Hence u _{1}^{n} and u_{2}^{n}
are each linear combinations of
cos(λn) and sin(λn).
Hence the solutions of Equation (3a) can be written as linear combinations: (5) p(n) = C cos(λn) + D sin(λn) where (of course) C and D are constants. The onset of oscillatory behaviour occurs for B If we believed this stuff, we'd look at recent market (or stock) oscillations and try to identify some critical
>Harumpf! Hindsight is 20/20. Let's see you predict the Okay, just for fun, eh? Here's what we'll do: - We'll look at the 5-day moving average of S&P500 closing values from May, 2000 to Jan 23, 2003.
(We'll use*5-day average*to get something a wee bit smooother than*daily*values.) - We'll pick a
**T**_{o}and a reduction factor**f**so that, with**T**=**T**_{n}=**T**_{o}f^{n}, Equation (3a) looks like:**magic equation****P**(n+1) = [ 2-{2π/**T**_{n}}^{2}]**P**(n) -**P**(n-1) + {2π/**T**_{n}}^{2}**P**_{0} - For
**P**_{0}we'll choose the 5-day moving average of*actual*S&P values. - We'll start with
**P**(1) and**P**(2) equal to the actual S&P 500 values. - Then we'll use the magic equation, above, and run through a bunch of values for
**T**_{o}and**f**and pick out the values that minimize the RMS error between the P(n) and the actual S&P values. (The Root-Mean-Square error is expressed as a percentage of the starting S&P value else an RMS error of 1.2345 is meaningless, eh?) - After identifying the "best" choice (for
**T**_{o}and**f**) we extrapolate, using the*predicted*values in the**magic equation**(for the**P**(n) and**P**_{0}). - We're now out on a limb. Our only numbers are the predicted values. The actual S&P 500 values are history. We're looking into the future. Moving averages can only be calculated using our predictions. The future evolution of the S&P ...
>Okay! Okay! So what do you get? Will there be a crash? >Yeah, yeah. So what do you get?
>So > Somebuddy uses the spreadsheet. >I promise NOT to sue and ... Well ... okay. RIGHT-clickhere and Save Target (such as it is) ...
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