grant
A couple of points if I may. I've never looked at historic vol but is there any correlation between hist and implied, and to what degree are comparisons/correlations valid? What time-frame(s) do you use.
Yes, they can correlate very closely, or conversely they can diverge significantly.
This divergence/convergence is the underlying mechanism within theoretical pricing models.
The time frame that most closely correlates with the trade being taken.
Using Frugi's example of 60 days, the time period that *in theory* be utilized is the last 60 day period.
However, as Profitaker has alluded, when SELLING, it is prudent to look at the highest volatility that has presented in at least the last year..........this represents a worst case scenario.
If BUYING, as long as there is some margin to the upside, then you can benefit should *direction* not be quite right, but volatility drifts upwards [if only slightly]
Of course now we need to look at *gamma* as in an ideal world, with low IV, I would like to see gamma on a rising curve.
It is generally accepted that implied is mean-reverting, therefore if there is a degree of correlation re hist vs implied, then there must be (a degree of) mean-reversion on the underlying. I would find this difficult to accept.
Absolutely correct. And, there is mean reversion on the underlying.
In 1913, a PhD was published by Bachelier in France, and demonstrated via rather complex mathematical equations that, price fluctuations grow in range and will be proportional to the square root of time
Stock prices in the United States over the last 100 years have 66% of the time fluctuated within a range of 5.9% on either side of their average.
The range in a course of a year has not been 72% or a multiple of 12 [year] rather, it has averaged around 20%
This is 3.5 times the monthly range.
The square root of 12 = 3.46
This data has stood the test of time, some 100yrs+, thus, the Option Pricing Models have been based on this proven mathematical theory, and form the rational basis underlying some of the assumptions.
Of course, if we want to move into slightly more esoteric ground, and that possibly forms a mathematical basis for Gann theory then;
This is chaos in the mathematical definition, not your standard day-to-day useage.
In it's simplest form chaos can be written;
4x{1 - x}
Computing the value of *x* of that expression for some initial value of *x* then substituting this answer back into the original expression starts a feedback loop.
Repeating this simple iterative process repetitively produces surprisingly complex, unpredictable mathematical behaviour.
The mathematical behaviour expresses the same kind of disorder produced by non-linear equations
The simplest non-linear equation;
Xn+1 = KXn - KXn(1 - Xn)
This equation determines the future value of the variable x at the time step n + 1 from the past value of x at time step n
This is known as the logistic equation
All well and good, but, what the hell is this to do with the Gannies?
Logistic equations are used in Medicine to predict population expansion, via Birth rates, Death rates, due in part to availability of food, water, arable land, disease etc.
It can also be used in ecology, for populations of insects, crops, etc.
Gann was interested in commodities.
Wait, there's more.
The logistic equation is a quadratic equation with a linear first term, and, a non-linear second term
It is the non-linear, or feedback component that is important.
For a given value of K once a starting point Xo is specified, the evolution of the system is fully determined. One step, inexorably leads to the next.
The whole process can be pictured on a graph.
It forms a parabola, that opens downwards.
There is a short-cut provided via the graphical representation, that avoids endless computations.
The addition of a 45 degree line up from the horizontal axis [representing the line Xn+1 = Xn]
The best course is to steer is from Xo vertically to the parabola to reach X1 then horizontally to the 45 degree line, and vertically back to the parabola.
These paths or Orbits give the first indication of which routes lead to the erratic behaviour of chaos
Whereas some orbits converge, on one particular value, others jump back and forth among a few possible values, and many roam, never settling anywhere.
When K is between 1 & 3, just about every route no matter where it starts, is eventually attracted to a specific value called a fixed point which occurs where the parabola intersects the 45 degree line at x = [k - 1]/k This corresponds to to a steady state or equilibrium
Therefore, taking the previous mathematical work performed by Bachelier, combined with a logistic equation, and you can reproduce seemingly Gann.
One may ask, if mean-reversion is OK for the implied, why not for the historical and by extension the underlying? I would suggest it's a question of risk. Implied reflects the markets perception of risk. We can't have unlimited risk as the price would be too high, and there would be no buyers; conversely, the concept of no risk doesn't sit too well - even options on government bonds have a risk premium.
As previously detailed, correlation exists for mean reversion.
Risk, is a somewhat different subject, and requires first a definition of risk, applicable to the general, and the specific.
jog on
d998
