Drag in the
Complex Plane

One formula, two signs.

Everything on this page falls out of a single identity. For X normal with mean μ and variance σ², and for any complex number s:

Put s = 1 and you get exp(μ + ½σ²) — the mean of a lognormal. Variance raises the average. That surplus is the thing every other page in this collection is about: the average is lifted above the path you actually walk, and the gap is ½σ².

Now put s = i. The same formula gives exp(iμ − ½σ²) — the characteristic function of a Gaussian. Variance now destroys the average, and it destroys it at exactly the rate it was previously building it up.

Between those two poles the term is complex, and it does something with no counterpart on the real line. Write s = a + bi:

So in general variance does not only shrink or grow a quantity — it rotates it. Uncertainty in the exponent becomes a systematic phase shift, proportional to σ² and present whenever s is neither purely real nor purely imaginary. Drag the dial below through a right angle and watch the surplus become a decay, with the rotation peaking exactly halfway between.

Log space becomes a cylinder.

The reason log space works for positive numbers is that log turns multiplication into addition. It does the same for complex numbers, with one complication: a complex number has a modulus and an argument, and only the modulus lives on a line.

Multiply a chain of complex numbers together and the two parts behave completely differently. The log-moduli add on a line, exactly as in every other page here. The arguments add on a circle, and wrap. So log space is no longer a line — it is a cylinder, and drag can act along either axis.

Two drags, not one

Take factors Z = r·eiθ with log r normal (spread σ_r) and θ normal (spread σ_θ), independent. There are now two separate penalties, and they point in opposite directions:

• Amplitude drag, +½σ_r². The familiar one. The average modulus sits above the typical modulus, because log is concave.
• Phase drag, −½σ_θ². The new one. Averaging arrows that point in different directions gives something shorter than the arrows themselves. Nothing about magnitudes; purely about direction.

Both are second-order in a spread, both carry the same ½, and in the ensemble mean of a product they simply add:

The mean can fall below the median.

On the positive reals this is impossible. The arithmetic mean is never below the geometric mean, the ensemble average is never below the typical path, and no amount of volatility will turn that around — Jensen's inequality forbids it, and the gap only ever opens in one direction.

In the complex plane it happens as soon as the phase spread exceeds the amplitude spread. The ensemble mean is pulled down by cancellation faster than convexity can lift it, and the average outcome ends up worse than the typical one.

Figure 3 shows the three quantities at once. Every dot is where one path ended. The circle is the typical modulus; the dashed circle is the mean modulus, always the larger of the two; and the arrow is the mean of the numbers themselves, which is shorter than both as soon as the dots stop pointing the same way.

No Jensen, and the same answer anyway

This is the part worth pausing on. The usual derivation of volatility drag is an appeal to Jensen's inequality: log is concave, so E[log X] ≤ log E[X], and the gap is the drag. That argument cannot be made here at all. Jensen needs an ordering, and the complex numbers are not ordered — there is no sense in which one complex number is less than another, so there is no concave function and no inequality to invoke.

The mechanism is different too. Amplitude drag is curvature: a concave function applied to a spread-out variable. Phase drag is cancellation: vectors pointing different ways adding up to less than their lengths. Those are not the same phenomenon dressed differently.

Which is a caution about the analogy in general. A correspondence this clean invites the assumption that the intuitions transfer. Some do. The next one does not.

The phase has no time average

ERGODIC turns on the difference between an ensemble average and a time average — what happens to everyone at once, versus what happens to you in sequence. The modulus of a complex product keeps that structure exactly: (1/n)·log|Zn| converges to E[log r], the same law of large numbers, the same time average.

The phase has neither. The argument is a random walk on a circle, and it does not converge to anything — it equidistributes. Ask what the long-run average phase is and the question has no answer: every angle is visited, in proportion, forever. There is no third number waiting to be discovered. There is a limiting distribution and no limiting value.

So the tidy dichotomy that runs through the rest of this collection — ensemble average here, time average there, mind the gap — does not survive the move to the complex plane intact. One coordinate keeps it. The other replaces it with something structurally different.

Where it already has a name.

Phase drag was not discovered by anyone thinking about volatility. It was found repeatedly, in fields with no interest in each other, and each time it was given a local name. In every one of these the quantity being computed is |E[eiθ]| and the answer is e−σ²/2.

The power-factor row is the one to hold on to, because it is the least abstract. A motor drawing current out of phase with the voltage still draws the current — the amplitude is all there — but the useful power is |V||I|cosφ, and the rest circulates without doing anything. You are billed for the amplitude and paid for the projection. That is phase drag, on a utility bill.

The sign problem, met head on

Everything above was checked numerically before it was written. Most of it agreed. One part did not, and the disagreement turned out to be the subject of the page rather than a mistake in it.

Estimating E[Z] by simulation means adding up a great many complex numbers that largely cancel. The answer shrinks like e−nσθ²/2, but the noise in the estimate does not shrink at all — it is set by E[|Z|²], which the phase never touches. Signal falls exponentially, noise stays put, and past a certain spread the measurement is pure noise.

Here is that happening, over 400,000 paths of 50 steps each with σ_r = 0.30:

Figure 3 above runs the same estimator on 20,000 paths and knows when to stop trusting itself. Sweeping it across forty settings and comparing every reading against the closed form, the error tracks the signal-to-noise ratio closely: above 8× the noise floor the median error is 1.2%, between 4× and 8× it is 3.3%, between 2× and 4× it jumps to 20%, and below 2× the readings are wrong by factors rather than percentages. The figure warns below 4×, which is where that calibration puts the cliff.

Which makes the failure worth more than the agreement. The rows that matched confirmed the algebra. The rows that missed demonstrated, on a laptop and by accident, why a well-funded field has been stuck on this for forty years.

What to carry away.

• One formula covers both. E[esX] = exp(sμ + ½s²σ²). Real s gives the lognormal surplus; imaginary s gives decoherence; the sign flip is i² = −1.
• Log space becomes a cylinder. Moduli add on a line, phases add on a circle. Drag can act along either.
• Variance rotates as well as scales. For s = a + bi the penalty is ½(a²−b²)σ² with a rotation abσ² alongside it. There is no real-line analogue of being turned by your own uncertainty.
• The mean can fall below the median, once σθ > σr. On the positive reals this is impossible.
• The same number, two mechanisms. Curvature on the reals, cancellation in the plane. They agree because both are the quadratic cumulant, not because they are the same idea.
• One intuition does not transfer. The phase has no time average at all — it equidistributes. Ensemble-versus-time survives in the modulus and dissolves in the argument.

The honest summary of the analogy: the algebra carries over completely, the arithmetic carries over with a sign, and the intuitions carry over about halfway. Which is a better result than either "it is all the same" or "it is a different subject", and it is worth knowing which half is which.