QEF15 – A quantum oscillator model of stock markets 2: q-variance and the q-distribution

Q-variance refers to the property, described previously in QEF14, that the expected variance of log returns x, corrected for drift, over a period T follows to good approximation the formula

V(z)=^2+z^2/2

where z=xT.

Q-variance follows because we model price change over a period as the displacement of a quantum harmonic oscillator (here we are not explicitly modelling the market maker).

A corollary is that the price change distribution follows a Poisson-weighted distribution of normal curves, where the Poisson rate parameter is =0.5, and the standard deviation of the n'th term is _n=(1+2n). This distribution, known as the q-distribution, looks like the figure below.

The q-distribution (solid line) is composed of a Poisson-weighted sum of Gaussians. The first three terms are shown by the thin lines.

The q-distribution can be derived in a number of ways. One is to note that the volatility of the perturbed oscillator is computed by measuring the energy level n. The result follows a Poisson distribution, with _n being the standard deviation associated with each level. An ensemble of oscillators, representing separate transactions over the period, then gives the q-distribution. Another way to derive the distribution is by demonstrating that the value =0.5 is consistent with q-variance.

The test of course is whether q-variance and the associated q-distribution are consistent with data. The figure below shows variance as a function of z=xT for 355 stocks for the period January 1992 to April 2025. For each stock, the results are compiled over periods T of one to 50 weeks. Companies included are currently in the S&P 500 index and have data for at least 75 percent of the period. Blue line is the mean, red line is the quantum model with a small shift to account for effects such as skew.

Q-variance

The left panel in the figure below shows the distribution of z for the same 355 stocks. Blue line is the mean. The right panel shows the same, after the results for each stock are scaled to bring them in line with the mean. The q-distribution, shown by the red dashed line, is almost indistinguishable from the mean.

Q-distribution

To summarise, the quantum oscillator model predicts that variance will show the property of q-variance, and price changes will follow the q-distribution. These are connected because the q-distribution is consistent with q-variance. The predictions have been tested against stock market data and are in excellent agreement with results. However neither q-variance nor the q-distribution are consistent with a classical random walk, which is probably why they have not previously been used in finance

For further reading, see:

Wilmott P and Orrell D (2025) Q-Variance: or, a Duet Concerning the Two Chief World Systems. Wilmott 2025(138).

Orrell D (2025) A Quantum of Variance, and the Challenge for Finance. Wilmott 2025(138).

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