Albert Einstein, 1926

Whether you can observe a thing or not depends on the theory which you use. It is the theory which decides what can be observed.

Mathematical models can be used to illuminate a system and make predictions about its behaviour, but they can also lead to a form of blindness.

A historic example is provided by supernovas, those massive stellar explosions which release a burst of radiation lasting months or even years. The first observations of such events by Western astronomers were in 1572 (recorded by the astronomer/alchemist Tycho Brahe) and then 1604 (recorded by his associate Johannes Kepler). However, Asian astronomers had known about them for centuries. The reason it took so long for the West to catch on was because astronomers there were blinded by Aristotelian science, which said that the planets rotated around the earth in spheres made of ether, and the heavens were immutable. Brahe also tracked a comet and showed that it would have smashed through those crystalline spheres, had they existed.

As I wrote in an article ten years ago, economics has its own versions of crystalline spheres which support its world view, and shape what is seen and not seen. One is Eugene Fama’s efficient market hypothesis, which states that markets immediately adjust to new information. The theory is reminiscent of Aristotelian physics, which assumed that in a vacuum changes take place instantaneously.

A related example is the random walk hypothesis, which assumes that asset prices are randomly perturbed at each time step, and which forms the basis of much of quantitative finance.

**The nervous walk**

The random walk model was first proposed by the French mathematician Louis Bachelier, whose 1900 dissertation on option pricing in the Paris Bourse described the behaviour of a stock’s price based only on its initial price, and the volatility (which Bachelier referred to as the “nervousness” of the stock). His thesis was initially ignored, but some 60 years later the economist Paul Samuelson found a copy “rotting in the library of the University of Paris,” and found it so interesting that he arranged for a translation. By 1973 the model had random walked its way to the heart of the developing field of quantitative finance, through the famous Black-Scholes model (also known as Black-Scholes-Merton or BSM).

The formula was based on the idea that one could construct a risk-free portfolio by dynamically buying and selling options and the underlying asset. Its “proof” relied on a number of simplifying assumptions, including efficient markets and the requirement that log prices follow the continuous version of a random walk with constant volatility. However its existence did seem to put option pricing onto a firm mathematical basis. Indeed, as Derman and Miller (2016) note, “the BSM model sounds so rational, and has such a strong grip on everyone’s imagination, that even people who don’t believe in its assumptions nevertheless use it to quote prices at which they are willing to trade.”

The continued strength of its hold is such that volatility in finance is usually assigned the dimension of inverse square-root of time (see for example Pohl et al., 2017) because this happens to hold for a random walk. But is this theoretical assumption justified?

**Quantum conundrum**

Unlike the classical model, the quantum model simulates price using a complex wave function which distorts when it is perturbed, leading to a change in volatility. It is therefore inconsistent with the classical assumption that volatility scales with the inverse square-root of time, which seems a bit of a conundrum until you remember that volatility is actually a relative (e.g. percentage) standard deviation so can be dimensionless.

I first became interested in this problem while investigating the question of how large transactions affect volatility. The quantum model predicts that the variance of price over the time T should be the sum of the normal volatility, which scales in the usual way with T, plus a term due to the order imbalance, which doesn’t:

Var = σ^{2} T + σ^{2} Q/(VT)

Here Q is the size of the excess order, V is volume per annum, σ is volatility, and T is the time period.

However when I checked the literature on market impact to see what else had been written on the topic, the best source I could find was a paper from a leading team of researchers which asserted that the variance in the square-root regime should follow the different formula

Var = σ^{2} T + σ^{2} a^{2} Q/V

with “a as the only fitting parameter (a~0.1)” (though when I queried this value it was corrected to “around 3”). Since Q/V has units of time, this version was consistent with the orthodox assumption that σ^{2} has units of inverse time.

A figure showed both “this prediction” and the actual variance as a function of Q/V for nine ranges of T, which seemed to give a good match – however the log-log scale of their graph made the results hard to interpret.

While I didn’t have direct access to the same data, the figure itself was already in the public domain, so after checking with the authors I digitalized the image to extract the points, and replotted with linear scaling (instead of log-log) as shown in Figure 1 below. With the linear scale it was obvious that the curves all have different slopes, with a variation from nearly 0 to around 5, and there is a distinct pattern where the slopes decrease with the duration T. The reason that the predictions lined up well with the data was because a different value of the tuning parameter was used for each line, so the model could effectively fit any slope at all. In other words, by any reasonable standard, the classical model clearly fails this empirical test (which of course has never been a deterrent to its use).

**Market impact, fixed**

Each curve in the original figure had a particular range of T, so using the mid-point of the range as the time duration, I plotted variance against Q/(VT) as per the quantum model (see Figure 2). Again, this version violates the assumption that volatility has dimensions of inverse square-root of time, however the slopes are now fairly constant with a mean of about 0.5. According to the quantum model this slope is not just a made-up fitting parameter but should provide an estimate for the volatility σ, so it is in the right range though somewhat higher. Given the inherent noisiness of the data, especially for smaller impacts, this confirms that the quantum model is capturing the underlying dynamics of market impact. I wrote the result up in a short note for Wilmott magazine.

While the volatility of market impact might seem like a rather specialised topic, the common assumption that volatility has dimensions of inverse square-root time is in many ways the lynchpin of quantitative finance – take it away and the rest of the structure starts to look shaky. For example, as mentioned already it is a key assumption of the Black-Scholes model which controls the pricing of derivatives. And a related demonstration of model blindness – which has a great deal of practical importance – is the fact that the volatility smile seen in options trading has long been treated as a subjective quirk of traders rather than recognised as an intrinsic property of markets. More generally, the dynamics of market impact are also informative about the dynamics of supply and demand.

As empirical signals go, the discovery that volatility diverges from classical theory isn’t quite as spectacular as a supernova, but perhaps it will open some eyes to the fact that quantitative finance – and economics in general – is in need of some novel ideas.

**References**

Bucci F, Mastromatteo I, Benzaquen M, Bouchaud JP (2019 ) Impact is not just volatility. *Quantitative Finance* 19(11):1763-6.

Derman E, Miller MB (2016) *The volatility smile*. Hoboken, NJ: John Wiley & Sons.

Orrell D (2022) Market impact through a quantum lens. *Wilmott* 2022(122): 50-52.

Orrell D (2022a) A Quantum Oscillator Model of Stock Markets. Available at SSRN.

Orrell D (2022b) Keep on Smiling: Market Imbalance, Option Pricing, and the Volatility Smile. Available at SSRN.

Pohl M, Ristig A, Schachermayer W, Tangpi L (2017) The amazing power of dimensional analysis: Quantifying market impact. *Market Microstructure and Liquidity* 3(03n04):1850004.

Wilmott P, Orrell D (2017) *The Money Formula: Dodgy Finance, Pseudo Science, and How Mathematicians Took Over the Markets*. Chichester: Wiley.