The Future of Everything

April 14, 2023

The uncertainty principle, finance version

Filed under: Economics, Quantum — Tags: , — David @ 3:58 pm

I’m a big fan of uncertainty. I’m more interested in what we don’t know than what we do know. In physics, that obviously leads naturally into quantum … where the idea of uncertainty is baked into the theory itself.
Shohini Ghose

In a Perimeter Institute podcast, quantum physicist Shohini Ghose discussed the role of uncertainty, applying it not just to physics, but to “questions around identity and society.” As she notes, “there is power to uncertainty, that the universe itself is telling us, stop with all of this precision measurements and stop with trying to know it all.” Despite the fact that quantum mechanics was developed over a century ago, “only recently have we really started exploring and digging deeply into the stranger properties of superposition, entanglement, all of which have quantum uncertainty underlying them.”

According to Ghose, there is a quantum revolution in the works which rivals the Industrial Revolution. The latter was shaped by a quest for precision and certainty in everything from production at scale to mass marketing. In contrast, “this whole new revolution with new quantum technologies” will reshape the way we think and behave. “Just like we move away in science from zero or one and go to zero and one, perhaps in society too, we will naturally start expanding our choices … and we will get to newer ways and newer approaches, which can influence so many aspects of our behavior.” (Listen to the full podcast here.)

While Ghose doesn’t discuss economics and finance, there is an interesting parallel with the world of quantitative finance.

This year marks the fiftieth anniversary of the Black-Scholes model, which is used to price financial options. The model has been described as “the most widely used formula, with embedded probabilities, in human history.” It is even more famous, though, for kickstarting the development of large-scale derivatives trading, by appearing to banish uncertainty from the pricing of options.

The Nobel-winning theory accomplished this by assuming that you can constantly buy and sell options and the underlying stock in such a way that the growth rate has to equal the rate of a risk-free instrument such as a government bond (so-called “dynamic hedging”). It was therefore unnecessary to make subjective and uncertain estimates of future growth. Key to the argument was that all the buying and selling will not incur excessive costs, and also that the volatility (standard deviation) of the price is a known constant.

In quantum economics, however, the bid/ask spread between buy and sell prices is not a technical detail, it is a fundamental level of uncertainty, and the main driver of volatility. The dynamic hedging proof therefore breaks down.

In a recent study that I carried out with hedge fund CTO Larry Richards, we showed that the key assumptions behind the Black-Scholes model do not hold up. Growth rates do matter, and they are uncertain. The volatility is not constant but depends on the degree of market imbalance, which affects the price change over an interval. The (empirically verified) result of all this is that, for commonly traded options, the Black-Scholes model is out by a factor about equal to the square-root of two.

The reason the Black-Scholes model has been so influential therefore is not because of its accuracy, but because it appeared to banish uncertainty from options trading, and transform it from what used to be considered a slightly disreputable form of gambling, into scientific risk management. In contrast, the new results came out of a quantum model of asset price behaviour which puts price uncertainty at its core.

In a sense, the two models therefore represent a different version of what we consider to be science. The Black-Scholes model appears to be rational and logical and certain. The quantum model takes uncertainty as its starting point. And the empirical evidence backs the latter.

So maybe, to borrow Ghose’s words, “the universe itself is telling us, stop with all of this precision measurements and stop with trying to know it all.” Finance is uncertain, and we can’t banish risk with an equation.


April 12, 2023

November 18, 2022

Model blindness

Filed under: Economics, Quantum — Tags: , — David @ 4:58 pm

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.

Albert Einstein, 1926

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 = σ2 T + σ2 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 = σ2 T + σ2 a2 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 σ2 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).

Figure 1. Variance as a function of Q/V for nine values of T from 3.5 to 345 minutes. The slopes show a clear inverse dependence on time T.

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.

Figure 2. Variance as a function of Q/(VT) for nine values of T from 3.5 to 345 minutes. The mean slope of the lines is about 0.5.

While the volatility of market impact might seem like a rather specialised topic, the common assumption that volatility can be treated as constant 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.


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.

September 25, 2022

QEF14 – Is the volatility smile real or imaginary?

The answer to this question, according to quantum finance, is both.

The volatility smile refers to the phenomenon in options trading where the implied volatility has a smile-like shape as a function of strike price (see Figure 1 below for an example). The volatility is lowest for at-the-money options where the strike price is the same (after discounting) as the current price, and is higher for out-of-the-money options.

The volatility smile is usually viewed as something of a conundrum, since it seems to violate the idea – which forms the basis of quantitative finance – that prices undergo a random walk with a volatility that can be described by a single number. The Black-Scholes model for option pricing, for example, assumes that prices follow a lognormal distribution with constant volatility.

This assumption is so widespread that in quantitative finance, volatility is usually expressed in terms of inverse square-root of time, because this happens to hold for a random walk where variance (i.e. volatility squared) increases linearly with time.

The quantum model differs from this classical random walk in a number of respects. In the classical model, a normal distribution is used to describe the possible range of prices at a particular time. This is an essentially static picture, which does not reflect the fact that price is the result of investors buying and selling, and assumes that markets are balanced. The quantum oscillator model replaces the normal distribution with a complex wave function, which rotates around the real axis. Since this is a dynamic model – the quantum version of a spring – it can be used to model this turnover process (it literally turns over in the imaginary plane), and capture what happens when markets are out of balance.

For example, a large order will perturb the price by an amount which depends on the square-root of the relative order size – which is just the well-known square-root law of market impact. However another effect is that the wave function also distorts in shape, leading to higher volatility.

In other words, market imbalance between buyers and sellers affects both price and volatility – so price and volatility are correlated, in a manner which happens to match the volatility smile seen in options trading.

Now, the volatility smile usually refers to the volatility that is implied by the price paid for options, so you could argue that it is based on traders’ subjective projections about the future, and is just a figment of their imagination.

But an easy way to test the hypothesis is to plot volatility versus price change for different time periods. For a lognormal distribution there is no correlation (and it is actually a bit tricky to produce an artificial data set which gives the right properties, but you can do it with the quantum model) but historical market data follows the predicted volatility equation.

Of course what we really care about here is the effect on option pricing. As mentioned, implied volatility usually refers to the volatility implied by the cost of options. But if the model is correct, then the cost should equal the expected payout. So another test is to ask, what volatility is implied – not by traders’ projections – but by after-the-fact market outcomes? In other words, what is the correct volatility to use in the Black-Scholes model so that the option cost calculated from the formula equals the expected payout, as determined from historical data?

The Black-Scholes model calculates option prices by assuming that the price distribution is lognormal. If the theory is correct, then the “market implied volatility” should be just the usual volatility, which does not depend on price change. The average option cost using this volatility should therefore equal the average payout (with of course an allowance for noise).

However the experiments show that while the Black-Scholes model does work for a lognormal data set – i.e. the model option cost equals the average payout – it produces systematic errors when historical data are used. It therefore fails a basic calibration test for a predictive model. Results can be improved by making volatility dependent on price change according to the smile equation.

So to summarize, the volatility smile is definitely real – even if the oscillations which produce it in the model take place in the imaginary plane.

For details, see the SSRN discussion papers:

A quantum oscillator model of stock markets

Keep on smiling: Market imbalance, option pricing, and the volatility smile

Previous: QEF13 – Quantum supply and demand

Playlist: Quantum Economics and Finance

June 6, 2022

Quantum economics FAQ

Filed under: Economics, Quantum, Quantum Economics and Finance — Tags: , — David @ 12:30 am

This post answers some questions that typically come up when discussing the quantum approach to economics and finance. For a list of broad objections (and responses) to the use of quantum probability outside of physics, see the post Ten reasons to (not) be quantum.

Why use quantum probability instead of classical probability?

The main difference between classical and quantum probability is that the former is based on yes or no, 0 or 1 logic, while the latter allows for superposition states (so yes and no, 0 and 1). This allows us to handle properties including interference and entanglement which characterize human interactions as much as they do the subatomic world. Another advantage of quantum probability is that it provides a useful framework for modelling probabilities that evolve dynamically (an example is the oscillator model of stock markets). Note also that quantum probability simulates a state using a complex-valued wave function, and much of its power comes from what has been called “the magic of complex numbers“.

How do quantum phenomena such as interference or entanglement occur in markets?

The field of quantum cognition models the decision-making process using the quantum formalism exactly because it can handle phenomena such as interference between incompatible beliefs, or entanglement between subjective context and objective calculations. Finance also has a more direct form of entanglement through things like debt contracts or the use of money.

Is this the same type of interference and entanglement as is seen in physical systems?

The point is that the same kind of model can be used for each. For example we can model a debt contract, including the potential decision to default, using an entanglement circuit on a quantum computer. The debtor’s decision is entangled with their subjective context; the creditor’s money is entangled with the debtor. Note that the entanglement involves information rather than macroscopic objects.

What are the practical applications of quantum economics?

Quantum economics offers an alternative to traditional economics that it is based on a different form of probability, and can be applied to a broad range of economic problems including decision making (quantum cognition), stock market analysis, option pricing, and the basics of supply and demand. More generally, it provides a mathematical framework for modelling properties such as subjectivity, interconnectedness, and power relationships which are downplayed or ignored in traditional economics.

Can the theory be used to make predictions?

The theory has been used to make a range of predictions (really postdictions, since the answer is known) including for cognitive effects of the sort studied in behavioural economics such as the order effect, the rate of strategic default on mortgages, the volume of options sold as a function of strike price, and the square-root law of market impact. One novel prediction was a relationship between price change and volatility that has important consequences for option pricing, since it violates a key assumption of the Black-Scholes formula.

Why is it appropriate to model social systems using concepts like force, mass and energy?

An advantage of quantum probability is that it provides a way to handle dynamical systems by quantizing forces. The entropic forces used in quantum economics are generated by propensity curves which specify the probability of an event such as a transaction. They are therefore just another way to describe a probability distribution, but they also serve as an intermediate step to create a quantum model. This in turn leads to natural definitions for concepts such as energy and mass, for example mass represents a resistance to change. Note that it is traditional in economics to talk about forces of supply and demand, but they are assumed to simply cancel out at equilibrium, so there is no need to describe something like mass.

Quantum systems are discrete, while observed systems are usually better described as continuous. For example a quantum harmonic oscillator has discrete energy levels, so how can we use that to model something like the price of a stock?

In the quantum model an oscillator represents a potential transaction. The energy level corresponds to the number of transactions over a time step, which is necessarily discrete (in a typical application the oscillator spends most of the time in the ground state, with transactions occurring every few steps). Indeed a defining feature of the economy is that it involves discrete transactions including money transfers.

What is the financial version of Planck’s constant?

In physics Planck’s constant is treated as an invariant quantity of nature, in quantum economics it is a parameter which decides the scaling for quantities such as mass.

Are quantum models more complicated than classical models?

The models used in traditional finance and economics are often very complicated because they need lots of bells and whistles in order to capture the complexities of the system. Quantum models do involve wave functions with an imaginary component, but the result can be simpler because they provide a more natural fit in the first place. For example in the oscillator model, the ground state is a wave function which rotates around the real axis and acts as a counter for transactions, which is only possible because it has an imaginary component. The only extra parameter is the oscillator frequency, which is needed in any case to describe the frequency of transactions.

Do you need a degree in quantum mechanics in order to work in this field?

No, most of the mathematics is basic linear algebra or calculus. In fact, while physicists tend to be the go-to experts for tricky technical problems, a training in physics sometimes seems to be a blocker – for example physicists often struggle with the idea of social or financial entanglement because they want to relate it to the behaviour of subatomic particles, instead of just looking at the math.

What is the difference between quantum economics as described here, and other quantum approaches?

Quantum economics starts with the idea that money has complex dualistic properties which are best handled using a quantum approach. It draws on ideas from quantum cognition and quantum finance, which developed independently. One approach to quantum finance is to see it just as a mathematical tool for solving hard problems from traditional quantitative finance (such as derivative valuation), without any attempt to incorporate effects such as interference or entanglement (for a critique see here). Another is the “quantum-like” approach which transposes models from physics, without necessarily trying to justify them from basic principles. Finally there is the two-state approach which focuses on price, and models stock markets in terms of a price operator with two states representing the bid and the offer. In this view, there is no concept of force or mass (instead mass is subsumed in the definition of the financial Planck’s constant). Quantum economics differs from the first in that it is concerned with quantum phenomena such as interference and entanglement; from the second (slightly) in that it derives models as far as possible from first principles rather than importing then from quantum physics; and from the third in that concepts such as force and mass are viewed as useful components of the model (however two-state models can be derived from it). Quantum economics is therefore broadly compatible with these other approaches, but treats mental and financial phenomena as quantum in their own right.

Does quantum economics assume a direct link with quantum mechanics, for example through quantum processes in the brain?

No, and even if consciousness turns out to rely on quantum processes we couldn’t infer from it that the economy should be modelled using wave equations. Similarly, the fact that quantum models are useful for modelling human cognition does not imply that the brain is quantum. In quantum economics, we take social properties such as interference and entanglement at face value rather than arguing that they are inherited from subatomic particles. The test of quantum probability in economics is not whether its use can be justified by physics; it is whether, if it had no known application in physics, we would still want to use it to model social systems.

Where can I receive training in quantum economics?

Quantum economics and finance has been chosen as a thesis topic by a number of students in higher education. Memorial University in Newfoundland has set up a Centre for Quantum Social and Cognitive Science whose remit includes quantum economics. People who wish to get into the area can check out a number of online resources including the papers here or for a general introduction this video series.

May 29, 2022

QEF13 – Quantum supply and demand

The neoclassical X-shaped supply and demand diagram is featured in every introductory textbook, is the basis for mathematical models of the economy, and has shaped our view of the economy for over a century, but as critics have pointed out many times it has a few basic problems (see Economyths for a summary).

It assumes static equilibrium so there is no dynamics. Supply and demand are assumed to be independent, when we know they are often coupled. And finally, there is no empirical validation for the diagram. For example demand curves involve hypothetical transactions which we can never observe. We also know that prices are not drawn to a stable equilibrium, but are subject to complex dynamics.

The quantum version starts by observing that supply and demand are two sides of the same coin, so what counts is the degree of imbalance – if both increase at the same time it has no effect on price (at least to first order). Modelling potential transactions with a quantum oscillator then leads to a version of the square-root formula derived for price impact in QEF12.

To start with a simple illustration, consider a scenario where there is one seller and one buyer for some perishable item such as a loaf of bread. The seller wants to sell one unit per day, and the buyer wants to buy one unit per day, so the system is balanced. We can model this using an oscillator in the ground state, where the wave function rotates around the real axis with a frequency of once per day, and the squared amplitude representing the joint propensity function is a normal curve.

Suppose now that another buyer enters the picture. The expected transaction rate will therefore double, and the supplier may not be able to keep up. In order to restore the original frequency, the required price increase from the formula is x=√2 σ where σ measures the degree of price flexibility. Making this change requires energy ∆E=ℏω∕2. As mentioned in QEF09 this is the base quantum of energy which allows for the possibility of a transaction between two people. The energy added by increasing the frequency is therefore balanced by the energy spent in increasing the price.

Left panel shows the probability distribution for the oscillator in the ground state. Adding an additional buyer shifts the oscillator to the right (as indicated by arrow). Right panel depicts the complex wave function rotating around the real axis.

In the general case, define the imbalance to be ι = (Nb – Na )/min⁡(Na,Nb ) where Na is the number of sellers and Nb is the number of buyers. As with the price impact result from QEF12, we then get a change in log price of x=±Yσ√|ι| where Y=2 and the sign depends on the sign of ι. This formula is shown in the right panel of the figure below.

Left panel shows the neoclassical supply and demand diagram. Source: Instant Economics. Right panel shows the quantum version with the uncertainty parameter set to σ=0.02. Change in log price has a square-root dependency on the demand/supply imbalance ι. The shaded area, representing price uncertainty, shows one standard deviation.

Another feature of the quantum oscillator model is that volatility is a function of the energy level, so if the energy state is En then the volatility will be σn=σ√(2n+1)=σ√(|ι|+1). It follows that observed volatility is not a constant, as usually assumed in finance, but varies depending on energy level. The uncertainty therefore increases with the degree of market imbalance, which is consistent with the large fluctuations seen during times of market stress.

If we assume the model is perturbed at each step by an amount ∆x=σ as in QEF12, then the energy of the system follows a Poisson distribution with average given by λ=1/4. This modifies the multiplicative constant for the price impact formula slightly to Y=√(4/3) which is in the correct range.

To summarise:

  • The quantum model treats supply and demand, not as fixed or independent, but as two coupled aspects of a single dynamic process
  • Transactions are inherently probabilistic – the uncertainty parameter σ is not an external noise term or addition, but is integral to the formula
  • The model responds dynamically to perturbations, and produces non-Gaussian statistics due to changing volatility
  • The model predicts (or postdicts) the square-root law of price impact, including a value for the multiplicative constant

For details, see the SSRN discussion paper Quantum impact and the supply-demand curve.

Previous: QEF12 – A quantum oscillator model of stock markets

Playlist: Quantum Economics and Finance

March 18, 2022

QEF12 – A quantum oscillator model of stock markets

Filed under: Quantum Economics and Finance — Tags: , — David @ 1:03 pm

We’ve seen how the quantum model of supply and demand can be used to model transactions and here we’re going to look at how we can use it to simulate the stock market.

With the quantum model supply and demand we have propensity functions for the buyer and the seller and the joint propensity function is equal to the product of those. A market maker would maximize profit if the spread between the buyer and the seller price, i.e. the distance between the two peaks, is equal to the standard deviation of the joint propensity.

This system is described by a linear entropic force but it does not behave like a classical oscillator because we know that the price is indeterminate and of course prices don’t actually oscillate. We therefore go to the quantum harmonic oscillator which is the quantum version of a spring system. The ground state is a normal distribution where the mass scales with the inverse variance. We’re going to define the frequency for the stock market in terms of turnover, so if the turnover is once per year then that will be the frequency.

One way that you can use the quantum harmonic oscillator to simulate the statistics of stock markets is to look at higher energy levels. This plot shows a tracing of the oscillations for the case where we have three energy levels. Most of the energy is in the ground state but there is also some energy in the next two energy levels.

We’re going to look at a slightly simpler version here but first of all we can compare with the actual sort of data that you get from a stock market. This plot is an order book for Apple stock over a single hour and so we’ve got the buyers putting up orders at lower prices and sellers at higher prices. These are not the same as for propensity functions because they don’t reveal all the preferences and of course orders in the middle region are going to clear so will disappear. However these orders do give an idea of the propensity functions which are shown by the shaded areas in the background.

Now, suppose that someone comes and makes a very large order and you want to know how much the price is going to change in response to that – it’s going to go up but by how much? According to the square-root law, which is an empirically derived relation from finance, the price change is given by the square-root of the size of the trade divided by the daily traded volume, all multiplied by the daily volatility, and then multiplied by a numerical constant Y of order unity.

In the oscillator model this makes sense because the restoring force is linear so the energy required to perturb the system is going to vary with the square of the displacement, or equivalently the displacement varies with the square-root of the energy. And of course the energy in a quantum harmonic oscillator is just a multiple of the frequency. When you have a large purchase that’s like boosting the frequency over that time period, which boosts the energy. Comparing these two formulas show that they’re in agreement if we have Y being of the order the square root of two. That sounds about right because in the oscillator model we’re assuming all the energy goes into lifting the price, so the actual price change might be a bit less than that.

This gives us a picture for how we could model prices in general. We can assume that we have an oscillator in its ground state, but then it’s displaced by perturbations which shift it from side to side. It is then going to oscillate in a coherent state, which means it stays in a normal distribution but it’s moving from side to side. The probability of being in a particular energy level varies with a Poisson distribution. We will assume that the spread between the bid and the ask price is equal to the standard deviation in order to maximize the profit.

So to summarise we have a normal distribution, which represents the joint propensity function, bouncing back and forth between the dashed lines in the figure. However there’s another participant in the market which is the market maker. They only transact at the particular ask/bid prices, shown by the vertical bars. We can therefore compute the probability of the ask or bid prices being selected.

If you look at the propensity for the ask price being obtained it follows a sinusoidal of plot. The dashed line is the actual result, the gray line in the background is the sinusoidal plot of the sort that you would obtain with a with a simple two-state quantum system for example. Similarly the propensity function for the bid would just be the opposite of that, so the propensity adds to one (we are assuming here that a transaction takes place at one of the prices).

Notice that for this particular setting the propensity varies from 0.25 to 0.75, so that’s plus or minus 25 percent which is consistent with the kind of shift that you might expect from the quarter law from quantum decision theory.

We can use this model to simulate stock prices. Here the squares are the ask price, circles are the bid price, and we are assuming that we have random perturbations, so unlike the previous plot the oscillations are not allowed to continue because random noise is being applied. We’re also going to have some noise added to the spread, i.e. the difference between the bid and the ask.

The noise in the spread turns out to be important because it’s a main contributor to the fat tails which you see when you look at the overall probability density for a long simulation shown here. This compares with the dash line which is the statistics for the Dow Jones Industrial Average.

To summarize, in neoclassical theory it is assumed that supply and demand cancel out in equilibrium, so there’s no concept of dynamics, force, mass, energy and so on. The quantum model in contrast is obtained by quantizing a linear entropic force. This linear restoring force is what explains the empirically derived square-root law of price impact, and the oscillator model can also be used to simulate the dynamics of stock markets such as price change distributions.

Further reading:

Orrell D (2022) Quantum oscillations in the stock market. Wilmott (forthcoming).

Previous: QEF11 – The money bomb

Next: QEF13 – Quantum supply and demand

Playlist: Quantum Economics and Finance

December 14, 2021

Quantum economics – the story so far

This piece gives a brief summary of my work to date (2016-2021) in quantum economics.

The idea that the financial system could best be represented as a quantum system came to me (dawned on me? evolved?) while working on The Evolution of Money (Columbia University Press, 2016). “Money objects bind the virtual to the real, and abstract number to the fuzzy idea of value, in a way similar to the particle/wave duality in quantum physics,” I offered. “Money serves as a means to quantify value, in the sense of reducing it to a mathematical quantity – but as in quantum measurement, the process is approximate.” Price is best seen as an emergent feature of the financial system. I summarised this theory in two papers for the journal Economic Thought: “A Quantum Theory of Money and Value” and “A Quantum Theory of Money and Value, Part 2: The Uncertainty Principle“.

While I had some background in quantum physics – I studied the topic in undergraduate university, taught a course on mathematical physics one year at UCL, and encountered quantum phenomena first-hand while working on the design of particle accelerators in my early career – my aim in the book (co-authored with Roman Chlupaty) was not to impose quantum ideas onto the economy. My primary research interest was in computational biology and forecasting and I had not touched quantum mechanics in many years. The dual real/virtual nature of money just had an obvious similarity to the dual nature of quantum entities, and in fact I was surprised that I appeared to have been the first to make this connection in a serious way and come up with a quantum theory of money.

I was aware that a number of researchers were working in applying quantum models to cognition and psychology, but it was only after finishing the book that I learned about the area of quantum finance (I also discovered a separate paper on “Quantum economics” by the physicist Asghar Qadir from 1978, which argued that the quantum formalism was well suited to modelling things like economic preferences). The reason I hadn’t come across these works in my research about money was because just like in neoclassical economics there was no discussion of that topic. Nor was there much discussion of quantum phenomena such as entanglement or interference. Instead the emphasis in quantum finance (as this paper notes) was on using quantum techniques to solve classical problems such as the Black-Scholes option-pricing algorithm, or portfolio optimisation.

My motivation was completely different. In books such as Economyths, and The Money Formula (with Paul Wilmott), I had investigated the drawbacks and limitations of these traditional models – so rather than invent more efficient ways of solving them, I wanted to replace them with something more realistic. Money was the the thing which linked finance and psychology, so a quantum theory of money could be a first step in developing a new approach to economics.

I sketched out the basic idea as an Economic Thought paper “Quantum economics” which served as a blueprint for my 2018 book of the same name. It tied together the quantum theory of money, with ideas from quantum finance, quantum cognition, quantum game theory, and the broader field of quantum social science. The ideas were also summarised in a piece for Aeon magazine – which was when I found out why no one had probably bothered to develop a quantum theory of money. The article was not well received, by economists but especially it seemed by physicists, some of whom went out of their way to trash the idea.

I was not new to having my work come under criticism. Indeed, much of my career has focused on pointing out the drawbacks and limitations of mathematical models, which has frequently brought me into conflict with people who don’t see it that way, starting with my D.Phil. thesis on model error in weather forecasting (see Apollo’s Arrow). My book Economyths also drew howls of outrage from some economists. However quantum economics felt different, and seemed to touch on a range of taboos, in particular from physicists who have long resisted the adoption of quantum ideas by other fields. But quantum mathematics is not owned by physicists, it is simply an alternative version of probability which was first used to model subatomic particles, but also can be used to describe phenomena such as uncertainty, entanglement, and interference which affect mental systems including the economy.

While writing the book I developed in parallel an online mathematical appendix which presented some key results from quantum cognition, finance, and game theory (an early version was translated into Russian). Because my aim was to develop a theory of quantum economics, I also started applying quantum methods to some key economic problems, including supply and demand, option pricing, stock market behaviour, and the debt relationship which underlies the creation of money. This online appendix later grew into my technical book Quantum Economics and Finance: An Applied Mathematics Introduction, first published in 2020 and now in its second edition.

For supply and demand, my idea was to model the buyer and seller in terms of a propensity function, which describes a probabilistic propensity to transact as a function of price. A simple choice is to describe the propensity function as a normal distribution. The joint propensity function is the product of the buyer and seller functions. The next step is to use the concept of entropic force to derive an expression for the forces which describes the tendency for each party to move the price closer to their preferred price point. The joint force is just the sum of the forces for the buyer and seller. However there is a contradiction because the probability distribution does not match that produced by an oscillation. To resolve this, we quantize the force to obtain a quantum harmonic oscillator whose ground state matches the joint propensity function. This model, which sounds elaborate but is actually quite minimal in terms of parameters, applies to economic transactions in general, so has numerous applications, including the stock market. The paper “A quantum model of supply and demand” was published in the journal Physica A in 2020.

Typical propensity functions for buyer (to the left), seller (right), and joint (shaded).

The question of how to price options is one of the oldest problems in finance. The modern method dates back to a 1900 thesis by Bachelier and is based on the concept of a random walk. For the quantum version, the logical place to start was with the quantum version of this which is a quantum walk. Instead of assuming that the log price will follow a normal distribution with a standard deviation that grows with the square-root of time, the model has two peaks which speed away from each other linearly in time. It therefore captures the psychological stance of an investor who has a bullish or bearish view on the asset (e.g. price might grow by 10 percent each year), but balances that with the possibility that the opposite might happen in order to obtain a fair price for the option. When coupled with the quantum model of supply and demand, the algorithm can be used to predict option price and volume. “A quantum walk model of financial options” was published in Wilmott magazine in 2021, and the theory was reported on the same year by the Economist in an article “A quantum walk down Wall Street“.

Probability distribution for a quantum walk (solid) versus random walk (dashed).

Finally a main question in quantum economics is the interaction between mind and money which underlies the debt relationship, and also the creation of money objects in the first place. Both of these topics are traditionally neglected in mainstream economics. In quantum economics it is easy to show that the debt relationship can be modelled as a simple circuit with two qubits, representing the debtor and creditor, entangled by a C-NOT gate which represents the loan contract. Interestingly, it turns out that the same circuit can be used to represent the decision-making process within the mind of a single person, where there is an interplay between a subjective context and the final decision. In quantum cognition, this is usually modelled as a two-stage process; however it can also be modelled using two entangled qubits, in which the context and the decision are separated out, as in the debt model. This result was published in a 2021 Frontiers in Artificial Intelligence paper, co-authored with Monireh Houshmand, called “Quantum propensity in economics“. A related paper published in Quantum Reports, that discusses applications including mortgage default, is “The color of money: threshold effects in quantum economics“. 

Two-qubit entanglement circuit for debt contract (A is debtor, B is creditor), or quantum cognition (A is context, B is decision).

For a full list of my research in quantum economics and finance, including links to these and other papers, please see the page Quantum Economics Resources. These findings and others are also presented in my technical book Quantum Economics and Finance: An Applied Mathematics Introduction, and for a general audience in Money, Magic, and How to Dismantle a Financial Bomb: Quantum Economics for the Real World (available 02/2022). The work continues! – if readers are interested in getting involved, please drop me a line here or through LinkedIn.

November 11, 2021

Ten reasons to (not) be quantum

Filed under: Economics, Quantum, Quantum Economics and Finance — Tags: , — David @ 11:23 pm

It’s not quantum physics.

Noam Chomsky, stating (correctly, in my view) that social relations are not the same as physics, but meaning (incorrectly, in my view) that they are simpler to understand, in his 2011 book How the World Works.

While the use of quantum models is becoming more popular in the social sciences including economics, it is still the case that when many people, especially those with a training in physics, hear the expressions “quantum economics” or “quantum finance” they immediately get confused, stop listening, and reach for some off-the-shelf arguments about why it must be nonsense (or some smelling salts). Here is a compilation of the usual ones, along with responses.

  1. Quantum mechanics was developed for subatomic particles, so it should not be applied to human systems. As one website claimed, “It’s only when you look at the tiniest quantum particles – atoms, electrons, photons and the like – that you see intriguing things like superposition and entanglement.” An article wonders why we “see the common-sense [classical] states but not the imponderable superpositions?”

Response: Bohr’s idea of superposition and complementarity was borrowed from psychology, as when we hold (or ponder) conflicting ideas in our heads at the same time, and the concepts of mental interference or entanglement are not so obscure. Also, many ideas from quantum mechanics such as the Hilbert space were invented independently by mathematicians. And calculus was developed for tracking the motion of celestial bodies but we don’t ban its application to other things.

  1. Quantum is too hard for non-physicists to understand. According to the physicist Sean Carroll, “No theory in the history of science has been more misused and abused by cranks and charlatans – and misunderstood by people struggling in good faith with difficult ideas – than quantum mechanics.”

Response: There is often a confusion between quantum probability, which is a mathematical tool, and quantum physics, which is about subatomic particles. Yes, the physics of subatomic particles is complicated – so are things like classical fluid dynamics. But quantum probability is just the next-simplest type of probability after the classical one. Most of the basic ideas involve simple linear algebra and some calculus. And the misuse of mathematical models which has created the most societal problems is the classical methods used in economics. As a side note, most people involved in quantum economics and quantum finance are physicists or (like me) mathematicians. Which brings us to:

  1. Quantum economics is physics envy, or an attempt to “appropriate the high prestige of physics” as one physicist put it.

Response: Mainstream economics is directly inspired by, and based on, concepts from classical mechanistic science. There is nothing inherently wrong with using the same mathematical tools for different areas, what is strange is when the tools used don’t change or adapt. As John Cleese said: “people like psychologists and biologists have still got physics envy but it’s envy of Newtonian physics and they haven’t really noticed what’s been happening the last 115 years.”

  1. Quantum is flaky, pretentious, pseudoscientific hype or woo. Sample usage: “As a quantum physicist, I’ve developed a reflexive eyeroll upon hearing the word applied to anything outside of physics. It’s used to describe homeopathy, dishwasher detergents and deodorant.” Cue nerd jokes about “quantum healing” or “quantum astrology”. In his description of what he called the Intellectual-Yet-Idiot, Nassim Taleb included anyone who “Has mentioned quantum mechanics at least twice in the past five years in conversations that had nothing to do with physics.” Science writers with no domain expertise in economics seem particularly eager to protect the faith.

Response: Quantum is a mathematical toolbox – it might come across as flaky or pretentious for a person to talk about it in the wrong context, but not to use it in their work. For example, complex numbers (i.e. those involving the square-root of negative one) are widely used in engineering to make some calculations easier. Quantum probability also gains much of its power from its use of complex numbers, so is more about the magic of imaginary numbers than the magic of subatomic particles.

  1. Entanglement is unique to special physical systems which can maintain quantum coherence. One science journalist told me that “Dollars don’t become quantum mechanically entangled. If they did, we’d be building quantum computers out of money.” A paper takes it as given that “one could not possibly idealize traders and investors as quantum objects exhibiting non-classical properties, like superposition or long-distance entanglement.” Physicists often conflate entanglement with Bell tests: one explained that “you can never violate a Bell inequality using systems like dice, dollars, or bank accounts. There is simply no way, and certainly no experiment has ever done so. (Maybe one or two ‘crackpot’ people claim otherwise, but they are not to be trusted.)” Another writes: “There is no chance that correlations in statistical economics will violate a Bell inequality … unless you fiddle the data.” In other words, using entanglement in an economics model is a sign of either gullibility, or scientific fraud.

Response: In mathematical terms, entanglement is a straightforward property of Hilbert spaces, and we can use it to model social and financial systems, including traders’ decisions or the behaviour of money. The key point is that it is being applied, not to macroscopic objects, but to information (see also #9 below). The Bell test is not a definition of entanglement, it is a way of teasing out a particular form of entanglement for subatomic particles. It is true that we can’t build quantum computers out of money, but nor can we build classical computers – does that mean money is not classical?

  1. Quantum is too complicated and mathematical – we need simpler models and less math. Variants: The economy cannot be reduced to equations, people are not subatomic particles. May quote Lin Yutang.

Response: The need for simple models is a theme of many of my books, however what counts is things like the number of parameters in a model. Quantum probability is more complicated than classical probability, but it is the simplest way to capture phenomena such as superposition, interference, and entanglement, which characterise many key mental and financial processes (for example, the quantum walk model for pricing options or the two-qubit model for quantum decisions or the quantum model of the volatility smile are not complicated). People are not subatomic particles, but nor are they classical particles, which doesn’t stop economists from using classical models, or talking about physics-like forces of supply and demand (they are just assumed to be at equilibrium, so cancel out). And while it is true that human behaviour cannot be reduced to equations of any sort, we use equations all the time to simulate the economy. Again, many of my books, such as Apollo’s Arrow, or Truth or Beauty, have criticised the overreach of mathematical models, but that is a separate issue and applies as much to classical models.

  1. Quantum is a forced analogy or a metaphor. As economist Paul Samuelson once wrote, “There is really nothing more pathetic than to have an economist or a retired engineer try to force analogies between the concepts of physics and the concepts of economics … and when an economist makes reference to a Heisenberg Principle of [quantum] indeterminacy in the social world, at best this must be regarded as a figure of speech or a play on words, rather than a valid application of the relations of quantum mechanics.” Some physicists seem to relax though with this interpretation because it is less threatening than a theory or a model.

Response: Quantum probability is a mathematical tool, which is not the same as an analogy or metaphor. The purpose of a metaphor is usually to describe something which is abstract and complicated in terms of something that is more concrete, so it would make more sense to go the other way and use human behaviour as a metaphor to help describe subatomic behaviour.

  1. The brain has not been shown to rely directly on quantum processes.

Response: Quantum effects appear to be exploited by biological systems in a number of processes (see quantum biology) but whether they are used in the brain or not makes no difference to economics. The argument is not that the economy inherits quantum properties from subatomic interactions in the brain, but that it can be modelled as a quantum system in its own right. For example, a debt contract can be expressed using a quantum circuit in a way which captures effects such as uncertainty, subjective context, power relationships, and so on.

  1. Markets are not quantum because there is no uncertainty. For example, something like a bank account, or an order book for a stock market, has clearly posted amounts and prices. One person compared her bank account to Schrödinger’s cat: “I am a PhD physicist, so for me the word quantum that gets thrown around is a bit ridiculous … So think about your bank account, it might be empty until you open it, so are you telling me that this is uh quantum finance or quantum economics okay you can have a million in your account or you can have zero we don’t know?”

Response: While it may be true that bank accounts are not like Schrödinger’s cat, I will let The Economist answer that one, from an article called “Schrödinger’s markets” in the print edition: “on a closer look finance bears a striking resemblance to the quantum world. A beam of light might seem continuous, but is in fact a stream of discrete packets of energy called photons. Cash flows come in similarly distinct chunks. Like the position of a particle, the true price of an asset is unknowable without making a measurement – a transaction – that in turn changes it. In both fields uncertainty, or risk, is best understood not as a peripheral source of error, but as the fundamental feature of the system.”

As computer scientist Scott Aaronson notes, quantum methods are adapted to handle “information and probabilities and observables, and how they relate to each other.” Since the financial system seems a pretty good example of information, probabilities, and observables (in this case through transactions) it seems like a suitable approach. Much of the confusion comes down to the fact that quantum economics is not quantum physics applied to the economy, but rather quantum mathematics applied to the economy (see figure below). Physicists often struggle with this because they tend to mistake their elegant models for reality (one even commented that “There is no quantum probability because quantum theory can’t be a theory created from probability”). But as Aaronson explains, “Quantum mechanics is what you would inevitably come up with if you started from probability theory, and then said, let’s try to generalize it so that the numbers we used to call ‘probabilities’ can be negative numbers. As such, the theory could have been invented by mathematicians in the 19th century without any input from experiment. It wasn’t, but it could have been.” Quantum mathematics should therefore be viewed as a mathematical toolbox that can be applied to either physical or social systems where appropriate.

The idea of quantum economics is not that physics can be directly applied to social behaviour as shown here …
… but instead that quantum mathematics can be applied to both physical and social systems. Figures from: Quantum Economics and Finance: An Applied Mathematics Introduction

People trained in physics tend to see quantum mechanics as a special theory with “many totally unintuitive predictions that makes it special,” as one put it. “Until at least one of them is borne out empirically, the onus is on you to convince us that QM is needed!” Entanglement for example is seen as a special property of subatomic particles – or “a surprising feature of the world” as another physicist emphasized (I get a lot of these emails) – and object that the two parties in a loan contract are not entangled in the same way. To understand the entanglement, it is necessary to lift the level of analysis from physical people, to mental constructs – which is entirely appropriate, given that money and value are mental constructs. From the perspective of the debt contract, if the debtor decides to default, then the state of the loan also changes immediately for both parties. And debt contracts are a feature of the world too (even if they are less remarkable or surprising than quarks or whatever). Perhaps the main point of quantum economics – and the hardest for people trained in physics to get, because it seems unintuitive to them – is that quantum properties can present in a way which seems familiar and intuitive (compare also the field of Quantum Natural Language Processing).

The above nine reasons for rejecting a quantum approach, which are the ones most commonly produced, are very superficial and are easily dismissed with a little reflection. (Skeptics sometimes prefer to say that they don’t understand or are “not convinced” without giving a specific reason, but my aim is not to convince people of anything, it is to lay out the facts as I see them and let others do their own research and come to their own conclusions.) Also, arguing against these reasons, as I have done above, will in my experience have absolutely no effect. One reason is that getting the quantum approach seems to involve something of an aha moment where it suddenly clicks into place. The other reason though is that they are not the real reason. So why is it that no one even tried to apply quantum methods to the economy until about a century after they were invented? This points to:

    10. Quantum economics touches on a range of taboo topics.

For the full picture read Money, Magic, and How to Dismantle a Financial Bomb: Quantum Economics for the Real World. Finally, given the numerous reasons to not take a quantum approach, I should point out that there also many reasons why the opposite is true, and the economy is amenable to a quantum treatment! In particular, quantum is the best framework for expressing in mathematical terms the complex interactions between mind and money. To see why, the best place to start is again with the books, or see this brief summary. For a mathematical treatment, see Quantum Economics and Finance: An Applied Mathematics Introduction.

[Update] My work in economics has seen me called a number of things including a conspiracy theorist, and the intellectual equivalent of a climate-change denier. More recently one physics professor read this piece and wrote, in a now-deleted tweet, that I was a charlatan who was ducking and weaving in order to avoid any criticism. I replied that he may have read the post, but he hadn’t understood it. He said “I judge you are not a crank. I judge you are a charlatan.” Then he thought about it (references to names redacted):

Any physicist worth their salt should agree with him that the only test is whether quantum math proves useful in modelling and prediction.

[Update] Further to the above statement about predictions, the reality is that most scientists – and certainly economists – care less about this than you might think (they are not all worth their salt). If a particular approach is initially judged as non-scientific then – as predicted by quantum decision theory – it is extremely difficult to overcome that judgement, which may involve strong subjective factors, by introducing empirical evidence. For example one physicist-turned-quant offered this assessment of my perpetual motion machine quantum model of price impact: “I unfortunately cannot follow your argument nor reproduce your train of thought. Our points of view about what is scientific and what is not seem too far apart, which is OK I guess.” He couldn’t quite put his finger on what was wrong, and was incurious about the fact that his own model was obviously incompatible with empirical evidence, while the quantum model gave accurate predictions – even though he had earlier written: “In the end, empirical observation must supersede all prejudices, so all ideas are a priori welcome. Time will tell.” Indeed. See: Orrell (2022) Market impact through a quantum lens. Wilmott 2022(122): 50-52.

May 17, 2021

Shifting the Economyths

Filed under: Economics, Quantum, Talks — David @ 3:29 pm

This is the text for my contribution to the online conference Beyond the False Dichotomy: Shifting the Narrative

I wrote Economyths a little over ten years ago, and in part the book was my response to the financial crisis. The thesis was that mainstream economics is based on a number of what I called economyths which were defined as beliefs or stories that have shaped economic thought. There were ten in total but to give you an idea here I will just mention four:

One said that the economy is made up of independent individuals. This is the idea that people are like the classical picture of a self-contained atom, and there is “no such thing as society” as Thatcher said.

Another is the idea that the economy is stable and self-correcting. This is the main story of economics from Adam Smith’s invisible hand, to modern equilibrium models.

A third is the idea that the economy is rational and efficient. The assumption that people make rational decisions to optimise utility is connected to the ideas of stability and efficiency. Corporations are defined to be an incarnation of rational economic man.

And then consistent with these is the next economyth which is that the economy is balanced and symmetrical. The issues of power and distribution are viewed as “soft” and peripheral to the subject, and therefore tend to be ignored. As one economist said in an interview, “economists are not good at what’s fair, right?”

Economists have always insisted that their theories are far more complex than these economyths would suggest, but the reality is that these assumptions have been extremely influential and in particular form the basis of the mathematical models used to simulate the economy and make policy recommendations.

In 2017 I did a revised version of the book which included an additional economyth, which was the idea that the economy boils down to barter. This is the myth which in many ways justifies the others, because it means that money isn’t important. Adam Smith for example focused on the “real” economy of labour and commodities and saw money as a kind of veil or a distraction. Economists since then have treated money as just a metric or an inert medium of exchange, and ignored its confounding properties – its dual real/virtual nature, its ability to entangle people through debt, its inherent instability, its tendency to cluster and create inequality, and its psychoactive effects on the human mind.

The drawbacks of omitting things like money and banks from the model became evident after the financial crisis of 2007, when it turned out – as one central banker explained ten years later – that “In the prevalent macro models, the financial sector was absent, considered to have a remote effect on the real economic activity.” And today, the continuing problems of financial instability, social inequality, and environmental degradation can all be largely traced to the money system, which is unstable, unfair, and is reliant on continuous growth to pay off debt.

One reason I called these ideas economyths is because, like myths, their legacy goes back a long way. We could start with post-war economists like Milton Friedman; or go back further to the neoclassical economists who first tried to establish economics as a kind of social physics in the late nineteenth century; or to Adam Smith, who didn’t try to quantify economics but was inspired by Newton. But I would argue that we can actually go back much further. My reason for saying this is that the ten economyths were based directly on a list of opposites, divided into good and evil, from the philosopher Pythagoras, who believed that the universe was based on number, and who lived at a time when coin money was changing the world of commerce. As he said, “number is all” and money is a way of putting numbers on the world.

In order to understand the narrative appeal of this model, it is interesting to compare it with another model which was very influential for a long time, and also reflected the Pythagorean ideals of symmetry and mathematical elegance, namely the Greek model of the cosmos. This model incorporated two main assumptions. The first was that the celestial bodies moved in circles, which were considered the most perfect and symmetrical of forms. The other assumption was that the circles were centered on the Earth. In Aristotle’s version, the planets and stars were thought to be encased in crystalline spheres which rotated around us at different speeds. The fact that planets did not follow perfect circles around the Earth, but sometimes tended to loop back on themselves, was handled by adding epicycles – circles around circles.

This geocentric model was complemented by a theory of physics. According to Aristotle, all matter consisted of the five elements Earth, Water, Air, Fire, and Ether which was reserved for the heavens. His theory was less a theory of motion, than a theory of stability: each element sought its own level, following the same order with Earth on the bottom and Ether on top, and in fact would do so instantaneously in a vacuum. Aristotle deduced from this that a vacuum could not exist: nature abhors a vacuum.

The Greek model lasted for well over a thousand years – it was adopted by the Church, and was not finally overturned until the Renaissance. How did it manage to last for such a long time? And what lessons are there for economics?

One reason for its durability was that it could make accurate predictions of important events such as eclipses. Another reason, though, is related to aesthetics and the fact that, as Aristotle put it, man is a political animal. There was a strong parallel between the perceived order of the cosmos and the order of society. Greek society was structured as a well-ordered hierarchy, with slaves at the base, followed in ascending order by ex-slaves, foreigners, artisans, and finally the land-owning, non-working upper class. These men alone could be citizens, and oversaw everything from above, like the stars in the firmament (women did not take part in political life and took their social class from their male partner). A model of the universe which suggested that everything has its natural place in a beautiful, geometrically-governed cosmic scheme therefore supported the status quo. For this reason it would certainly have appealed to the male leisure class that ruled ancient Athens, and later to the Catholic Church.

The first cracks in the model appeared in 1543, when Copernicus proposed that the Earth might go around the Sun, rather than vice versa. European astronomers began to observe comets which passed between the planets, so if Aristotle’s crystalline spheres had actually existed, they would have broken through them. Finally, in the late seventeenth century, Isaac Newton derived his three laws of motion and the law of gravity. The static circles of classical geometry were replaced with dynamical equations, which had a different but equally powerful aesthetic appeal. Sometimes it takes a model to defeat a model.

So what does this have to do with economics? Well, there is an obvious parallel between the Greek circle model, and our modern model of the economy, because it too pictures the world as rational, stable, ordered, and efficient, and therefore favours the elite. Indeed mainstream economics, in its obsession with rationality and efficiency, sometimes sounds like the PR wing of the financial sector. The model can’t make predictions of things like crises, or indeed the economy in general, but it goes a step further by predicting that it can’t predict, as per efficient market theory. The main difference is that while Aristotle thought that a vacuum could not exist, because otherwise things would find their equilibrium immediately, efficient market theory assumes that prices do reach equilibrium instantaneously – so a vacuum does exist, and it is the market.

Both the Greek circle model and the economics model also show the strength of a mathematical model. Even something like the financial crisis only made a dent. As Paul Krugman wrote in 2018, “Neither the financial crisis nor the Great Recession that followed required a rethinking of basic ideas.” So like the Greek model, the orthodox model of the economy is incredibly resilient.

I think if there was a specific point where this narrative became compelling in economics, it was when economists changed their theory of value, while leaving the rest of the theory mostly the same. The switch was announced 150 years ago by William Stanley Jevons, in the second paragraph of his 1871 Theory of Political Economy, where he wrote that “Repeated reflection and inquiry have led me to the somewhat novel opinion that value depends entirely upon utility.” Here utility was a kind of energy-like quality which roughly equated to happiness. Classical economists such as Smith had followed a labour theory of value which acknowledged the role of power. Utility created a new narrative which flipped this on its head. Economics was now about pleasure and good times.

Of course utility couldn’t be measured directly. Another approach though was to simply assume that utility is reflected by price. Or as Jevons put it: “just as we measure gravity by its effects in the motion of a pendulum, so we may estimate the equality or inequality of feelings by the decisions of the human mind. The will is our pendulum, and its oscillations are minutely registered in the price lists of the markets.”

A side-effect of this emphasis on subjective utility, ironically, was that by reducing value to a number, subjective things like emotion or social power or dignity or ethics were usurped by the theory and thus stripped of all weight. As Tomáš Sedláček wrote for example in his 2011 book The Economics of Good and Evil, “The issue of good and evil was dominant in classical debates, yet today it is almost heretical to even talk about it.” Another thing which also of course didn’t fit into this rational utility approach – it doesn’t compute – was money, with its economyth-defying properties of entanglement, instability, and irrationality.

But still, why is it that more than a decade after the crisis economics still remains rooted in the past? Why is the orthodox narrative so resistant to change?

One reason again is that as with Aristotelian physics, the economyths form part of a connected structure. The story they tell is consistent and self-reinforcing – markets are efficient, stable, rational – and you can’t change one part without changing them all.

Instead, what happens is that economists attempt to fold in new ideas from other areas such as complexity or behavioural psychology, without changing the structure too much. Behavioural economics for example has in my view won acceptance exactly because it can be incorporated in this way, and viewed as an epicycle that can be wheeled out for particular situations. Economists also try to add in so-called “frictions” to their equilibrium models, but assume the equilibrium exists in the first place. Paul Krugman again: “We start with rational behaviour and market equilibrium as a baseline, and try to get economic dysfunction by tweaking that baseline at the edges.” A few more epicycles and it will be perfect, the thinking goes.

The story also offers a powerful restoration narrative, because it says that crises and upsets are caused by external events, and economic forces bring the economy back to this imagined equilibrium. It therefore taps into the human desire for order which is important in politics but also in fiction. And the idea that markets are rational and efficient also justifies the powerful position of the financial sector and the wealthy, while at the same time distracting from the workings of power and the role of money.

To change the narrative it isn’t enough to modify details of the model, instead we need to go back as the neoclassicals did to the fundamental idea of value and its relation with price – in other words, the question of how much something is worth, its numerical cost. There are a number of ways of going about this, but I would argue that a good place to start is with the math.

This might seem counter-intuitive. After all, it is something of a cliché to say that economics is too mathematical. As one recent book put it, “today mainstream economics follows a path of great mathematical rigor that . . . does not make much room for other accounts of economic life.” However rigor isn’t useful if you are using the wrong kind of mathematics in the first place. And while Keynes wrote that “practical men … are usually the slaves of some defunct economist” we could also say that those economists are themselves slave to beliefs about number and value that are inherited in large part from a mathematical model.

Orthodox theory for example is based on the core idea that markets are stable and price is a measure of inherent value. It is best represented by the X-shaped figure of supply and demand. This plots supply as a line which increases with price, and demand as a line which decreases with price. The point where they match in an X represents the stable equilibrium where the market clears. This diagram appears in all introductory economics textbooks, but it is also there in the mathematical models used to simulate the economy.

An odd feature of the graph is that price appears on the vertical axis, and is assumed to be determined uniquely and passively by unknown forces of supply and demand which are in balance, so cancel out. While neoclassical economics is often described as Newtonian, it assumes equilibrium and has no real concept of mass or dynamics or force. The problem though is we never observe supply or demand independently, we only observe transactions. Like the crystalline spheres, these are imaginary constructs. Indeed the whole idea of representing a complex dynamic system as the stable intersection of two lines is very strange and is not done in other areas such as biology, where the only things that are at equilibrium are dead.

To make the diagram more scientific, a first step is to flip it around so that price is the independent variable on the horizontal axis. This seems a trivial change but is actually key to the whole story. Instead of utility, we can then plot the propensity curves for the buyer and seller, which represent the probability of transactions as a function of price. The fact that price is somewhat arbitrary and decisions are subject to effects such as context means these curves are inherently probabilistic. The X-shaped lines of supply and demand from the classical diagram are therefore replaced by probabilistic waves which only collapse down to a particular price during a transaction. This model therefore acknowledges that value is a soft and fuzzy concept, while price is a measure which is subject to intrinsic uncertainty.

The next step is to acknowledge that people are not separate atoms who only communicate by bouncing off one another, instead they are entangled beings who talk to each other, and share things like social norms. An example is the prisoner’s dilemma game, another staple from the textbooks: in classical theory everyone defects and rats out on the other person, but experimental results show people choose to collaborate between a third and a half of the time, which suggests a high degree of entanglement.

We also need to address the fact that people make decisions based on a mix of objective and subjective factors which are entangled in the mind and may interfere with one another. And above all we need to include the dynamics of money, which behaves more like a kind of information than a classical physical object.

The correct mathematical framework for this theory has already been developed by a group of radical thinkers over a century ago. Unfortunately this mathematics has until recently been reserved for the esoteric area of subatomic particles. I’m talking of course about the quantum formalism. For our purposes this refers, not to quantum physics, but to a kind of logic and probability which allows for effects such as interference, entanglement, and the idea of a measurement procedure. The word quantum is from the Latin for “how much” which applies naturally to the economy, where prices are measured through transactions. The point is not that there is a direct map between subatomic particles and humans, but that we can use similar mathematical tools to model each, which is a subtle but important distinction.

While quantum ideas have been around for a while, they are only now starting to reach critical mass in the social sciences. This new quantum narrative is under construction, in economics and finance but also in other areas such as psychology and political science. For example the Carnegie Foundation is funding a series of quantum bootcamps for social scientists at Ohio University starting this summer. There is a new anthology coming up this year from Oxford University Press on Quantizing International Relations. Danah Zohar has been bringing quantum principles into management theory for some time. The area which is seeing the most rapid adoption of quantum ideas is finance, because of quantum computing. As far as I know you can’t take a university course in quantum finance, but you can get a job in it right now in financial centers such as Paris, New York, Toronto, and so on. The development of classical computers in the post-war era changed the way we model and think about systems including the economy, and quantum computers – which have entanglement built in – are now doing the same thing.

What counts for the purposes of today’s discussion though is not the math, but the story told by the math. To summarise, the core narrative of mainstream economics is that people behave like classical atoms: hard, independent, stable. The economy can therefore be modelled as an equilibrium system. The main message of quantum economics is that people are entangled: with their own subjective feelings, with other people, with what they read in the news, and above all through the money system. The economy is a complex, dynamic, living system which can be modelled using a mix of techniques, such as ones from complexity science or systems dynamics, so long as they respect the indeterministic and entangling nature of both mind and money; and particularly the ability of the money system to scale up cognitive and financial entanglements to the societal level.

So as a one word description of the new narrative I would choose quantum or maybe entanglement. If you don’t want to do quantum mathematics, which is understandable, it doesn’t matter because what counts is the idea that the economy is best seen as a complex system which is entangled through a mix of financial and social effects. In practical terms this means that all the so-called “soft” ideas such as subjectivity, emotion, social dynamics, power, money, value, fairness and ethics that have been exiled from economics are now back in play. Debtors and creditors are entangled, shareholders and stakeholders are entangled, and we are all entangled with the climate system. Obviously the quantum approach doesn’t have all the answers, but its built-in emphasis on uncertainty can be liberating, and encourages a pluralistic response. Perhaps it is the model which teaches us to sometimes at least let go of models, because they can’t capture enough of the complex reality. And my hope is that the quantum approach and the idea of entanglement resonates with some of the other ideas and narratives discussed today.

Again, the idea that a new narrative should begin with our system of logic and probability may seem strange but history shows that mathematical models have great influence. And sometimes, as mentioned, it takes a model to beat a model. A new narrative which marks the next evolutionary step of capitalism is going to need a new mathematical framework, if only to better define its language, and help to do an audit on what ideas one may inherited, perhaps unconsciously, from the classical model. It is ironic that we live in an age characterised by volatility, uncertainty, complexity, and ambiguity but our economic theory assumes a deterministic state of placid equilibrium. It is therefore well overdue for an upgrade.

Further reading: quantum economics resources

Older Posts »

Blog at