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.
Figure 1. Right panel shows volatility as measured by the VIX index versus monthly price change for the S&P 500 January 1992 to August 2022. Left panel is same for an artificial lognormal distribution. Red line shows average volatility, black line is the theoretical smile curve. The lognormal distribution shows no smile, while the market data smile is well approximated by the quantum formula.
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.
Figure 2. Left panel shows Black-Scholes model cost (blue line) for 1-month straddle options, versus average payout for a lognormal distribution. Right panel shows the same for S&P 500 historical data. The red line shows the quantum model where volatility follows the smile curve. The Black-Scholes model is well calibrated for the artificial lognormal data, as expected, but the quantum model is a better fit for the market data.
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.
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 quantumapproaches?
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.
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 Economythsfor 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
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).
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).
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.
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.
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:
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.”
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.
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. 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?
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.
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.
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.
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 …
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.
[Update] My work in economics has seen me called a number of things including a conspiracytheorist, 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.
These are the notes to the first of a series of presentations on quantum economics and finance. For the video version see here.
Quantum economics and finance uses quantum mathematics to model phenomena including cognition financial transactions and the dynamics of money and credit. In these talks we’ll be talking about topics including: why quantum in the first place; quantum probability and logic; basics of quantum computing; quantum cognition ; quantum walk; quantum game theory; quantum supply and demand; threshold effects; option pricing; and the money bomb.
So why quantum in the first place? The quantum revolution in physics was born when physicists found that at the subatomic level energy was always exchanged in terms of discrete parcels which they called quanta, from the Latin for “how much”. In economics the equivalent is financial transactions, like when you buy an ice cream in Italy and you say “quanto costa” which makes the quantum connection a little clearer. So money behaves in some ways like an object but not a classical one. It shows signature properties of quantum systems such as discreteness, indeterminacy, entanglement, duality, interference and so on.
Perhaps the most obvious such property is the way that money jumps. In physics Erwin Schrodinger said “If we have to go on with these damned quantum jumps then I’m sorry that I ever got involved” but with financial transactions of course the same thing happens all the time. For example when you tap your card at a store, the money doesn’t flow out continuously, it just jumps.
In physics the position of a particle is fundamentally indeterminate and is in a sense constructed by the measurement procedure. It’s the same thing in markets. If you put your house up for sale you will have a fuzzy idea of the price but the actual monetary value is only determined at the moment of the sale.
And this is money’s job: it’s a way to collapse value down to number. Theorists often talk about money but one thing that isn’t often emphasized is the most basic feature which is is its connection with numbers. If you look at a US dollar bill for example you see it has a numerical 1 and a word “one” in each corner, so it’s got quite a few ones, and then it’s got “one dollar” down at the bottom, and a big “one” in the middle, and there’s a lot more ones on the other side, so they’re really emphasizing the connection with one, and that is money’s most basic property – that it combines the properties of a real owned thing with a virtual number.
These dual real/virtual properties are reflected in the two main historical theories of money which are bullionism – money is gold and nothing else as JP Morgan said – and chartalism, which is the idea that credit alone is money as Alfred Mitchell-Innes said the next year. But then Bitcoin comes along and on the one hand it seems to be completely virtual, but on the other hand it’s also real as you’ll notice if you happen to lose the hard drive in which your bitcoins are located.
The duality of money is therefore similar to the duality of light. Wave-particle complementarity has been reflected in theories of light that go back millennia – Aristotle thought light was a wave, and Newton thought it was particles, and this bounced back and forth until finally the quantum theory came along and showed that it has properties of both at the same time. It’s the same with money.
In economics we’re used to treating preferences as something like fixed and known objects, adjusted for some cognitive biases, but often our preferences are made up in response to questions which act like a kind of a measurement event. So thoughts and ideas behave in some ways like objects but they’re not classical objects. In physics, Bohr’s theory of wave particle complementarity was actually inspired by the observation from psychologists that we can hold opposite ideas in the mind at the same time in superposition, and in fact it’s these interference terms which play a very important role in quantum cognition as we’ll see.
In physics, particles can mysteriously become entangled so they act as a single system. In the financial system there is a much more direct form of entanglement where financial assets and virtual liabilities have these quantum characteristics of entanglement.
In economics there’s this idea of rational economic man, who is like a kind of robot. The picture which is emerging from quantum social science is a quantum economic person who’s entangled, indeterminate, dynamic, paradoxical and alive. As the philosopher Slavoy Zizek said, a fact rarely noticed is that quantum physics appears to defy our common sense view of material reality, but it seems to apply somewhat better to human reality where the human spirit encounters itself outside itself.
Further reading:
Orrell D and Chlupatý R (2016) The Evolution of Money. New York: Columbia University Press.
What does it mean when we say we want to quantize economics? Well, it doesn’t mean that we’re emulating (or somehow abusing) quantum physics. Quantum mathematics is about information and probabilities and observables and how they relate to each other, as Scott Aaronson said, and money is a form of information that does not behave like a classical object.
Now when we try to apply quantum ideas to areas outside of physics we encounter a lot of obstacles. One is that we’ve constantly been told by famous physicists and mathematicians that quantum mechanics is fundamentally incomprehensible, so we’ll never really understand it. We have these weird phenomena like Schrodinger’s cat which can be alive or dead at the same time. We’re often told that concepts such as superposition and entanglement only apply to the tiniest quantum particles and you’ll never encounter them in your everyday life. And then finally there’s this impenetrable mathematics. But let’s just forget physics for the time being, and think of something much simpler like a coin toss.
If we wanted to model the state of a coin toss where we don’t know the outcome, there are two possible outcomes heads or tails, so we would want a two-dimensional space. And we could represent the state of the coin, if it was a balanced coin, with a diagonal line (see screenshot of video below).
That balance between the two states is really the idea of quantum probability. We can represent heads as an up arrow in Dirac notation, which is used in quantum mechanics, or as a vector (1,0)T. Tails can be a down arrow or a vector (0,1)T. Our superposed state is a mix of these two, a balanced combination of up and down.
In order to get the probabilities we’ll just take the projections. The projection on the horizontal axis, when squared, is going to give the probability of heads, and the probability of tails is going to be the square of the projection onto the vertical axis. The reason we are squaring them is because we want the projections to always add to one, and by the Pythagorean theorem that will be the case.
So the basic difference between classical and quantum probability is that classical probability uses what’s called the 1-norm, so the options are heads or tails, while quantum uses this 2-norm which involves the square. Making this change from a 1-norm to a 2-norm leads to all these different quantum concepts such as superposition, negative probability, interference, and entanglement.
So what do we mean by negative probability. Well if we just flip our state over, mirroring it around the vertical axis, then now we’ve got a negative probability for heads. The size of the probability when we take the norm, because we’re squaring it, is obviously positive again so nothing has really changed, except that when we add probabilities together we can get a plus and a minus canceling out in what are called interference effects.
We can also think of more complicated systems. Imagine for example that we had two coins, so now there are four different things that we need to keep track of: there’s heads and tails for one coin, and heads and tells for the other coin. That’s four dimensions which is obviously not easy to draw. But quantum coins can be entangled so for example you might have the only possibilities as heads-heads or tails-tails. So what this means is that just by making this switch from a one norm to a two norm we’re allowing all of these quantum phenomena such as superposition, interference, entanglement and so on.
Further reading:
Aaronson S (2013) Quantum Computing Since Democritus. Cambridge: Cambridge University Press.
The approach which we’ll be following is based more on quantum computing than on quantum mechanics per se, so let’s have a look at how these quantum computers work.
Quantum computers perform their operations using qubits which are in a superposed state, so one big difference between them and classical computers is that they’re inherently probabilistic – you have to run a simulation many times and then sample the output .
The basic mathematical tool we’ll be using is the Hilbert space. This can be viewed as a generalization of Euclidean space, with the difference that there are complex coefficients. Quantum probability involves negative probabilities (in the sense of projections) so we need complex numbers for example when we take the square root of a probability.
The dual state is the complex conjugate of the transpose of , also called the Hermitian conjugate, which is written as .
The inner product between two elements and is denoted , and is analogous to the dot product in a normal vector space, with the difference that the result can again be complex. The norm of is given by which is a non-negative real number.
The outer product is denoted , and is like multiplying a column vector by a row vector.
A unitary matrix is one whose inverse is its conjugate transpose. Unitary matrices preserve probabilities and play a big role in quantum computing, where logic gates are represented by unitary matrices that act on qubits. An example of a gate is the NOT gate which has the effect of flipping a qubit, so if the input is then the output will be . There’s the Hadamard gate which we’ve already seen. It takes a an input of say and puts it into a superposed state. There’s a rotation gate which simply rotates the qubit by an angle , so if the input is then the output will be a superposed state which has the probability of being measured in and of being measured in . The symbol for measurement is the gauge symbol.
Multiple qubits are denoted as a tensor product , so two qubits gives you a column vector with four elements.
A state in a Hilbert space is entangled if it does not factor as a tensor product of the form where and . For example , which has an equal probability of being observed in the state or , is not entangled because it can be written as a tensor product; however the state , which has an equal probability of being observed in the state or , cannot be similarly decomposed, so is entangled.
Gates can act on multiple qubits, an example is the C-NOT gate which here flips the state of the lower qubit depending on the state of the upper qubit: , , , .
C-NOT gate
The Toffoli gate on the left is similar but acts on three qubits: it flips the state of the first qubit depending on the state of the two control qubits. The figure below shows a quantum circuit with three qubits. The first gate is a Toffoli gate, the middle gate is a C-NOT gate which acts on two qubits, and the last gate on the right is a NOT gate which flips the bottom qubit.
A quantum circuit to increment a counter
If we analyse the effect of this circuit, we find that , , and so on. The circuit therefore has the effect of incrementing a binary counter.
Another circuit that we’re going to be using quite a lot is the two-qubit figure one shown in the video screenshot below. We have a unitary matrix acting on the first qubit which might for example rotate that qubit around in a particular direction, and the same thing for the lower qubit. Then the top qubit is going to be acting as a control on the second one. As we’ll see this circuit can be used in quantum cognition, where the top qubit can represent a context for a decision, or also in quantum finance where we’re going to be using this to model the debt relationship.
in Dirac notation, which is used in quantum mechanics, or as a vector (1,0)T. Tails can be a down arrow 
or a vector (0,1)T. Our superposed state 
is a mix of these two, a balanced combination of up and down.
in a Hilbert space 
is entangled if it does not factor as a tensor product of the form 
where 
and 
. For example 
, which has an equal probability of being observed in the state 
or 
, is not entangled because it can be written as a tensor product; however the state 
, which has an equal probability of being observed in the state 
, cannot be similarly decomposed, so is entangled.
, 
, 
.
, and so on. The circuit therefore has the effect of incrementing a binary counter.
The Future of Everything 