A Quantum Oscillator Model of Stock Markets

Abstract

Classical oscillators have long been used to model financial time series, but suffer from the drawback that the underlying system does not behave like a mechanical spring: there are no regular oscillations, and price is not a well-defined mechanical quantity but always has a degree of uncertainty. In recent years a number of authors have attempted to address these problems by basing their models on quantum harmonic oscillators, which always have a non-zero volatility, however the role of other features that are characteristic of quantum systems, such as discrete energy levels, has not been clear. This paper develops a quantum model in which the energy level corresponds to an integer number of transactions. The model is derived by quantizing entropic forces which represent the intentions of buyers and sellers to transact as a function of price. It is shown that the model captures the non-Gaussian nature of financial statistics, and correctly predicts empirical phenomena including the square-root law of price impact, along with its associated variance.

1. Introduction

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.

William Stanley Jevons (1905)

Since at least the time of Adam Smith, mainstream economists have emphasised the notion that markets drive prices to an equilibrium value which refect intrinsic value. And yet one of the most persistent features of the economy is that prices are unstable, as shown by the phenomenon of financial booms and busts.

One approach to modelling such events, is to assume that prices are subject to external or internal perturbations, but are returned to equilibrium by the forces of supply and demand. William Stanley Jevons (1909) for example argued that financial crises were driven by variations in sunspot activity, which affected global temperatures and therefore the price of wheat. Jevons’ contemporary, the French economist Clement Juglar, independently found a boom/bust cycle in investment of 9–25 years, and identified four phases: prosperity, crisis, liquidation and recession (Juglar, 1862). Other such cycles identified by economists included the Kitchin cycle for inventory of 40 months (Kitchin, 1923); the Kuznets infrastructure cycle of 15–25 years (Kuznets, 1930); and the Kondratiev technology wave of 45–60 years (Kondratiev, 1925). In his 1939 book Business Cycles, the economist Joseph Schumpeter proposed that economic fluctuations were caused by all these different cycles adding together, as he believed occurred at the start of the Great Depression.

The idea that the economy has regular cycles assumes the existence of a restoring force (Smith’s invisible hand) that tends to pull prices back towards a central equilibrium. The fact that oscillations are not perfectly regular can be accomodated by assuming that perturbations have a random quality, and by including a degree of damping so that oscillations do not continue indefinitely. The Norwegian economist Ragnar Frisch (who also coined the term econometrics) was jointly awarded the first Nobel Memorial Prize in economics for his work on the dynamics of damped oscillator models (Frisch, 1933, 1936).

Oscillator models fell out of favour during the post-war era, and were inconsistent with the efficient market hypothesis (Fama, 1965). This assumed that prices are restored to equilibrium immediately, and viewed price changes as following a random walk with perturbations caused by external news. It therefore made no sense to model markets in terms of classical springs, unless perhaps the springs were infinitely stiff; so those perceived market oscillations were all in the imagination. The random walk picture was key to the Black and Scholes (1973) option-pricing algorithm, where it was assumed that asset prices fluctuate in a random fashion with a certain known volatility.

While versions of the random walk picture continue to dominate the field of quantitative finance, it has long been known that market statistics do not perfectly follow the Gaussian distribution predicted by the theory. Another drawback is that it assumes that price at any time is a well-defined quantity, while in fact the actual price is always indeterminate except at the exact time of a transaction. If an asset hasn’t traded for a long time, then its price is uncertain, especially if market conditions have changed (Sarkassian, 2020).

Motivated by these considerations, a number of authors have investigated the use of quantum harmonic oscillators. As discussed further below, a quantum harmonic oscillator is the quantum version of a classical oscillator or spring system, and can be obtained by quantizing the equation for a linear restoring force. The resulting wave function has a ground state which is a normal distribution, along with higher-order modes with more complex distributions.

Perhaps the first paper to take this approach was Ye and Huang (2008), which assumed that the quantum harmonic oscillator was in its ground state most of the time, but had an additional oscillatory term so that the expected price swung between highs and lows, plus an attenuation term for damping. Meng, Zhang, Xu and Guo (2015) used a version of the model which included a boundary condition on the wave function, while Meng, Zhang, and Guo (2016) extended the approach to a stock index. Ahn et al. (2017) derived a model which included additional damping terms, while Gao and Chen (2017) and Lee (2021) used quantum anharmonic oscillators to explore the dynamics introduced by different actors such as momentum and fundamental traders.

In all these studies, it was found that the quantum model could recreate the basic properties of financial statistics, which perhaps is not surprising since they contained enough tuneable parameters to fit the data. However questions remained about whether the quantum approach was appropriate in the first place. If markets do not behave like classical oscillators, then they do not on the surface always seem to resemble quantum oscillators either: for example, what does it mean to say that the market has discrete energy levels? And more generally, what is the justification for adopting a quantum approach?

In this paper, we consider a particular quantum model, based on a previously published model of supply and demand, in which the propensity of agents to transact is viewed as being the product of entropic forces. Switching to a quantum framework yields a quantum oscillator describing a probabilistic wave function. The energy level is interpreted as a number of representative transactions over a time step. The model therefore integrates the dynamical view of markets as having oscillatory properties, with the discrete probabilistic picture common in finance. The use of the model is justified by its ability to explain and predict a variety of financial phenomena, including price impact and the relationship between price change and volatility.

The outline for the rest of the paper is as follows. Section 2 develops a quantum model to be applied to stock markets. Section 3 explores the connection between uncertainty and market spread. Section 4 shows how the model relates the concepts of volatility and price impact, and draws on previously published work to compare the model predictions with empirical results. Section 5 discusses the use of quantum models in finance, and summarises the results.

2. The Quantum Spring

Consider first the case of a single buyer and seller, who are negotiating the price of a single unit of stock. As discussed in (Orrell, 2020) we can model the probability of a transaction occuring at a particular price, where price $x=\log \left(p\right)$ is a logarithmic variable, using a normal distribution. The price will therefore vary randomly, as in the random walk model of efficient market theory, while the total probability of a transaction taking place within a certain time interval depends on factors including the spread between the buy and sell prices.

This model, which is based on classical probability, is similar to one that was developed by Kondratenko (2015) and validated against stock market data (Kondratenko, 2021). However the classical approach has a number of limitations. One is that it treats the price as being only subject to random noise, as opposed to being the product of a fundamentally inderministic process which is measured through transactions. In fact the model doesn’t say anything about transactions at all, other than giving a probability that one might occur. The standard deviation is assumed to be constant, while we know that the volatility of financial data changes with time. Also, it is based on classical probability so is not well-suited for handling the interplay between complex dynamics and discrete exchanges, along with effects such as entanglement and interference, which characterise the financial system and social interactions in general.

We therefore make the modelling choice to switch to a quantum framework, and model the propensity as the product of a complex-valued wave function. A first step is to note that the propensity curve can be viewed as being the product of linear entropic forces (Sokolov, 2010), which act as a restoring force on perturbations to the price (Orrell, 2020). The entropic force of the buyer and that of the seller pull in opposite directions, and the net force acts on price in a manner similar to the restoring force of a mechanical spring. The energy associated with these forces is related to information, so a distribution with a narrower spread (more information) will have a higher energy than one which is more dispersed.

The next step is to quantize the entropic force, following the rules of quantum probability which preserve probability for a dynamic system. The result is a quantum harmonic oscillator whose ground state matches the joint propensity curve (Orrell, 2020). The corresponding wave function has a complex component, and rotates around the real axis with a frequency $\omega$ . The associated mass is given by $m=\hslash /\left(2\omega {\sigma }^2\right)$ where $\hslash$ is the financial version of Planck’s constant. The number $\hslash$ is a measure of uncertainty that is not universally fixed as in physics, but can be treated as constant for a particular simulation. The mass is therefore a measure of inverse variance which provides a way to describe different levels of flexibility in price negotiations.

The result of this process is a quantum oscillator model, derived from a general model of transactions, which represents a single potential transaction between a representative buyer and a seller for one unit of stock over a time step of length $∆t$ . The variable $\omega$ relates to the turnover rate. A typical frequency for a stock market would be of the order of once per year, which is slow compared to the representative transaction time $∆t$ .

As mentioned earlier, a key difference between a classical oscillator and a quantum oscillator is that the latter has discrete energy levels, with a base energy $E_0=\hslash \omega /2$ , and higher levels with energy $E_n=\left(2n+1\right)E_0$ . The base energy $E_0$ can be viewed as the energy (or the information) required to change a person’s mind about the transaction (Orrell, 2021; see also Khrennikova, 2019 for a discussion of information processing in quantum cognition). In a similar way, we interpret the energy of the oscillator in terms of the number of transactions that occur, for that oscillator, in a time step (Orrell, 2020). This can be compared with the Fock state interpretation from quantum mechanics, which identifies the energy of an oscillator with the number of associated particles. The energy is therefore a discrete quantity when measured over a single time step (which makes sense since there is also a finite number of transactions), but will appear to be continuous when measured over many time steps (or over an ensemble of oscillators in a more complicated model).

3. Uncertainty and Market Spread

A distinguishing feature of stock markets is that a stock never has a single price; instead there is a range of ask prices from sellers, and bid prices from buyers (Sarkassian, 2020). The spread $\mu$ , defined as the difference between the highest bid and the lowest ask, therefore represents a fundamental level of uncertainty. In order to incorporate this effect, we set the ground state volatility equal to the half-spread so $\sigma =\mu /2$ . Because price evolves by toggling to either the bid or the ask, we assume that at the start of each time step the oscillator is in its ground state, and the price is then perturbed by an amount $∆x=\pm \sqrt{\lambda }\mu$ , in a quantum version of a random price change.

If we set $\lambda =1/4$ , then the price change $∆x$ equals the half-spread $\mu /2$ , which would correspond to a case where the bid/ask spread is responsible for all of the uncertainty. We will use that as the default value, however a higher value such as $\lambda =1/2$ can be used to incorporate additional sources of uncertainty, or distortions to the volatility estimate as discussed below. Note that another modelling approach is to modify the potential, as for the anharmonic oscillator, or explicitly model the role of market-makers (Orrell, 2022, 2022a), but the method used here has the advantage of simplicity. The wave function is effectively stable over the time step, and is collapsed down at the end when measured through transactions.

The energy state number $n$ of the system when measured is the same as for a coherent state, and follows a Poisson distribution with average given by

$〈n〉=m\frac{\omega {∆x}^2}{2\hslash }=\frac{\hslash }{2\omega {\sigma }^2}\frac{\omega {∆x}^2}{2\hslash }=\frac{{∆x}^2}{4{\sigma }^2}=\frac{\lambda {\hspace{0.17em}\mu }^2}{4{\sigma }^2}=\lambda$

The expected number of transactions $〈n〉$ in a particular step is therefore equal to the parameter $\lambda$ , which reflects the average energy of the system. Since the time step $∆t$ is small relative to the period of oscillation, we can treat the state as being essentially unchanged over a single step.

The volatility of a quantum oscillator in energy state $n$ is given by the equation ${\sigma }_n=\sigma \sqrt{2n+1}$ . The expected RMS volatility for a system with average energy level $\lambda$ is therefore

${\sigma }_a=\sigma \sqrt{2\lambda +1}$

(1)

If the average energy level is set to the default value $\lambda =1/4$ then the RMS volatility is $\sqrt{3/2}\sigma$ , while if we choose a higher energy level $\lambda =1/2$ then the RMS volatility is $\sqrt{2}\sigma$ . The case with $\lambda =0$ corresponds to the situation where the volatility is constant. In a model where market makers are included, the volatility can also be controlled by adjusting the size of the spread (Orrell, 2022, 2022a).

The choice of ground state volatility will depend on the time step being used. Suppose for example that a commonly-traded stock has a relative spread of $\mu =2∆x=0.0007$ (i.e., .07%), and an average turnover of 1 unit every $\tau =0.0006$ trading days (about 14 seconds). One choice would be to use the oscillator to represent any potential transaction. In this case we would set $∆t=0.00015$ since the probability of a transaction in each step is only one in four (assuming $\lambda =1/4$ ). The corresponding volatility over the period will be $\sigma =∆x=0.00035$ .

The oscillator with these parameters would effectively represent the entire market, since it models all potential transactions. On the other hand if we use a time period of one day, then the oscillator would represent only a proportion of potential transactions (since the corresponding rate of actual transactions should now be slower). The volatility is then scaled accordingly, so in this case ${\sigma }_d=\sigma /\sqrt{∆t}=0.03$ . In general, the number of oscillators is $N=V_d∆t/\lambda$ where $V_d$ is the average daily volume. The total mass of the $N$ oscillators is then $M=Nm=V_d∆t\hslash /\left(2\omega {\sigma }^2\lambda \right)=\hslash V_d/\left(2\omega {\sigma }_d^2\lambda \right)$ which is independent of the time step.

Since we are setting $\mu =2\sigma$ and $\sigma ={\sigma }_d\sqrt{∆t}$ , it follows that the spread can be written as

$\mu =2{\sigma }_d\sqrt{∆t}={\sigma }_d\sqrt{\tau }$

(2)

where $\tau$ is the average time between transactions (which reflects the volume of the stock). While this scaling is adopted here in the context of a single representative transaction, it is interesting to note that a similar scaling appears to be a stable property of markets. According to (Sarkissian, 2020), the relationship between spread, volatility, and time step can be written in the same form $\mu =c{\sigma }_d\sqrt{\tau }$ with the multiplicative constant $c$ in the range of $1.6$ (from a theoretical estimate) and $3.5$ (from an empirical estimate) which, given the uncertainties involved (such as choosing an appropriate time step), are quite close to the value used here.

The top panels of Figure 1 compare the resulting price change distribution with historical index data for intervals of 1, 2, 4 and 8 weeks, where the energy level (so volatility ${\sigma }_n$ ) is observed at each period and is assumed to remain constant over the period, and the corresponding price change is drawn from a normal distribution with that volatility. The oscillator model therefore naturally produces the characteristic non-Gaussian behaviour of financial assets, independently of the time scale used. The scaling of the horizontal axis by inverse square-root of time comes about because in the oscillator model, the energy change corresponding to a price shift of $x$ depends on $m{x}^2=\hslash /\left(2\omega {\sigma }^2\right)x^2$ and ${\sigma }^2$ scales with time. Since we are looking mostly at daily index data, we will assume unless otherwise specified that $∆t=1$ day so $\sigma ={\sigma }_d$ .

Figure 1.

Top panels show density plots of log price changes, normalised by the square-root of time in years, for the S&P 500 (left) and Dow Jones Industrial Average (right) over the years 1992–2021 (source Yahoo Finance). The solid lines, from darkest to lightest shades of grey, are for periods $T$ of 1, 2, 4 and 8 weeks. Dashed line is the oscillator model with $\lambda =1/4$ . Bottom panels show RMS annualised volatility, plotted against normalised price change. Dashed line is the quantum curve given by equation (8).

4. Volatility and Price Impact

The previous section showed that the dynamic nature of the oscillator model allows it to capture basic market phenomena such as the relationship between spread and uncertainty, and the fat-tailed distribution of price changes. We now relate this to some previously published empirical studies, in order to explore how the model responds to specific perturbations caused by market imbalance.

As seen in Section 2, there is a relation in the quantum model between displacement of the oscillator by an amount $∆x$ , and an input of energy $∆E$ . This makes sense given that the model assumes an entropic force which resists displacement. As with a classical spring, the energy required varies with the square of the displacement. We can apply the same principle to larger displacements caused by an imbalance between buyers and sellers.

To see this, we define the market imbalance as

$\iota =\frac{\left(N_b-N_a\right)}{\min \left(N_a,N_b\right)}$

where $N_b$ and $N_a$ are the numbers of buyers and sellers for a unit, and model each potential transaction using a separate identical oscillator. If $N_a=N_b=N$ then the system is balanced. We associate this with the ground state, so the number of transactions is theoretically zero (a transaction would have the effect of changing the price and therefore the energy). In this model, for a transaction to take place, we therefore need a non-zero probability that another person will arrive to break the deadlock. In terms of quantum cognition, this can be thought of as contributing the energy required in order to offset the endowment effect (Khrennikova, 2019; Orrell, 2021).

In the case of a large trade, if the excess size of the trade over a period $∆t$ is $Q$ and the daily volume is $V_d$ then the average imbalance over the same period $∆t$ can be written as

$\iota =\frac{Q}{V_d∆t}$

(3)

The trade will therefore increase the oscillator frequency by an amount $∆E/E_0=\iota$ so the energy is $E=\left(1+\iota \right)E_0$ . Comparing $E_n=\left(2n+1\right)E_0$ it follows that for an oscillator in energy level $n$ we have $\iota =2n$ . Also

${\sigma }_n=\sigma \sqrt{2n+1}=\sigma \sqrt{\iota +1}$

(4)

which connects the oscillator’s volatility, number of transactions per step, and imbalance in a single equation.

On the other hand, if we use a displacement operator to shift a quantum harmonic oscillator of ground energy $E_0=\hslash \omega /2$ centered at position 0 by an amount $x$ , the energy change $∆E$ satisfies the equation $x=\sqrt{2}\sigma \sqrt{∆E/E_0}=\sqrt{2}\sigma \sqrt{\iota }$ . The price change $x$ is therefore the amount required in order to restore the oscillator to its original frequency, so the rate of purchases slows down to match the rate of sales. If demand instead decreases so $\iota <0$ , then the energy perturbation will act to reduce the price, so $x=-\sqrt{2}\sigma \sqrt{-\iota }$ , and in general we have

$x=\pm \sqrt{2}\sigma \sqrt{\left|\iota \right|}=\pm \sqrt{2}\sigma \sqrt{\frac{Q}{V_d∆t}}$

Using equation (1) and scaling for time gives ${\sigma }_d=\sigma \sqrt{2\lambda +1}/\sqrt{∆t}$ so

$x=\pm \sqrt{\frac{2}{2\lambda +1}}{\sigma }_d\sqrt{\frac{Q}{V_d}}\cong \pm {\sigma }_d\sqrt{\frac{Q}{V_d}}$

(5)

where the expression is exact for $\lambda =1/2$ and within about 15% for $\lambda =1/4$ . This result compares with the so-called “square-root law” for price impact, which states that the relative price change is $∆p/p=\pm Y{\sigma }_d\sqrt{Q/V_d}$ . According to (Tóth et al., 2011) the numerical constant $Y$ is “of order unity” which is in good agreement with the quantum model.

The term $Q/V_d$ has units of time and corresponds to the number of days that it would take to clear the excess order at the regular rate of transactions. Equation (5), which gives the price change $x$ due to market imbalance over a particular time period, is therefore similar in structure to equation (2), which gives the spread due to market imbalance over a time step. When the excess order is complete, the price evolution depends on the degree to which the large order has shifted investors’ beliefs about the central price. One proviso is that the square-root formula has been derived by shifting the oscillator wave function, however smaller displacements with $x\ll ∆x$ can also be accommodated by shifting the buy/sell probabilities, which implies a linear dependence on price. We would therefore expect to see a transition from the linear to the square-root regime to occur for some point smaller than $∆x$ . Indeed, this transition point is seen in the data and occurs for an imbalance of around .01 although it varies depending on the duration of the order (Bucci et al., 2019).

The variance of the impact measured over a model time step is given by the equation

$Var\left(∆t\right)={\sigma }^2\left(1+\left|\iota \right|\right)={\sigma }^2+{{\sigma }_{\iota }}^2$

(6)

where ${{\sigma }_{\iota }}^2={\sigma }^2\left|\iota \right|=x^2/2$ is due to the imbalance. We can therefore think of the variance as being caused by two types of volatility. The first is the usual type which corresponds to a balanced (so $\iota =0$ ) random walk, while the second can be viewed as a scaling factor ${\sigma }^2$ related to inverse mass, multiplied by $\left|\iota \right|$ which relates to the oscillator energy level.

If we wish to compute the effect on variance of a large order $Q_T$ which takes place over time $T$ , then what counts is not the total size of the order, which is known only at the end, but the rate $R_T=Q/T$ at which it occurs, as compared with the usual daily rate which is $V_d$ . Setting the time step equal to $T$ and using equation (5) for the displacement $x$ at that time gives a variance of

$Var\left(T\right)={{\sigma }_T}^2+{{\sigma }_T}^2\frac{R_T}{V_d}={\sigma }^{2}T+\frac{x^2}{2}$

(7)

Scaling to one day, it follows that the displacement-dependent volatility satisfies

${\sigma }_x^2={\sigma }^2+\frac{x^2}{2T}$

(8)

This equation describes a smile-shaped curve, similar to the volatility smile seen with implied volatility in options trading. (Note that if endogenous growth occurs under balanced conditions at a rate $\alpha$ , we would replace $x$ with a term $x-\alpha T$ corrected for this growth, which introduces a degree of skew to the smile.) The bottom panels in Figure 1 compare equation (8) with historical index data. The curve’s minimum value of $\sigma ={\sigma }_a/\sqrt{2\lambda +1}=\sqrt{2/3}{\sigma }_a$ is significantly smaller than the average volatility, which has implications for option pricing. For a discussion of option pricing using the quantum model see (Orrell, 2024).

The second term $x^2/2$ in equation (7), which reflects the variance due to imbalance, is independent of time. Again this is because what counts is not the order duration but the rate $R_T$ which traders will compare with the usual daily rate $V_d$ (which has units of inverse days). Even if we set $T=0$ , although there is no time for actual transactions to occur, there is still the uncertainty due to what might be called the “virtual transactions” associated with the oscillator energy level. We can therefore write this term as ${\sigma }^2{R}_T/V_d$ where $\sigma$ is the daily volatility, so equation (7) becomes

$Var\left(T\right)={\sigma }^{2}T+{\sigma }^2\frac{R_T}{V_d}\cong {\sigma }^{2}T+{\sigma }^2\frac{Q_T}{V_{d}T}$

(9)

In the classical view, volatility is assumed to scale inversely with time (see for example Pohl et al., 2017); and according to (Lillo, 2023, p. 122), “the variance of impact depends linearly on T, as expected by the diffusivity of price.” That paper cites a result from (Bucci et al., 2019), who used a proprietary data set to explore price impact. However as seen in Figure 2, when plotted as a function of imbalance the variance in the data from that paper actually shows the relationship predicted by equation (9) (Orrell, 2022b). While the formula is perhaps not surprising with hindsight, it was only after testing it that this pattern in the variance was discovered in the data. The fit ignores points with very small imbalance $\iota <0.02$ (where the effect of impact may be harder to measure) and uses only a single parameter which estimates the volatility $\sigma$ .

Figure 2.

Comparison of classical and quantum models of price impact. Left panel shows variance plotted against the ratio of order size to daily volume $Q/V$ , with order time $T$ ranging from 3.5 minutes (black) to 345 minutes (light grey). According to the classical model (Bucci et al., 2019), where variance is assumed to scale inversely with time, the slopes should be approximately constant, but in fact range from about 0 to 5. The right panel shows variance plotted against $Q/\left(VT\right)$ along with fits using equation (9). The single parameter $\sigma$ estimates the volatility. The data was obtained from a figure in (Bucci et al., 2019); see (Orrell, 2022b) for a discussion.

5. Imaginary Oscillations

The quantum model therefore captures a range of phenomena that characterise stock markets. One reason for preferring the quantum approach over classical alternatives is that it provides a simple model where there is no need for extra dynamical terms or made-up parameters. Another, though, is that it provides a rich way of thinking about the financial system in terms of physical analogies with things of which we have direct experience such as force, mass and energy (as opposed to things we can’t see like subatomic particles). The effect of market impact for example can be understood more easily if we think of a quantum spring that is being perturbed by an input of energy. Jevons (1905) wrote that “The will is our pendulum, and its oscillations are minutely registered in the price lists of the markets” but instead of mechanical oscillations that bounce the price from side to side, the oscillations in the quantum model are in a sense imaginary (they are rotations in the imaginary plane) and present as fluctuations in volatility as well as price.

While mathematical models are related to analogy or metaphor (Orrell, 2012), they differ in that they are expressed in terms of equations which demand a degree of internal consistency. In classical theories, the main dimensions are price and time, but by extending these to include mass the quantum approach allows one to express economic versions of force and energy in consistent units.

A common misapprehension about quantum probability is that it was developed in order to model strange and spooky phenomena of the subatomic world such as superposition and entanglement. In this view, the onus is on the modeller to show that financial markets exhibit similarly bizarre and counterintuitive behaviour. However, perhaps the key message of quantum social science and quantum economics is that quantum phenomena can often present as familiar and intuitive. As Aaronson (2013) notes it is a historical accident that quantum probability, which is a mathematical construct, was discovered by physicists rather than by mathematicians. The main advantage of its use in finance is simply that it provides a way to handle properties that are both probabilistic and dynamic in a unified manner. In the oscillator model presented here the key quantum properties that are exploited are a base level of fundamental uncertainty, which is related to the bid/ask spread; the existence of discrete energy levels, which correspond to the number of transactions per step; and the dynamics of the complex wave function, which describes the evolving superposition of price states. In principle such effects could be modelled using classical probability, just as one can emulate a quantum circuit using a classical computer; however there would be no advantage because, lacking the simple structure of the quantum model (which is just a quantum version of a spring), the resulting model would require extra made-up parameters and be more complicated.

The ultimate test of a model, after all, is not whether it is a good analogy or has the right units or can be coerced to fit data, but whether it is useful for explaining and predicting a system. This paper has presented a quantum oscillator model which captures key empirically-observed features of stock markets, using a minimal set of assumptions and parameters. To summarise the main points:

• The model is derived from a basic model of transactions which expresses probabilities in terms of entropic forces

• The model accounts for spread and correctly captures the inherent uncertainty of financial markets.

• Non-constant volatility leads to the non-Gaussian statistics seen in market data.

• The model naturally produces the square-root law of price impact, and correctly predicts that the numerical constant should be close to unity.

• The model also correctly predicts a simple empirical relationship for the variance of price impact, and more generally the relationship between volatility and price change.

When oscillators were first used almost a century ago to simulate financial markets, the models were based on classical mechanics rather than the quantum version which was being developed by physicists and mathematicians around the same time. One consequence was that concepts such as mass and energy seemed to have no obvious analogues in finance. The developing field of quantitative finance therefore focused on random walks and downplayed the role of dynamics. The efficient market hypothesis for example assumes that information is perfect and prices are instantly restored to equilibrium, so the system is effectively independent of time. The quantum approach therefore offers a way to bring the related concepts of energy, information, and entropy back into finance, in a manner that is consistent with observed market behaviour.