The Inverted Parabola World of Classical Quantitative Finance: Non-Equilibrium and Non-Perturbative Finance Perspective
TThe Inverted Parabola World of Classical Quantitative Finance:Non-Equilibrium and Non-Perturbative Finance Perspective
Igor Halperin
NYU Tandon School of Engineering e-mail: ighalp @ gmail.com August 11, 2020
Abstract:
Classical quantitative finance models such as the Geometric Brownian Motion or its later ex-tensions such as local or stochastic volatility models do not make sense when seen from aphysics-based perspective, as they are all equivalent to a negative mass oscillator with a noise.This paper presents an alternative formulation based on insights from physics.
I would like to thank Peter Carr for critical remarks. All possible errors are my own. a r X i v : . [ q -f i n . GN ] A ug Introduction
The novel ”The Inverted World” by Christopher Priest paints a fascinating image of a worldwhere a city called the City of Earth slowly travels on railway tracks across an alien planet.The city’s engineers keep laying a fresh track for the city, and pick up the old track as it moves.The city must move on to stay within 10 miles of the Optimum - which is a location where thegravitational field is not distorted, and matches the gravitational field of the planet Earth.But the world of the novel is not Earth. In this world, the ground is in a constant movefrom the north to the south, as a result of some sort of a global gravitational catastrophe thathappened at some point in the past. Even as the Optimum stays in the same position, theCity will drift away from the Optimum if it does not move all the time. If the City of Earthfinds itself too far from the Optimum, gravitational distortions become too strong as a result ofmoving grounds and their drift to the south. This is what makes the City crawls to the Northall the time. If it ever stops, it will eventually be pulled to the South, and destroyed at theend by gravitational distortion forces, along with all its citizens. So it has to move forwardthrough a devastated land full of hostile tribes. The only alternative to a constant move in suchan inverted world, where the grounds are moving and the Sun looks like a rotating parabola, isdeath.The resolution of the many puzzles and gaps of history of the Inverted World comes onlytowards the end of the book. The City of Earth was crawling the planet of Earth, never leavingit. Moving grounds, a parabolic Sun, and other related puzzles of the Inverted World was causedby side neurological effects of a UV radiation that was produced by the city’s power generator.The generator was based on an alternative energy method that was developed by a founder ofthe city.This novel, which I first read many years ago, came repeatedly to my mind when I workedon a model of asset price dynamic in an open and non-equilibrium market called the QuantumEquilibrium-Disequilibrium (QED) model [5]. The QED model generalizes the Geometric Brow-nian Motion (GBM) model by introducing two additional parameters, along with a non-linearextension of a diffusion equation driving the dynamics of the model, see also [4]. Surprisingly,the QED model suggests that the notions of a market growth and market stability in this modeland in the GBM model are essentially opposite .What was ’equilibrium’ dynamics in the GBM model becomes non-equilibrium dynamicsfrom the perspective of the QED model. What was the commonly excepted average exponential growth of asset prices becomes a fall from a point of instability towards a point of local stability.This sounds much like an Inverted World vs a Normal World, the only question is which one isthe Inverted World?This paper offers a non-technical introduction to the QED model, along with a reasoningwhy it corresponds to a ’Normal World’, while the GBM model, along with its multiple directdescendants such as local or stochastic volatility models, describes an ’Inverted World’. AsI will try to argue, assumptions of closed-system dynamics, (quasi-)stationarity and linearitymade in classical financial models do not adequately capture realities of real world financialmarkets, and in a sense can be viewed as ‘wrong limits’ of a (yet unknown) ‘right’ theory. ThenI show how these deficiencies are addressed in the QED model that treats markets as openand non-linear systems, and does not rely on a linearization of dynamics within a perturbationtheory to treat non-linearities. Instead, the QED model presents a non-perturbative approach tohandle non-linearities. This position paper discusses how insights from modern non-equilibriumand non-perturbative physics can be fruitfully used for financial modeling with non-linear andnon-equilibrium models such as the QED to better capture the true market dynamics.2
GBM, Langevin equation, and Inverted Parabola
Since the groundbreaking work of Samuelson in 1965 [11], Geometric Brownian Motion (GBM)model, also known as the the log-normal asset return model, dX t = µX t dt + σX t dW t (1)remains the main work-horse of financial engineering. In Eq.(1), X t is an asset price at time t , µ is the stock drift, σ is the stock volatility, and W t is a standard Brownian motion. For whatfollows, we can view the GBM model as a special linear case of a more general model called It´o’sdiffusion dX t = µ ( X t ) dt + σ ( X t ) dW t (2)with a linear drift function µ ( X t ) = µX t and a multiplicative (i.e. proportional to X t ) diffusionfunction σ ( X t ) = σX t . Samuelson proposed the GBM model (1) as an improvement over anArithmetic Brownian Motion (ABM) model suggested by Bachelier in 1900 [1]. His objectivewas to modify the ABM model to ensure non-negativity of stock prices. Note that the ABMmodel itself can be viewed as a model with a constant drift and volatility terms.A few years after Bachelier published his 1900 thesis that gave birth to the ABM model,Paul Langevin proposed in 1908 an equation that later became known as the Langevin equation.Langevin’s work focused on a simplified analysis of overdamped Brownian particles within theEinstein-Smoluchovski theory of classical diffusion in the presence of an external potential field U ( X ). Such a field can represent an impact of heavy molecules, a external gravitational orelectromagnetic field, etc. [8]. The Langevin equation can be written in similar terms to It´o’sdiffusion (2), except that in the Langevin dynamics, a drift term is given by the negative gradientof the potential U ( X ). The (overdamped) Langevin equation with a multiplicative noise reads dX t = − ∂U ( X t ) ∂X t dt + σX t dW t (3)Therefore, the Langevin equation that is rooted in physics provides an interpretation to the driftterm µ ( X t ) in the mathematical construction of It´o’s diffusion (2): any drift function µ ( X t ) canbe viewed as a negative gradient of a potential U ( X t ) in the equivalent Langevin dynamics (3).In particular, when the potential U ( X t ) has a minimum, the Langevin equation describes astochastic relaxation towards this minimum, where the gradient of the potential vanishes.While this observation applies to any drift function µ ( X t ) in Eq.(2), it is of particular interestto explore its consequences for the GBM model (1). Comparing Eqs.(3) and (1), we observethat the GBM model corresponds to a special case of a more general Langevin equation for thefollowing choice of the force potential U ( X ): U GBM ( X ) = − µ X (4)When the drift µ is positive, this is the potential of an inverted harmonic oscillator with ’mass’ µ ! Such a potential has a maximum at X = 0 and no minimum, see Fig. 1. From elementaryphysics, a classical motion in such potential describes an unstable system. A similar analysis can be performed by changing to log-prices y t := log X t . When expressed in terms of log-prices y t , the noise term becomes additive, while the y -space potential is V ( y ) = − (cid:16) µ − σ (cid:17) y . When µ > σ ,this potential corresponding to a uniform force pushing the particle away from the negative infinity y = −∞ corresponding to a financial distress or a bankruptcy. On the other hand, taking µ < σ would produce a linearattraction to y = −∞ and defaults (bankruptcies) happening too fast. U ( x ) corresponding to the GBM model with µ >
0. Red dotscorrespond to a “particle” representing the firm, coordinate X t being the firm’s stock price.This is the potential of a harmonic oscillator with a negative mass. Such a system is globallyunstable, and the default state X t = 0 is unreachable as the force of the negative gradient ofthe potential pushes the particle away from the default boundary X t = 0 for any value X t > The inverted parabola potential (4) thus describes the most classical example of an unstablesystem in physics - an inverted (negative mass) harmonic oscillator. Of course, the fact thatthe GBM model is non-stationary for µ (cid:54) = 0 is evident and well known in the literature. Onthe other hand, ’classical’ financial inter-temporal models (e.g. the Black-Scholes model [2,10] or inter-temporal versions of the CAMP model [12]) often work under assumptions of ageneral equilibrium or competitive market equilibrium. In these approaches, one assumes adynamic market equilibrium between rational financial agents having instantaneous access toeither symmetric or asymmetric information. Here the concept of a market equilibrium refersnot to a price process, but rather to an equilibrium of supply and demand given a price level(recall the Equilibrium in ”Inverted World”). Thus, the stock price is considered a ’referenceframe’ to describe the supply-demand balance equations. As the stock price dynamics is non-stationary, the same holds for the full system given by both financial agents (traders) and pricedynamics. This means that the concept of a market equilibrium under a non-stationary priceprocess can be at best applied only approximately for short times, as a local approximation .To illustrate this point, imagine you step into an elevator on a top floor of a skyscraper. Allof a sudden, the elevator cable breaks, and now the elevator is in a free fall, with you trappedinside. According to elementary physics, as long as the elevator continues to freely fall, youwill be levitating inside of the elevator, being in a state of a local ‘equilibrium’. This is becausebeing inside of a freely falling elevator is equivalent to residing in a non-inertial reference frame,where a fictitious ’anti-gravitational’ force exactly cancels the gravitational force. This simpleexample illustrates the point that from the point of view of an external observer who observes both the elevator and you inside of the elevator, the dynamics of the full system most definitely cannot be described as equilibrium dynamics proceeding indefinitely in time.Far from being a purely theoretical observation, the inverted parabola potential (4) implies a Of course, this is largely a hypothetical scenario, see e.g. https://science.howstuffworks.com/science-vs-myth/everyday-myths/question730.htm regarding practical safety measures to prevent it from happening. While the fact the the zero-price level X = 0 is not attainable in the GBM model is wellknown, the nature of this mechanism is rarely discussed. A critical observation is that not onlythe GBM model is incompatible with corporate defaults due to its inability to reach the zerolevel X = 0, but rather that non-negative prices are obtained in the GBM model at the cost ofintroducing a completely fictitious and absurd force, due to the negative gradient of the GBMpotential, that somehow saves a firm from default once it gets close to the zero level. Morethan that, this force becomes unboundedly stronger as the price increases! The presence of suchan absurd ever-growing force for all positive prices appears to be too steep a price to pay fornon-negativity of stock prices, which was the original motivation for the GBM model.Interestingly, the last observation that the repelling force actually increasses rather than decreases as the price moves away from the default boundary X t = 0 also implies the modelbehavior should also become progressively less trustworthy for large values of X t , that could beexpected in a long run for the GBM dynamics (1). The GBM model predicts that on average,the price of a given stock should grow exponentially in time, but empirically, a very few stockshave observable prices for a long period of, say, 100 years. Most of stocks live much shorterthan this, and often end their life via mergers, acquisitions, or corporate bankruptcies. Recallthat none of such events should be possible according to the GBM model, again suggesting thatit contradicts the reality. Also note that it would not be fair to use a long history of marketindex portfolios such as e.g. Dow Jones or S&P500 as an evidence of an average exponentiallong-term growth for individual stocks. Due to the fact that the composition of such marketindex portfolios continuously changes, it embeds a survivorship bias that allows it to ignore thefact that stocks can default, and proceed away with the implicit assumption that stocks areimmortal.An analogy with the Inverted World mentioned in the introduction should become moretransparent to the reader at this point. An unlimited fall in an unbounded potential (4) de-scribing a small noise dynamics of the GBM model and all its descendants can only be perceivedas an exponential growth only if the observer is somehow ‘inverted’ as well. When viewed fromthe perspective of the Langevin dynamics, an unbounded average exponential growth of assetsaccording to the GBM model turns out to be an unbounded fall in the inverted parabolic po-tential. This is obviously a catastrophic scenario for most models in physics except dedicatedmodels designed to describe short-lived unstable systems (e.g. in some cosmological models). Analysis of a small noise limit is useful in order to not get ’fooled by randomness’. Note that this is not the same as setting volatility σ to zero exactly . In such a strict limit, a stock becomes a riskless asset thatshould earn a risk-free rate r according to a no-arbitrage argument, and thus should be the same as cash ina bank account. As the exponential growth e rt follows as a solution of a compound interest equation, financialmathematicians typically have no issue with taking the exponential law e rt on its face value, and formally applyingit for arbitrary times t → ∞ arguing that ’we believe this will continue in the next 100 years or so, giventhe past experience’. I believe that an appeal to the money bank account law e rt as a ‘justification’ for anunbounded average exponential growth for stocks would be erroneous both financially and mathematically. Itis wrong financially because stocks are not cash, they can default, while a bank account is protected by stateregulations. It is also wrong mathematically because the limit σ = 0 is a singular limit: corporate defaults becomemathematically impossible in this limit, while their probability may remain small but non-zero for arbitrarily smallbut non-zero values σ > I would like to thank Peter Carr for pointing out one such stock: Sotheby’s (BID). While this only becomes evident in a long run, it does not mean that the GBM model is only ‘asymptoticallywrong’ rather than being ‘qualitatively wrong’. As I argued above, a fictitious ever-growing force equal to thenegative gradient of the GBM potential is absurd on the whole positive semi-axis X ≥
0, that is, at each timemoment. On the other hand, the simplicity of the GBM potential (4) does not justify invoking of an asymptoticanalysis to identify regions of the state space X ≥
5s an unbounded exponential expansion never occurs in most of other natural systems known tophysics, real ‘physical’ markets should have mechanisms that eventually stop such an unboundedexpansion. As I argue next, such stabilization can arise from taking into account interactionsand non-linearities in the market dynamics.
Both the classical GBM model and a majority of models used in quantitative trading are linear models, in the sense that they have a linear (or constant) drift. A linear or a constantspecification of a drift term for stock dynamics might appear a simplest reasonable choice, giventhat a drift is harder to measure at small time steps ∆ t than a diffusion term .As is known in physics, linear dynamic systems can typically be only considered approxi-mations to real-world dynamics of natural systems, which are often non-linear . Non-linearitiescapture interactions in physical systems, that could be produced either by interactions betweendifferent elements of a system, or interactions with some external potential. In particular, withinthe Langevin approach, the most common approach to incorporate complex interactions in aphysical system is to consider more complex potentials than a harmonic oscillator potential. Onepopular choice are potentials expressed as polynomials in a state variable X . While the use ofa general polynomial potential can be justified as a Taylor expansion of an arbitrary potential,for most systems encountered in statistical and quantum physics it usually suffices to considerpolynomial potentials up to the fourth degree [7] (see also references in [5]).One of the most popular non-linear potentials describing many systems in physics is theso-called quartic potential U ( X ) = − θX + 13 κX + 14 gX , (5)where parameters θ , κ and g may depend on time through their dependence on various predictors,but can be taken constant in a simplest case. Obviously, if we set κ = g = 0 and θ = µ , thispotential recovers the GBM potential (4). Otherwise, for non-zero values of κ, g , the GBMpotential (4) can serve as a local approximation, valid for small values of X , to the non-linearpotential (5).The fact that linear models such as the GBM model are only approximations to more generalnon-linear models that incorporate market impact, transaction costs etc. is well known in theliterature. However, there exists a common belief among both market practitioners and aca-demics that such non-linear effects are only important for large players who ‘move the market’,while for small trades the standard linear models that neglect market impact can still be used.I propose that even for classical quantitative finance models such as the GBM that do not assume a large trader, capturing non-linear effects of market interactions is critically importantfor constructing more reasonable models that would not produce the absurd ever-acceleratingunbounded decay in the inverted parabola potential (4) of the GBM model.Indeed, assume for a moment that the quartic potential (5) is a ‘right’ model of the world(I will argue later in favor of this choice, so that the example is not hypothetical). When X is sufficiently small, the cubic and quartic terms can be neglected, and upon setting θ = µ , we recover the GBM potential (4) as a small-field approximation to the quartic potential Excluding more specialized models specifically addressing market impact, for example for optimal stockexecution. This is because a drift and diffusion term scale as O (∆ t ) and O (cid:16) √ ∆ t (cid:17) , respectively, therefore the secondterm dominates when ∆ t → X t . Clearly, if we set parameters κ and g to zero exactly , then the two potentials are identical on the whole semi-axis X ≥ κ and g that control, respectively,the cubic and quartic non-linear terms in the potential U ( x ), the latter can produce a wide varietyof shapes, depending on the values of parameters, as illustrated in Fig. 2.Figure 2: Under different parameter choices in the quartic potential U ( x ) of Eq.(5), it can takedifferent forms. A stable state of the system corresponds to a minimum of the potential. Thepotential on the left describes a metastable system with a local minimum at zero and a globalminimum at x = 3 .
3. If a particle is initially released near the global minimum, most of the timeit will experience a small diffusive relaxation towards the global minimum, which, with a smallprobability, can be replaced at each instance by a large sudden jump across the potential barrierseparating the two minima. The minimum at zero corresponds to the default state. For thepotential in the center, the state x = 3 . x = 0 is metastable.The potential on the right has two symmetric minima, and the particle can choose any of themto minimize its energy. Such a scenario is called “spontaneous symmetry breaking” in physics.As was argued in [5], it is the potential in the left graph in Fig. 2 that leads to the mostinteresting dynamics of a stock market price. Instead of unstable dynamics of the GBM model,with such a potential, dynamics can rather be metastable . Such metastable dynamics are dif-ferent from globally stable dynamics such as e.g. the harmonic oscillator dynamics in that they eventually change, though the time for this change to occur may be long, or very long, dependingon the parameters. In between of such infrequent transitions, dynamics are approximately equi-librium (stationary) or quasi-equilibrium. Changes of the dynamics correspond to rare eventsof transitions between local minima of the potential.While an explanation of how this happens will be given momentarily, it is very important toemphasize a critical role of a non-vanishing noise σ > σ >
0. If σ is large,fluctuations become stronger and transitions happen more often, but in the strict opposite limit σ = 0, any fluctuations die off, and transitions between local minima of the potential are nolonger possible. Dynamics obtained in the strict limit σ = 0 are qualitatively different fromdynamics obtained for non-zero values σ >
0, even though the actual numerical value of σ maybe very small numerically. This is the reason why appealing to an exponential bank accountlaw as a justification for a similar average behavior for stocks would be mathematically wrong7 as was mentioned in Sect. 2.2, the limit σ → The potential shown on the left of Fig. 2 has a potential barrier between a metastable pointat the bottom of the local well, and the part of the potential for small values of x , where themotion against the gradient of the potential means a fall to the zero price level x = 0. Dueto noise-induced fluctuations, a particle representing a stock with value x t at time t placedinitially to the right of the barrier, can hop over to the left of the barrier. In physics, solutionsof dynamics equations that describe such “barrier-hopping” transitions are called instantons .The reason for this nomenclature is that the transitions between the meta-stable state and theregime of instability (a “fall” to the zero level x = 0) happens almost instantaneously in time.What might take a long time though is the time for this hopping to occur: depending on modelparameters, the waiting time can in principle even exceed the age of the observed universe. SeeFig. 3 for examples of an instanton, anti-instanton (an instanton going backward in time), anda bounce (an instanton-anti-instanton pair, i.e. an instanton followed by an anti-instanton)Figure 3: Instanton, anti-instanton, and bounce solutions. The instanton hops from the rightof a global maximum to the left of it, the anti-instanton proceeds in an opposite order, and thebounce is made of the instanton followed by the anti-instanton.In financial terms, an event of hopping over the barrier en route to the zero level at x = 0corresponds to a corporate bankruptcy (default). As the GBM model corresponds to the invertedharmonic potential where the point x = 0 is unattainable, corporate defaults cannot be capturedby the GBM model. In contrast, with the quartic potential shown on the left of Fig. 2, corporatedefaults are perfectly possible, and correspond to the instanton-type hopping transitions betweendifferent local minima of the meta-stable potential. Note that both the drift (the negativegradient of the potential (5)) and volatility vanish at X = 0. This means the the zero level X = 0 is an absorbing state: once the particle reaches this point, it stays there forever. Thisis a highly desirable model behavior as it captures corporate default in a simple diffusion-basedstock price model, in a sharp contrast with a failure of the GBM model to produce a defaultableequity model, in addition to unrealistic dynamics for X > A popular example of a non-analytical dependence on a model parameter is given by the function f ( g ) = e − A/g , where g ≥ A > g >
0, it does not have a Taylor expansion around the point g = 0, which means that this limit is singular. The “Quantum Equilibrium-Disequilibrium” (QED) model
Unlike the GBM model, the QED model [5] incorporates capital inflows and outflows in themarket, along with capturing their price impact in the model construction. As I will showbelow, capturing these phenomena using simple function approximations effectively producesthe Langevin dynamics with the quartic potential (5), thus offering a plausible mechanism forstabilization of market dynamics by non-linearities as described in Sect. 2.3. Before providing amathematical formulation of the model, it is helpful to discuss empirical data.
Traditional classical finance models such as the GBM model of Samuelson [11], the Black-Scholesmodel [2], the CAPM model [12] etc. typically all assume that a market is a closed system thatdoes not exchange cash with outside investors (an “outside world”). A common assumption forstock dividends often made for modeling stock prices is that any dividends paid by a companyare immediately re-invested back into the stock by the shareholders. However, in addition tocurrent investors in a given stock at any point in time, the normal regime of the market isthat on average, there is an approximately continuous rate of cash inflows into the market from new investors, mainly due to various retirement plans programs. In other words, money is not conserved in the market due to continuous inflows (and outflows) of new market participants.Fig. 4 demonstrates the dynamics of combined inflows into equity, bond, and hybrid funds[3]. It shows that on average, there was a steady inflow of around $325bn annually into the USfunds between 2004 and 2016, with a local drop around 2009 as a result of the economic crisis.Assuming as a rough estimate that about two thirds of these inflows are invested in stocks, thisgives rise to about $200bn injected every year into the stock market. The main origin of suchcash injection are retirement plans of the US workers.Figure 4: Combined inflows into equity, bond, and hybrid funds. The annual rate is approxi-mately constant at the level of $325bn [3].Should an annual injection of $200bn in the capital market be considered a large or a neg-ligible effect? The total market capitalization of all stocks in the S&P500 index is about $25.5trillion, or $25,500bn, so the inflows are of the order of 1% of the total index value, which maynot be a numerically insignificant effect. In addition, the answer depends on how exactly these9nflows are distributed across different stocks. If retails or institutional investors are massivelydriven to invest in a particular “hot” stock, after a relatively short period of increased returnsdriven “mechanically” by the momentum, a long term impact of such investor “crowding” in thestock normally amounts to diminishing long-term returns. The latter phenomenon is known asthe “dumb money” effect [9].Therefore, to model the impact of investors flows and their impact on stock returns, we shouldsimultaneously incorporate two things into the modeling framework, which are both missing inmost conventional classical models such as the GBM: capital inflows, and saturation/marketfriction effects. As we will see next, the QED model incorporates both these effects, and moreoverit provides an explanation why these effect are critically important to ensure a long-term stability(or, more accurately, meta-stability, as will be more clear below) of the resulting dynamics, nomatter how small these effects may be numerically.
Let X t be a total capitalization of a firm at time t , rescaled to a dimensionless quantity of theorder of one X t ∼
1, e.g. by dividing by a mean capitalization over the observation period. Weconsider discrete-time dynamics described, in general form, by the following equations: X t +∆ t = (1 + r t +∆ t ∆ t )( X t − cX t ∆ t + u t X t ∆ t ) ,r t +∆ t = r f + w T z t + f ( u t ) + σ √ ∆ t ε t , (6)where ∆ t is a time step, r f is a risk-free rate, c is a dividend rate (assumed constant here), z t isa vector of predictors with weights w , u t ≡ u t ( X t , z t ) is a percentage rate of cash inflow/outflowfrom outside investors , f ( u t ) is a market impact factor, and ε t ∼ N ( ·| ,
1) is white noise. Herethe first equation defines the change of the total market cap in the time step [ t, t + ∆ t ] asa composition of two changes to its time- t value X t . First, at the beginning of the interval, adividend cX t ∆ t is paid to the investors, while they also may inject the amount u t X t ∆ t of capitalin the stock. After that, the new capital value X t + ( u t − c ) X t ∆ t grows at rate r t +∆ t . The latteris given by the second of Eqs.(6), where the term f ( u t ) describes the price impact of the moneyinflow or outflow. In [5], we used a simple linear trade impact specification f ( u t ) = − µu t (7)where µ is a market impact parameter. Assuming that µ >
0, the chosen sign conventioncorresponds to a market saturation effect, which may be a proper setting for long-term assetreturns. On the other hand, µ < positive impact of money inflow u t >
0, whichmay be a relevant setting to describe a short-term impact of money inflows due to momentumeffects. Note that u t can be either zero or non-zero, including both positive values and negativevalues, with a ‘normal’ market corresponding to u t >
0. Another possible specification of theimpact function f ( u t ) will be presented below, after we introduce the basic setting.The reason that the same quantity u t appears in both equations in (6) is simple. In the firstequation, u t enters as a capital injection u t X t ∆ t , while in the second equation it enters via themarket impact term f ( u t ) because adding capital u t X t ∆ t means trading a quantity of the stockthat is proportional to u t . Using a linear impact approximation, this produces the impact term f ( u t ) = − µu t . Note that here we define cash inflows u t as multiples of the total market cap (or equivalently, of the stockmarket price) X t , while it was defined in the absolute terms in [5]. or, equivalently, the stock price, if the number of outstanding shares is kept constant.
10n general, the rate of capital injection u t injected by investors in the market at time t shoulddepend on the current market capitalization X t (or current returns), plus possibly other factors(e.g. alpha signals). In [5], we considered a simple quadratic choice for u t u t = ¯ u + φX t + λX t (8)with three parameters ¯ u , φ and λ . Note that Eq.(8) implies that the total money flow u t X t → X t →
0. This ensures that no investor would invest in a stock with a strictlyzero price. Also note that the Eq.(8) can always be viewed as a leading-order Taylor expansionof a more general nonlinear “capital supply” function u ( X t , z t ) that can depend on both X t andsignals z t . (alternatively, the capital supply u can be made a function of returns rather thanprices [6]). Respectively, parameters ¯ u , φ and λ could be slowly varying functions of signals z t .Here we consider a limiting case when they are treated as fixed parameters, which may be areasonable assumption for time periods when an economic regime does not change too much.Substituting Eq.(8) into Eqs.(6), neglecting terms O (∆ t ) and taking the continuous timelimit ∆ t → dt we obtain the “Quantum Equilibrium-Disequilibrium” (QED) model: dX t = κX t (cid:18) θ + w T z t κ − X t − gκ X t (cid:19) dt + σX t dW t , (9)where W t is the standard Brownian motion, and parameters are defined as follows: θ = r f − c + ¯ u, κ = ( µ − φ, g = ( µ − λ. (10)The dynamics of the QED model is therefore given by the Langevin equation (3) with thefollowing potential U ( X ) = − (cid:0) θ + w T z t (cid:1) X + κ X + g X (11)When the signals z t are turned off, this is exactly the quartic potential of Eq.(5). As discussed inSect. 2.3, for some choices of model parameters, this potential leads to stabilization of dynamicsaround a metastable potential minimum that prevents the stock price from an indefinite growth,while also allowing for corporate defaults (Fig. 2). The latter proceed via instanton transitionsthat correspond to sudden thermally induced jumps over the top of a potential barrier separatingdifferent local minima of the potential (11). Instanton solutions in the QED model are illustratedin Fig. 3.It should be noted that the linear impact function (7) may be overly simplistic. Indeed,assuming that µ > u t >
0, it implies that when new money are invested in the stock, itproduces an immediate negative impact on the next-period returns. This goes contrary to thepresence of momentum effects in the markets that predict that, unless the stock is “saturated”or “crowded”, an injection of the new money increases the demand and should increase ratherthan decrease returns. As shown in [6], instead of a linear impact model, a quadratic model withtime-dependent parameters can better capture the ’dumb money’ effect [9] that predicts thatan initial flow into a stock should increase expected returns, but a continuous buildup of inflowsinto the stock leads (crowding) leads to diminishing long-term returns. With such a choice ofthe impact function, we can retain only the linear term in Eq.(8) to come up with the sameQED dynamics (9) and the potential (11), albeit with different expression for parameters θ, κ and g in terms of original model parameters entering Eq.(6). A slight difference between Eq.(8) and a similar formula presented in [5] is because here we define u t as arate, rather than in absolute terms as was defined in [5]. This difference is inessential as it only produces re-defined parameter values of the final model, given by Eq.(9) below, in terms of original model parameters enteringequations (6) and (8). .3 QED model and instantons: non-perturbative finance For some physical systems, non-linearities can be handled approximately, by treating them assmall perturbations around a linear regime, using e.g. a perturbation theory in a small parameterthat quantifies the stength of non-linearity. However, in many other cases arising in the naturalsciences, non-linearities should be treated as key ingredients of the dynamics.For example, non-linearity is critical for self-organizing systems which cannot be describedusing a perturbation theory around a linear regime. Another well-known example is provided byinstantons - barrier transition phenomena in statistical and quantum physics discussed above.Probabilities of such barrier transitions cannot be obtained at any finite order of a perturba-tion theory in a small parameter controlling the non-linearity. They are examples of so-called non-perturbative phenomena. While instantons and other non-perturbative phenomena are veryimportant in many models of statistical physics and quantum field theory , they are not trace-able using tools of perturbation theory, see e.g. references cited in [5].Similarly, instantons in the QED model (see Fig. 3) are non-perturbative phenomena inparameters κ, g , and thus could not be seen at any finite order of a perturbation theory con-structed around a strict limit κ = 0 , g = 0 of the QED dynamics. As in this strict limit theQED model would be identical to the GBM model, this means that while the latter could for-mally be considered as a ’baseline’, unperturbed model for construction of such a perturbativeexpansion, instantons (and hence corporate defaults) would be entirely lost in such a scheme.Non-perturbative methods to compute instanton-induced transition probabilities associated withprobabilities of corporate defaults are presented in [5]. As was illustrated in [5], this enables asimultaneous calibration of the QED model to equity and credit markets, by a joint fit to equityreturns and credit default swaps (CDS) spreads. In its turn, it enables using data from creditmarkets to produce information on a long-term equity returns. The QED model is therefore afirst defaultable equity model that captures corporate defaults without introducing additionaldegrees of freedom such as hazard rates. To summarize, starting with Samuelson’s GBM model, many models used by practitioners formodeling stock prices and derivatives prices, such as local or stochastic volatility models, reliedon the assumption of a linear (and typically positive) drift of a price process, or equivalently aconstant drift of a log-price process. In this paper I showed that, when interpreted in physicsterms, these models describe an oscillator with a negative mass (or equivalently a particle in aninverted parabolic potential ) subject to noise, where differences between specific models amountto different ways of modeling noise. This makes them all models of stochastic dynamics inan unstable potential, and conflicts with conventional ways of analysis of natural systems inphysics where models typically describe fluctuations around some stable or metastable state.A qualitatively wrong behavior describing an unlimited fall in such an unbounded potential isobtained as a result. Samuelson’s solution of the problem of negative prices in the ABM modelof Bachelier is unsatisfactory as it leads to a conflict with basic physics.I argued that such a pathological behavior can be avoided if the market is modeled as anopen system with a possible exchange of money with an outside world, along with a priceimpact of the new money on stock prices. For a single-stock market, this produces a simple non-linear two-parametric extension of the GBM model, with new parameters κ, g , called ”Quantum Including e.g. quantum chromodynamics (QCD), the modern theory of strong interactions. To explain thevery existence of protons and neutrons, QCD needs to go beyond perturbation theory. κ, g →
0. With non-zero parameters, it produces a qualitatively dif-ferent behavior: while the GBM model describes unstable dynamics, the QED model describes metastable dynamics where a diffusive relaxation to a metastable state is followed by a rare largenegative move describing a transition to a distressed state or corporate bankruptcy. Such rarelarge moves are due to noise-induced solutions of the model called instantons. Similarly to in-stantons in physics, instantons in the QED model are non-perturbative phenomena: they cannotbe seen in a perturbative expansion of the model that could be attempted when parameters κ, g are small but non-zero. In particular, instanton disappear in the strict ’GBM limit’ κ = g = 0.The QED model offers a few important theoretical insights. While classical financial modelshave traditionally focused on modeling volatility while keeping simple linear assumptions ofthe drift, the QED model suggests that the drift should instead be non-linear , and shouldbe identified prior to analyzing volatility patterns. In particular, it would be interesting toreconsider various stochastic volatility and ‘rough volatility’ models after fixing the drift functionalone the lines suggested in this paper.The QED model can be extended along multiple dimensions. In particular, it can be extendedto a market with multiple assets, producing the IQED (“Interacting-assets QED”) model [6].Other possible extensions can make the quartic potential random - for example, by allowing adependence of parameter θ on signals z t as in Eq.(11). This may make the dynamics of themodel more realistic and avoid a possible negative long-term drift that might be obtained in themodel if the potential is kept static. Clearly, for applying the model for a long term modeling,one should better make some or all model parameters dependent on signals z t , assuming thatthe latter carry information on a contemporaneous market environment. Proceeding in suchway would effectively promote the potential to a random quartic potential.Obviously, an important practical question is that assuming that the QED model is a ‘right’model, how important are non-perturbative effects implied by the model? Can we still relyon traditional financial engineering models that are all based on the assumption of a linearor constant drift? If yes, when can we still rely on them? Such questions can (and should)be answered for any particular stock market and any traditional model by comparing resultsobtained with that model versus the QED model. In general, non-linear and non-perturbativeeffects are not expected to be critically important for small local price fluctuations. However,it is a comparison with a more general non-linear model such as the QED model that shouldanswer the question about a range of prices and times where traditional linear models can stillbe used. The QED model could also be used for option pricing, and its predictions could beanalyzed and compared with traditional models both numerically and analytically using variousapproximations. Results of such analysis will be presented elsewhere. References [1] L. Bachelier, “Thorie de la speculation”,
Annales Scientifiques de L?cole Normale Su-prieure , , 21-86 (1900). (English translation by A. J. Boness in P.H. Cootner (Editor): The Random Character of Stock Market Prices , p. 17?75. Cambridge, MA: MIT Press(1964).[2] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities”, Journal ofPolitical Economy, Vol. 81(3), 637-654, 1973.[3] Deutsche Bank, “Grassroots Crowding Measures” (2016).134] M. Dixon, I. Halperin, and P. Bilokon,
Machine Learning in Finance: from Theory toPractice , Springer 2020.[5] I. Halperin and M.F. Dixon, “Quantum Equilibrium-Disequilibrium: Asset Price Dynam-ics, Symmetry Breaking, and Defaults as Dissipative Instantons”,
Physica A , 122187,https://doi.org/10.1016/j.physa.2019.122187 (2020).[6] I. Halperin, “Dumb Money, The Origin of Factors, Skew and Rare Events: Interacting-assets“Quantum Equilibrium-Disequilibrium””, forthcoming (2020).[7] L.D. Landau and E.M. Lifschitz,
Statistical Physics , Elsevier (1980).[8] P. Langevin, “Sur la Th´eorie du Mouvement Brownien”,
Comps Rendus Acad. Sci. (Paris)146, 530-533 (1908).[9] A. Frazzini and O.A. Lamont, “Dumb Money: Mutual Fund Flows and the Cross-Section ofStock Returns”,
Journal of Financial Economics , Elsevier, vol. 88(2), pages 299-322 (2008).[10] R. Merton, “Theory of Rational Option Pricing”, Bell Journal of Economics and Manage-ment Science, Vol.4(1), 141-183, 1974.[11] P. Samuelson, “Rational theory of warrant pricing”,
Industrial Management Review , (Spring), 13-32 (1965).[12] W.F. Sharpe, “Capital asset prices: A theory of market equilibrium under conditions ofrisk”, Journal of Finance ,19