"Quantum Equilibrium-Disequilibrium": Asset Price Dynamics, Symmetry Breaking, and Defaults as Dissipative Instantons
““Quantum Equilibrium-Disequilibrium”: Asset Price Dynamics,Symmetry Breaking, and Defaults as Dissipative Instantons
Igor Halperin and Matthew Dixon May 29, 2019
Abstract:
We propose a simple non-equilibrium model of a financial market as an open system with apossible exchange of money with an outside world and market frictions (trade impacts) incorpo-rated into asset price dynamics via a feedback mechanism. Using a linear market impact model,this produces a non-linear two-parametric extension of the classical Geometric Brownian Motion(GBM) model, that we call the “Quantum Equilibrium-Disequilibrium” (QED) model.The QED model incorporates non-linear mean-reverting dynamics, broken scale invariance, andcorporate defaults. In the simplest one-stock (1D) formulation, our parsimonious model has onlyone degree of freedom, yet calibrates to both equity returns and credit default swap spreads.Defaults and market crashes are associated with dissipative tunneling events, and correspond toinstanton (saddle-point) solutions of the model. When market frictions and inflows/outflows ofmoney are neglected altogether, “classical” GBM scale-invariant dynamics with an exponentialasset growth and without defaults are formally recovered from the QED dynamics. However, weargue that this is only a formal mathematical limit, and in reality the GBM limit is non-analytic due to non-linear effects that produce both defaults and divergence of perturbation theory in asmall market friction parameter. NYU Tandon School of Engineering. E-mail: [email protected] Department of Applied Math, Illinois Institute of Technology. Email: [email protected] would like to thank Eric Berger, Jean-Philippe Bouchaud, Sergei Esipov, Andrey Itkin, Andrei Lopatin,Sergey Malinin and Rossen Roussev for useful comments. a r X i v : . [ q -f i n . S T ] M a y Introduction
Since the groundbreaking work of Samuelson in 1965 [48], the log-normal asset return model,also known as the Geometric Brownian Motion (GBM) model dX t = ( r f + w T z t ) X 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 , r f is a risk-free rate of return, z t are return predictors (“alpha”-signals), w are their weights,and W t is a standard Brownian motion. For what follows, we can view the GBM model as amodel with linear drift f ( X t ) = ( r f + w T z t ) X t and multiplicative (i.e. proportional to X t ) noise.The GBM model (1), which was in itself an improvement over an Arithmetic BrownianMotion (ABM) model suggested by Bachelier in 1900 [2], ensured non-negativity of model prices.The ABM model can also be viewed as a model with a constant drift and volatility terms.Many classical results in finance are based on linear asset return models similar to Eq.(1).Most notable examples include the Capital Asset Pricing Model (CAPM) [49] and the Black-Scholes option pricing model [4], [41].On the other hand, these classical models are widely recognized to be in conflict with manyempirical facts about financial markets, such as volatility clustering, market frictions, feedbackeffects, market crashes and “6-sigma events”, corporate defaults etc. Countless models in theliterature tried to improve the log-normal dynamics of Eq.(1) to make it more consistent withmarket data, essentially by modifying one or more elements in Eq.(1): (i) extend a set ofpredictors z t [18]; (ii) include non-linear dependencies on predictors z t ; and (iii) allow for adifferent state-dependent (and possibly stochastic) noise amplitude in (1) [28].A common feature of such extensions is that they all keep a linear (or constant) drift term ofthe GBM model. Such a linear drift term corresponds to a free diffusion, i.e. describes dynamicswithout interactions, as will be explained in more detail below. Generally, both the presence andimportance of non-linear effects (in particular, due to market frictions) for market dynamics iswidely recognized in the literature. Yet at the same time it is also commonly believed that, eventhough non-linearities exist, the GBM model could still be used as a “zero-order” approximationto the “true” behavior of financial markets . Furthermore, another drawback of the GBM modeland its extensions is that they are incapable of being simultaneously calibrated to both equityand credit markets without violation of modeling assumptions. One principal challenge is how torepresent corporate bankruptcy and default in the GBM model - the point X t = 0 correspondingto such events is not attainable.The classical Merton corporate default model [40] belongs to a class of “structural models”that simply bypass this problem by identifying a credit event as a first crossing event for somenon-zero boundary level ¯ X > firm value process, rather than an observablestock price process .This is problematic on multiple accounts. Assumptions of a well-defined default boundaryat some fixed (and exactly known) value ¯ X > estimate would be subject to noise. Uncertainty of the default boundary would lead to uncertainty of thedefault time itself. This means that if we strictly adhere to the model, at each moment in timewe can’t confidently determine the occurrence of default, as the exact location of the boundaryis unknown. One of the rare exceptions is Dash [11] who suggested that “true” dynamics cannot be as simple as the GBMmodel, and proposed Reggeon Field Theory (RFT) as a better candidate model for the stock price dynamics Structural models are linear models in the drift and are often formulated in the risk-neutral measure whereit becomes redundant. interacting and non-linear dynamical systems, rather than as non-interactingstochastic systems. Our model is derived from the Langevin approach to stochastic dynamicsthat generalizes free diffusion dynamics to a diffusion of interacting particles in an externalpotential which is generally non-linear [36]. In our approach, such a potential has a specificquartic polynomial form.
Simultaneously , non-linearity of the model produces a new mechanismof default, which ensures that stock prices mostly follow diffusion but sometimes may also crashto zero due to a corporate bankruptcy. As a result, we obtain a model of a defaultable equity,with only one degree-of-freedom given by the equity price itself, that can be calibrated to bothequity and credit default swap (CDS) markets. To the best of our knowledge, this is a propertythat is not shared by any other equity model.Such a non-linear extension of the GBM model, with two additional parameters κ and g ,controlling the amount of non-linearity, is obtained by modeling two aspects of real marketdynamics that are absent in the GBM model and its descendants. We incorporate the fact thatmarkets are open, rather than closed, systems and we include the market feedback effects oftrading via market impact mechanisms.In this work, we focus on the simplest possible approach where these effects are treated usingsimple linear or quadratic functions. In particular, we use a simple linear model of market im-pact. In choosing such function approximations, we rely on physics-inspired analyses of analyticproperties and symmetries of resulting price dynamics. Such analyses suggest that other, morecomplex non-linear extensions of the GBM dynamics should converge to our model in a regimeof “mild” non-linearities. Our model, that we call the “Quantum Equilibrium-Disequilibrium”(QED) model, thus offers a simple non-linear extension of the GBM model that captures themost essential characteristics of real markets such as non-linearities of dynamics due to marketfrictions, and the presence of corporate defaults.Importantly, a simple model formulation does not mean simple model dynamics . Dynamicsof the QED model can be highly complex due to a joint effect of non-linearity and noise. Aswill be discussed later in the paper, the model captures a combination of non-linearities, noise and symmetries that play a key role in defining a dynamic behavior of stock prices.In its reliance on analyticity and analysis of symmetries, our approach bears some similarityto the classical phenomenological Ginzburg-Landau (GL) approach to equilibrium phase tran-sitions [35]. As will be shown below, in the QED model, corporate defaults (or market crashes,in a multi-stock formulation of the model) can be described as phase transitions.Phase transitions in our model occur as events of non-equilibrium noise-induced barriercrossing phenomena for “particles” representing firms, where a barrier is created, for certainvalues of model parameters, by a combination of a process of capital inflow/outflow in themarket and a linear feedback mechanism. Such transitions are well studied in statistical andquantum physics, where they are known as instantons . In our model, corporate defaults andmarket crash events are described by instantons.As their name suggests, instanton transitions occur nearly instantaneously in time, thereforethey might be good candidates to represent sudden large price moves such as those occurringduring market crashes or corporate defaults. While such effects are often described by additionaljump processes or by modifications of a diffusion term in a dynamic equation, in our frameworkthey can be produced endogenously , as a result of interactions of market non-linearities withrandom market noise using only one degree of freedom, and a white noise model.By focusing on low-order non-linearities of stock price dynamics, our model appeal to physics-rooted ideas of universality of phase transitions, which suggest that qualitative features of phasetransitions are determined by symmetries of a system, rather than details of a microscopic3amiltonian [35]. Similarly to the classical Ginzburg-Landau theory [35], our model uses aquartic potential (see below) to capture dynamics of the phase transition.Differently from the GL theory, phase transitions in our model are in universality classes ofnon-equilibrium phase transitions to absorbing states [29]. Our model identifies these absorbingstates with a zero-price level corresponding to a corporate bankruptcy or default.When two additional parameters κ and g of the QED model are set to zero, it reduces to theoriginal GBM model. This seems to conform to a perception commonly held by practitionersand academics alike that while friction effects are important, in most practical cases , they canbe treated as “first-order” effects, while the original friction-free GBM model can still be usedas a reasonable “zero-order” approximation to the dynamics of actual markets. The underlyingidea here is that as a linear function can always be viewed as a linear approximation to a non-linear function, the GBM model could also be viewed as a result of such a Taylor-truncationof underlying non-linear dynamics. When frictions effects (e.g. due to market impact) becomeimportant, they could be computed using, say, perturbative methods.However, on closer inspection we observe that the limit κ, g → qualitatively differentfrom a behavior obtained at non-zero values of these parameters. As will be shown in detailin Section 3, for certain values of parameters, the resulting dynamics are metastable , with anoise-induced barrier transition to a state of zero price associated with a corporate default. Onthe other hand, the dynamics in the GBM limit κ, g → unstable (see in Sect. 3), anddo not allow for corporate defaults. This suggests that the GBM limit may be non-smooth(non-analytic).In statistical physics, non-analyticity often arises in phase transition phenomena. In par-ticular, non-analyticity of a free energy of a statistical system leads to a second order phasetransition, see e.g. Landau and Lifshitz [35]. The Lee-Yang theory of first order phase transi-tions relates them to zeros of a partition function in a complex plane of model parameters (seee.g. [30], Sect. 3.2). In our model, corporate defaults are associated with noise-induced phasetransitions of barrier crossing via an instanton mechanism. This suggests that our model maysimilarly be non-analytic in a complex plane of parameters κ, g . Such non-analyticity would pro-duce a non-analytic GBM limit. In other words, the exact GBM limit κ, g = 0 for computablequantities would not correspond to any continuous limit of a smooth function that would beobtained with non-zero values of this parameters in a fuller model such as the QED model.This might have important implications, as non-analyticity implies that certain measurablequantities, such as prices of derivative instruments, may have corrections that are non-analytic(non-perturbative) in parameter κ, g . In particular, instantons in physics are non-perturbative phenomena: they cannot be seen at any finite order of a perturbative expansion around a non-interacting limit (which would correspond in our case to the limit κ, g = 0). In our model,corporate defaults are described by instanton solutions. The presence of instanton-induced cor-rections invalidates the idea of a Taylor truncation of the true dynamics, where the GBM modelwould become a “zero-order” approximation. Instead, this implies that the right “zero-order”approximation should already include non-zero friction parameters κ, g (which can neverthelessbe numerically small), and treat them non-perturbatively, rather than as small corrections toan ideal friction-free limit.The contribution of this paper is two-fold. On the theoretical side, we present the QEDmodel and introduce multiple arguments in favor of non-analyticity of this model (or similar non- that is those who do not specifically target modeling market impact and other market friction effects, whichis unavoidable for some applications, for example for optimal stock execution, or e.g. for portfolio optimizationwith transaction costs. κ, g . Sect. 3.4 identifies a qualitatively different behaviorfor positive and negative values of κ , in the spirit of Dyson’s argument about divergence ofperturbative expansions in quantum electrodynamics [15]. In the same Sect. 3.4 we obtainan explicit solution of the noiseless “Verhulst limit” that demonstrates non-analytic behavior.Furthermore, we show qualitative differences in a long-term model behavior in the GBM limitand outside of this limit. When noise is turned on, we show non-analyticity of a normalizationcoefficient of a stable solution of the Fokker-Planck equation, which is analogous to the Lee-Yangtheory of phase transition (see Appendix B.3). And the last but not least, in Sect. 5 we obtainexplicit instanton solutions in our model, showing that they are non-analytic in the frictionparameters.On the computational side, we develop numerical approaches for calibrating the QED modelto univariate market data beyond using the path integral Langevin formulation. We show howthe Fokker Planck Equation (FPE) approach is used in the classical Kramers method to deriveour model calibration technique. We also describe the reduction to the Schrodinger equationas a potential technique for multi-variate calibration in finance and leave it to future work toimplement this approach.The remainder of this paper is organized as follows. In Sect. 2, we provide a short overviewof related work. Sect. 3 derives our QED model. Sect. 4 discusses methods for solving themodel using the Langevin equation and its path integral formulation, in addition to introducingLangevin instantons. Sect. 5 describes defaults as tunneling events in the equivalent quantummechanical formulation of the model. Sect. 6 describes our numerical experiments. Finally,Sect. 7 provides a short summary and directions for future research. The GBM model (1) is often used within models based on the competitive market equilibriumparadigm, such as e.g. the CAPM and the Black-Scholes models [13]. Within this paradigm,a market is considered a closed system fluctuating around a state of perfect thermodynamicequilibrium, without any exchange of money and information with an outside world.An alternative to a closed-market concept of competitive market equilibrium paradigm isproposed by Amihud et. al [1]. Instead of competitive market equilibrium, the authors arguethat a better paradigm should acknowledge the very existence of financial markets. Indeed,markets are facilitated by market makers who process outside information and provide liquidityin the market in an amount that is optimal for them . This has an impact on market prices,which in turn impacts decisions of investors to inject or withdraw capital in the market. Theirtrades impact the market via a feedback mechanism.Under “normal” market conditions, such a scenario implies a dynamically stable state of amarket, which Amihud et. al referred to as an “Equilibrium-Disequilibrium” [1]. In physics,such states of physical systems are usually referred to as non-equilibrium steady states, see e.g.[35]. Our “Quantum Equilibrium-Disequilibrium” model implements the market dynamics inaccord to the approach of Amihud et. al . For a special case when we exclude signals and set g = 0, our model produces the so-called Geometric Mean Reversion (GMR) process which wasused in the literature for modeling commodities and real options, see e.g. Dixit and Pindyck[12]. Its properties for κ > et. al. [8], that weextend here by adding linear market impact, and applying it to a market portfolio , instead ofan individual investor portfolio, following the approach of [26]. Our previous model in [26] was5 single-agent model where a bounded-rational agent aggregates all traders in the market.The Reinforcement Learning (RL) - based model of Ref. [26] provides a mesoscopic descrip-tion of market dynamics when viewed from the prospective of a market agent. Here we presentan alternative, and perhaps more “physically” clear view of the same dynamics, but seen thistime from outside of the market.The reason for adopting such a view is as follows. In the approach of [26], agents’ actions u t are adjustments of all positions (by all traders in the market) on a given stock at the beginningof the interval [ t, t + ∆ t ]. Because now we look at all traders in the market at once, unlikethe individual-investor setting of Boyd et. al. , it is natural to interpret u t as an amount of new capital injected (or withdrawn, if it is negative) in the market by outside investors at thebeginning of the interval [ t, t + ∆ t ]. As we will show later, such a “dual” view generalizes thesimple market dynamics model suggested in [26] to describe not only a stable “growth” marketphase (as implicitly assumed in both [8] and [26]), but also more realistic market regimes withcorporate defaults and market crashes.A model obtained in this way amounts to non-linear Langevin dynamics with multiplicativenoise that contains a white noise and colored noise components. The colored noise term describessignals used by investors. Such dynamics and related phase transitions are well studied inphysics, see e.g. [50], [53], [46], [29].Approaches to modeling stock markets based on Langevin dynamics were previously con-sidered in the econophysics literature, in particular by Bouchaud and Cont [5], Bouchaud andPotters [6], and Sornette [51, 52]. In particular, Bouchaud and Cont used the Kramers escaperate formula (which also appears in our model, see Sect. 4.2 below) to describe market crashes.While we similarly use the non-linear Langevin equation for dynamics of stock prices, here wefocus on phenomena that were not the focus in these previous works.More specifically, we concentrate on (i) comparison of analytical properties of the GBM andQED models; (ii) analysis of their different symmetries ; and (iii) the existence of a possiblephase transition that differentiates these two models. Unlike the GBM model, which does notadmit defaults (in the sense of transitions into a truly absorbing state), our QED model leadsto defaults described as phase transitions into absorbing states at the zero price level X = 0. 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 )( X t − cX t ∆ t + u t ∆ t ) ,r t = r f + w T z t − µu t + σ √ ∆ t ε t , (2)where ∆ t is a time step, r f is a risk-free rate, c is a dividend rate (assumed constant here), z t is a vector of predictors with weights w , µ is a market impact parameter with a linear impactspecification, u t ≡ u t ( X t , z t ) is a cash inflow/outflow from outside investors, and ε t ∼ N ( ·| , in the time Such an agent embodies a “collective wisdom” of the market, that can also be related to various versions ofan “Invisible Hand”-type market mechanisms popular in economics since Adam Smith who coined the term itself. or, equivalently, the stock price, if the number of outstanding shares is kept constant. t, t + ∆ t ] as a composition of two changes to its time- t value X t . First, at the beginning ofthe interval, a dividend cX t ∆ t is paid to the investors, while they also may inject the amount u t ∆ t of capital in the stock. After that, the new capital value X t − cX t ∆ t + u t ∆ t grows at rate r t . The latter is given by the second of Eqs.(2), where the term µu t describes a linear tradeimpact effect. Note that u t can be either zero or non-zero.The reason that the same quantity u t appears in both equations in (2) is simple. In the firstequation, u t enters as a capital injection u t ∆ t , while in the second equation it enters via themarket impact term µu t because adding capital u t ∆ t means trading a quantity of the stock thatis proportional to u t . Using a linear impact approximation, this produces the impact term µu t .As will be shown below, this term is critical even for very small values of µ because the limit µ → non-analytic .In general, the amount of capital u t ∆ t injected by investors in the market at time t shoulddepend on the current market capitalization X t , plus possibly other factors (e.g. alpha signals).We consider a simplest possible functional form of u t , without signals, u t = φX t + λX t , (3)with two parameters φ and λ . Note the absence of a constant term in this expression, whichensures that no investor would invest in a stock with a strictly zero price. Also note that theEq.(3) can always be viewed as a leading-order Taylor expansion of a more general nonlinear“capital supply” function u ( X t , z t ) that can depend on both X t and signals z t . Respectively,parameters φ and λ could be slowly varying functions of signals z t . Here we consider a limitingcase when they are treated as fixed parameters, which may be a reasonable assumption whenan economic regime does not change too much for an observational period in data.In what follows, we assume that the second parameter is positive and very small, i.e. 0 <λ (cid:28)
1. This implies that the second term in (3) provides only a small correction to the firstterm in a parametrically wide region | X t | (cid:28) | φ/λ | . As will be more clear below, the main roleof the quadratic term in (3) is to regularize large fields | X t | ∼ | φ/λ | .Thus, in the parametric region | X t | (cid:28) | φ/λ | the capital supply is mostly determined by thevalue of parameter φ in Eq.(3), that can be either positive or negative. If φ >
0, capital isinjected in the market (a growth period), while if φ <
0, capital is withdrawn (i.e. the marketcontracts).As will be clear below, while for positive values of φ a contribution the second term ∼ X could be neglected when λ →
0, a non-zero value λ > negative values of φ . This implies that when φ is negative (more accurately, when φ < λ/µ ),large fluctuations becomes critically important.In a model proposed in [26], the decision variable u t is considered an action of a goal-directed bounded-rational agent that aggregates all traders in the market. Such agent can be viewed asan embodiment of an “Invisible Hand”-type market mechanism. Assuming that such market-agent sequentially optimizes risk-adjusted returns of a market portfolio, the model developed in[26] suggests that an optimal action u t is a constrained Gaussian random variable whose meanis linear in the state X t . If we take a deterministic limit in this approach and omit a possibledependence on signals z t , this leads to Eq.(3) for a specific choice λ = 0.The reason that the optimal control of such market-wide agent implementing an “InvisibleHand” market mechanism in [26] is obtained as a linear control of the form (3) is that the modeldeveloped in [26] uses a quadratic (Markowitz) utility function and assumes that the marketis in a “normal” regime. For non-quadratic utilities, a non-linear optimal control, instead of alinear specification, would be obtained. These adjustments could be interpreted as higher-orderterms of a Taylor expansion of a non-linear “optimal capital supply” function u ( X t , z t ).7s will be clear below, the strict limit λ → φ ≥ λ/µ . In this case, a non-vanishing value of λ would simply produce a small proportional shiftin physical quantities computed with the model such as future price distribution or correlationfunctions. In other words, the quadratic term in (3) is irrelevant in the growing phase with φ ≥ λ/µ . The model of [26] implicitly assumes that the market is in a stable “growth” mode,therefore it can be understood as a strict limit λ → , φ ≥ λ > φ < λ/µ that can describe meta-stable rather thanstable market dynamics.Note that all this assumes that market friction µ >
0, as it creates a feedback loop of impactof added or subtracted equity on market returns. As we will see next, a non-zero friction µ > non-linear .Substituting Eq.(3) into Eqs.(2), 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) θκ − X t − gκ X t (cid:19) dt + σX t (cid:0) dW t + w T z t (cid:1) , (4)where we introduced parameters g = µλ, κ = µφ − λ, θ = r f − c + φ. (5)If we keep µ > κ can be of either sign, depending onthe values of φ and λ . If φ < λ/µ , then κ <
0, otherwise one for φ ≥ λ/µ we get κ ≥ g = 0 is known in physics and biology as the Verhulst population growth modelwith a multiplicative noise [25], where it is usually written in an equivalent form that can beobtained by a linear rescaling of the dependent variable X t that makes the coefficient in frontof term X t equal one.Note that the higher order terms in the drift in (4) are responsible for a possible saturation ofthe process. In population dynamics, this corresponds to a population competing for a boundedfood resource. In a financial context, this spells a limited total wealth in a market without aninjection of capital from the outside world. Eq.(4) is a special case of the Langevin equation [36] d x t = − U (cid:48) ( x ) dt + σx t dW t , (6)which describes an overdamped Brownian particle in a potential U ( x ) whose negative gradientgives a drift term in the equation, in the presence of a multiplicative noise.We start with an analysis of a deterministic (noiseless) limit of Eq.(4) with σ → z t = 0.Comparing with Eq.(6), this is the same as a deterministic limit of the Langevin equation: dX t = κX t (cid:18) θκ − X t − gκ X t (cid:19) dt ≡ − ∂U ( X t ) ∂X t dt. (7)where the potential U ( x ) is a quartic polynomial: U ( x ) = − θx + 13 κx + 14 gx , (8)8n what follows, we will refer to (8) as the classical potential, to emphasize the fact that it gets“dressed” by noise, similarly to how classical potentials in quantum physics become modifiedby quantum corrections proportional to powers of the Planck constant (cid:126) . In our case, the noisevariance σ will play the role of (cid:126) in quantum physics.The quartic potential (8) can describe a number of different dynamic scenarios dependingon the signs and values of parameters involved. Note a cubic term in this expression, whichappears because the dynamics in our model is strongly asymmetric in time (see below). Suchasymmetry is caused by the presence of an absorbing state at x = 0 in the dynamics. This isquite different from the Ginzburg-Landau theory of equilibrium second-order phase transitionswhere a time reversal symmetry prohibits odd powers in a polynomial expansion of a free energy[35].The classic potential (8) can also be parametrized differently in terms of zero level points a and b : U ( x ) = − θx (cid:16) − xa (cid:17) (cid:16) − xb (cid:17) , (9)where parameters in Eq.(8) and (9) are related as follows: κθ = 32 a + bab , gθ = − ab . (10)The potential (8) or (9) has a degenerate extremum ¯ x at zero and two other extrema ¯ x , atnon-zero values¯ x = 0 , ¯ x , = − κ ± (cid:112) κ + 4 gθ g = 38 (cid:32) a + b ± (cid:114) ( a + b ) − ab (cid:33) , (11)where ¯ x and ¯ x correspond to the plus and minus signs, respectively.The roots are real-valued if the discriminant is non-negative, which produces a constraint θ ≥ − κ g . (12)Different shapes of the classical potential that can be obtained with different parameters inEq.(9) are illustrated in Fig. 1.For small values g →
0, the roots (11) can be approximated as follows:¯ x = θκ (cid:18) − gθκ (cid:19) + O ( g ) , ¯ x = − κg − θκ (cid:18) − gθκ (cid:19) + O ( g ) . (13)Note that the first root ¯ x is non-perturbative in κ and perturbative in g , while the second rootin non-perturbative in both κ and g .When θ > g > x > x < κ >
0. As the second root ¯ x < g → x < x ≥ κ <
0, we have ¯ x < x >
0. In this case, taking the limit g → x approaches a positive infinity in the physical region. This behavior suggeststhat in this regime, the behavior of the system is sensitive to the specific choice of value of g .9igure 1: Classical potentials for different choices of parameters in (9). The graph on the leftis obtained with θ = − , a = 2 . b = 4 .
0. This describes a bimodal system with twometastable states and possible absorption at zero. The graph in the center corresponds to θ = − , a = − . b = 4 .
0, and describes instability with a metastable state at x = 0.The graph on the right corresponds θ = 1 , a = − . b = 2 . κ = 0), and describesa potential that leads to spontaneous symmetry breaking, where the point x = 0 becomes amaximum rather than a local minimum.Depending on the values of parameters, potentials shown in Fig. 1 may therefore lead toquite different dynamic scenarios. The most insightful scenario is presented on the left graph.In this case, there exists a global minimum of the potential at a non-zero value of X , while theorigin X = 0 is a local minimum. If a particle hops to the well centered at the origin from theright well, this can be interpreted as a large drop of the stock price. Following this, the stockprice can continue to diffuse to the left towards the origin at zero. The even of reaching the zerolevel can be identified as a default event.Note that in the noiseless limit considered in this section, the first jump to the left well isnot possible (it requires adding noise, see below), but the process of sliding to the minimum atzero while in the left-most well has a well-defined zero-noise limit given by Eq.(7). In particular,it shows that the stock price cannot become negative if we start with a positive initial value: asthe gradient of the potential is zero at x = 0, the price will stay at this level forever once it isreached. Note for what follows that such irreversibility of defaults persists in the noisy case aswell, as long as noise is multiplicative as in Eq.(4).Therefore, a potential such as shown on the left graph in Fig. 1 can simultaneously describesmall diffusion in the right well, and large drops and defaults via thermally-induced tunneling.Reintroducing the signals z t will result in a fluctuating barrier.On the other hand, two other scenarios shown in the middle and the right figure are ofmarginal interest as they do not provide insight into defaults. The figure on the right describes ascenario with spontaneous symmetry breaking, where the zero-level point becomes the repellingboundary, as the potential in the vicinity of this point can be approximated by the invertedharmonic oscillator potential: U ( x ) = − θx , θ > . (14)Such scenario can only be realized in our setting if κ = 0, which means a market with zero netinflux of capital. Using this as an approximation to the potential around zero, we obtain an10pproximation to Eq.(4) for small values of X t (we omit signals z t here): dX t = θX t dt + σX t dW t , (15)which is the same as the GBM model (1). Therefore, the latter can be formally considered aseither the formal limit of κ, λ → U ( x ), or as an approximation for X t (cid:28) θκ , λ (cid:28) . However, thisscenario loses defaults: the point x = 0 is now repelling rather than attracting, as indicated byEq.(14).This can be compared with the scenario in the left figure. In this case, the point x = 0 is alocal minimum of the potential, therefore it becomes an attractive point for a particle, once ithops to the left well from the right well.On the other hand, the same GBM approximation (15) can also be obtained in this scenarioas well. From the perspective of local dynamics for large prices, the only difference from thefigure on the right is that now the potential is locally quadratic around a non-zero value. Buta big difference from the spontaneous symmetry breaking scenario is that defaults are still bothpossible, and also have a chance to be modeled. The latter is because in the figure on theleft, they are associated with exponentially small tunneling probabilities, also known as escapeprobabilities.Integrating Eq.(15) in the limit σ →
0, we obtain the conventional exponential asset growth,but this time only for sufficiently small deviations from a minimum of the potential U ( X ): X t = X e θt , X t (cid:28) . (16)Therefore, the exponential growth (16) that is often used in classical “financial equilibrium”models, i.e. equilibrium in a financial rather than a physical context, is associated within theQED model (4) with an initial exponential growth during the process of relaxation to a truevacuum . In the next section we will return to this scenario, after we analyze a time-dependentsolution of the model in the deterministic limit.The same exponential growth (16) is also obtained with a potential with spontaneous sym-metry breaking, though in this case it corresponds to an expansion around a maximum ratherthan a minimum at X = 0, and defaults would not be allowed classically,To summarize so far, out of many possible scenarios for dynamics that can be obtained withthe quartic potential (8) or (9), the graph on the left in Fig. 1 is the most interesting scenario.In what follows, we will be mostly concerned with the type of dynamics corresponding to sucha scenario. CPT -symmetry
Consider the classical Newtonian’s second law of dynamics for a particle in the potential (8): d xdt = − ∂U∂x = θx − κx − gx . (17) A potential with spontaneous symmetry breaking as a model for asset prices was discussed by Sornette [51]. In cosmological models used in physics, such behavior is often referred to as an initial inflation period.
11t is instructive to consider symmetries of Eq.(17) under the following set of transformationsthat we will refer to as the
CPT transformations, using an analogy with physics: C -parity: κ → − κ P -parity: x → − x (18) T -parity (Time reversal): t → − t. It is easy to see that the Newtonian second law equation (17) is separately symmetric withrespect to the time reversal T and the joint CP -inversion (while the latter is also equivalentto flipping the sign of the potential). As a consequence, it is also invariant with respect to asimultaneous CPT transformation.However, unlike the Langevin equation (6), Newton’s second law (17) contains the secondderivative in time in the left-hand side. This difference is important because it implies thatfor classical mechanics we need both an initial and terminal conditions to define two arbitraryconstants of integration, while for the Langevin dynamics we have only one. This results in theLangevin dynamics having different symmetries from the symmetries of classical mechanics.It is therefore interesting to see how the Langevin dynamics can arise from the classicaldynamics. To this end, we have to add in Eq.(17) a dissipative force term that is proportional toparticle velocity ˙ x , with a friction coefficient γ , and in addition, make parameter θ fluctuating: d xdt = ( θ + ε t ) x − γ dxdt − κx − gx . (19)To obtain a limiting behavior of this system in the limit of large friction γ → ∞ , note that wecan re-write Eq.(19) by introducing dimensionless time τ = t/γ :1 γ d xdτ = ( θ + ε t ) x − dxdτ − κx − gx . (20)Taking here the formal limit γ → ∞ (called the overdamped limit), the term in the left-handside vanishes, and the remaining terms exactly reproduce the Langevin equation (28) in time τ : dxdτ = − ∂U∂x + σxdW t , U ( x ) ≡ − θx + 13 x + 14 x . (21)The acceleration term in the left-hand side of Eq.(20) is dropped in the overdamped limit, whichmeans that in the overdamped limit, we assume that the velocity is nearly constant, hence itsfluctuations are neglected, resulting in zero acecleration.The Langevin equation is no longer invariant under time reversal T , though it maintainsinvariance with respect to the joint CP -inversion. As the time reversal symmetry is broken, thesimultaneous CPT invariance does not hold. An interesting property of Eq.(21) is that it impliesthe a backward motion with τ → − τ in a potential U is mathematically equivalent to a forward motion in an inverted potential − U . We will use this observation below. In this section, we temporarily set λ = 0, and hence g = 0, to simplify the analysis of thedeterministic limit. The solution of Eq.(7) in this limit is X t = X e θt κθ X ( e θt −
1) = θκ − (cid:16) − θκX (cid:17) e − θt . (22)12his solution is strongly asymmetric in time. Indeed, for an arbitrarily small starting point X >
0, the process (7) becomes singular in a finite time t ∞ given by the following expression: t ∞ = 1 θ log (cid:18) − θκX (cid:19) . (23)Note, however, that in an ‘expanding” regime with κ > t ∞ is actually negative . It can beinterpreted as an emergence from a negative singularity at time t ∞ < positive singularity immediately prior to that). For any given initial condition, X , Eq.(23)specifies a time in the past when the current market emerged from a negative singularity. Forpositive times, the solution remains positive and bounded for κ > X > κ <
0, i.e. capital is being withdrawn from the market,so that φ <
0. In this case, t ∞ is positive , i.e. the Verhulst dynamics (22) exhibits a positive singularity in a finite time 0 ≤ t ∞ < ∞ starting from an arbitrarily small value X >
0. Afterthat, it re-emerges from a negative singularity and remains on a “non-physical” solution branchwith X t < κ > κ <
0, respectively, is a consequence of the
CPT -symmetry (18) of the dynamics. Examples of solution of the “Verhulst limit” g → σ → g = 0 of the QED model for positive and negativevalues of κ . The following parameters are used: r f = 0 . , c = 0 . , µ = 0 . cubic term in Eq.(4) with a positive coefficient g (cid:28)
1. However small the value of g is, it regulates the model for very large fields | X | ∼ /g , while producing only very smallcorrections to the behavior for small fields | X | (cid:28) /g . A non-zero value of g smoothes out thesingularity as it becomes the second root ¯ x in Eqs.(13) when g (cid:54) = 0.Keeping g > λ >
0) arbitrarily small but finite, ensures that the limit κ → κ itself is adjusted by λ as k = µφ − λ ) is non-analytic in the product µφ . if wekeep µ > κ or φ . We prefer to use the parameter κ to describe analyticity as it directly enters theQED diffusion (4).Physical quantities can only be defined in a complex κ plane with a branch cut singularityalong the negative κ axis. Therefore, any perturbative expansion developed for κ > κ = 0 would be a divergent expansions .Therefore, when κ <
0, the only stable steady solution of Eq.(7) is ¯ X = 0. At κ = 0, thissolution becomes unstable, and a new stable steady state solution¯ X = θκ , κ > , (24)bifurcates at κ ≥
0. Recall that in this section we temporarily set g = 0. When g >
0, thesteady state solution (24) is corrected by terms of ∼ O ( g ), see Eqs.(13).This solution (or (13)) is non-perturbative in κ , i.e. this branch emerges in a continuousbut non-differentiable way. This means that in the deterministic limit σ → z t = 0, the system undergoes a second-order phase transition at κ = 0, with a non-vanishingorder parameter (24) that emerges for κ > κ = 0. As we will see below, the noisy problemis mathematically equivalent to a problem of quantum mechanical tunneling, where a conceptof a dynamic absorbing phase transition, as a special kind of a second order phase transition,becomes less formal and more interesting.For small values of κ and short times such that θt (cid:28)
1, we can approximate the solution(22) as follows: X t (cid:39) X e θt (cid:18) − kθ X (cid:16) e θt − (cid:17)(cid:19) (cid:39) X e θt (1 − κX t ) (cid:39) X e ( θ − κX ) t = X e ( r f − c + φ − κX ) t . (25)Therefore, the return on equity R e is given by the following expression: R e = r f − c + φ − κX (cid:39) r f − c + φ. (26)(the last relation holds as long as κX (cid:28) R e is correct only for sufficiently short times, where approximations leading to Eq.(25) arejustified. In a long-term limit, the asset price converges to the stable solution (24):lim t →∞ X t = θκ (cid:20) − (cid:18) θκX − (cid:19) e − θt (cid:21) . (27)This is one of the key differences between the QED model and the GBM model, where an expo-nential asset growth continues indefinitely. As the steady state solution (24) is non-perturbative (non-analytic) in κ , it could not be recovered with either the GBM model or any perturbativescheme of handling frictions that would be still based on the GBM model (1).Assume for the moment that the QED model (4) is the “true” model of the market, butan observer is unaware of this, and uses instead the GBM model (1). These dynamics can beviewed as fluctuations around the unstable steady state solution (cid:104) X (cid:105) = 0. Such fluctuations canonly be about a local equilibrium at best, simply because their “ground state”, (cid:104) X (cid:105) = 0, itselfbecomes unstable for κ >
0. If one starts with such an unstable state, it will relax into thestable state (24). This is similar to the famous argument by Dyson about divergence of perturbation theory in Quantum Electro-Dynamics (QED) [15]. In our case, negative values of κ would produce a positive feedback loop from investing inthe market as long as u t >
0, leading directly to an explosion at a finite time for a negative κ . opposite to its meaning in classical “competitive market equilibrium” models [13]. The expo-nential asset growth, assumed to be an attribute of equilibrium dynamics in these models, turnsout to be just an initial (non-equilibrium) exponential growth stage of a relaxation into a stablestate in our model.Thus, both the GBM and QED models produce an exponential asset growth in a shortterm, but strongly differ in their long-term predictions. Yet, depending on the values of modelparameters, and given their possible non-stationarity, testing for boundedness vs unboundednessof an exponential asset growth may not be an easy task. Fortunately, the two models (1) and(4) can be discerned by their behavior. More specifically, while the GBM model does not admitdefaults as a process of “falling to the origin”, X = 0, in the QED model such events becomepossible as noise-activated barrier penetration transitions. The following sections consider thesephenomena in more details. When we keep the white noise (the Brownian motion W t ) in Eq.(4), but omit (or take averagesof) signals z t , it can be viewed as a Langevin equation with multiplicative noise: d x t = − U (cid:48) ( x ) dt + σx t dW t , (28)where the classical quartic potential U ( x ) is defined in Eq.(8), and U (cid:48) ( x ) stands for the derivativeof the potential with respect to x .Instead of working with multiplicative noise, Eq.(28) can be transformed into a Langevinequation with additive noise by defining a log-price variable y t = log x t . Using Itˆo’s lemma, weobtain dy t = ∂y t ∂x t dx t + 12 ∂ y t ∂x t ( dx t ) = (cid:18) − U (cid:48) ( x ) x − σ (cid:19) dt + σdW t . (29)This produces an equivalent Langevin equation for the log-price (more accurately, the log-cap): dy t = − V (cid:48) ( y ) dt + σdW t , V (cid:48) ( y ) ≡ ∂V ( y ) ∂y = − (cid:18) θ − σ (cid:19) + κe y + ge y , (30)where the potential V ( y ) in the log-space is defined as follows: V ( y ) = − ¯ θy + κe y + 12 ge y , ¯ θ ≡ θ − σ . (31)Note that this is no longer a classical potential, as now it acquires a “quantum” Itˆo’s correction ∼ σ in the linear term ¯ θy .Different shapes of the y -space potential that can be obtained with different parameters inEq.(9) are shown in Fig. 3.The Langevin equation (30) in the log-space y = log x , can also be written in a form moreconventional in physics (crudely, by dividing both sides of (30) by dt ): dydt = − ∂V∂y + ξ t (cid:104) ξ t ξ t (cid:105) = σ δ ( t − t ) , (32)15igure 3: Potentials V ( y ) in the log-variables y = log x for the same choices of parameters in (9)as used in Fig. 1, with σ = 0 .
25. The graph on the left describes metastability corresponding toa bimodal system with two metastable states in Fig. 1. Absorption at zero is replaced by escapeto negative infinity in the y -space.where ξ t is Gaussian white noise. Note that the form of noise correlation in Eq.(32) suggeststhat the combination σ can be identified with an effective temperature T of the system, i.e. T = σ .Another important point to note is the behavior of the potential V ( y ) at positive and negativeinfinities in the y -space. For y → ∞ , we obtain V → + ∞ if g ≥
0. On the other hand, thebehavior at a negative infinity depends on the sign of ¯ θ : V = sign(¯ θ ) ∞ .Therefore, the potential V ( y ) leads to a bounded motion on the real y -axis only for θ ≥ σ .On the other hand, when θ ≤ σ (i.e. ¯ θ ≤ decreases as y → −∞ . This meansthat a particle placed in such potential will escape to a negative infinity.The escape to y = −∞ is the same as a “fall to the origin” x = 0 in the original x -space,which can be interpreted as corporate defaults. The latter are therefore identified in our modelwith a decay of a metastable state and escape to y = −∞ in the log-space.Metastability of an initial state is ensured by the presence of a barrier separating this initialstate and a ‘free fall” regime for y → −∞ . This happens when parameters are in a certain rangethat will be specified momentarily. For other values of parameters (but such that the constraint θ ≤ σ still holds), there would be no barrier, and escape to y → −∞ would correspond to adecay of an unstable initial state.Two extrema of the potential (31) are given by¯ y , = log − κ ± (cid:112) κ + 4 g ¯ θ g . (33)We are interested in the range of parameters when both extrema ¯ y , are real-valued. Thisrequires that κ < θ <
0. Using the definitions of κ and θ in terms of φ , this produces thefollowing range of parameter φ for a metastable decay to occur in the model: φ ≤ φ ≤ ¯ φ, ¯ φ ≡ σ − r f + c, φ ≡ − λ + (cid:112) µλ ¯ φµ . (34)16or yet smaller values φ < φ , the barrier disappears, and the process becomes unstable, ratherthan metastable. This can describe a regime of a “free fall” to default, or a total market collapsein a multivariate context.For the height of the barrier E b ≡ V (¯ y ) − V (¯ y ), we obtain E b = ¯ θ log 2 g ¯ θ + κ − κ (cid:112) κ + 4 g ¯ θ g ¯ θ + κ g (cid:113) κ + 4 g ¯ θ. (35)For small values of ¯ θ , this yields E b = ¯ θ log g ¯ θκ + O (cid:0) ¯ θ (cid:1) . (36)Note non-analyticity of this expression in the Itˆo-adjusted drift ¯ θ . If we view this expression ina complex plane of z = θ , it has a branch cut singularity on the negative semi-axis z ∈ ( −∞ , κ and fixed θ , Eq.(35) gives rise to another approximateformula: E b = κ g + O (cid:32) κ (cid:115) κ + 4 g (cid:18) θ − σ (cid:19)(cid:33) . (37)These formula shows that the barrier has a non-perturbative origin (it depends on 1 / √ g ), andits height vanishes in the strict limit κ = 0. Also, note the effect of noise volatility σ on thebarrier: as σ increases, the barrier becomes smaller.An additional point to note in Eq.(37) (or the full formula (35)) is that it is non-analytic at θ = σ . As is shown in Appendix C, this non-analyticity is related to the fact that a stationarystate of the Fokker-Planck equation exists only for θ ≥ σ (for the Itˆo prescription), or for θ ≥ σ (for the Stratonovich interpretation). A potential such as shown in Fig. 3 enables a noise-activated escape of a particle initially neara bottom of a local minimum of the potential (the point ¯ y (cid:39) . P (jump) = Γ exp (cid:18) − E b T (cid:19) ≡ Γ B, (38)where Γ is a pre-factor and B the Arrhenius factor B = exp ( − E b /T ) , (39)where E b is the barrier height, and T = σ is the temperature. The pre-factor Γ can be inter-preted the probability per unit time of finding the particle hitting the barrier while oscillatingin the right well. For a potential with one minimum at y (cid:63) and one maximum at y (cid:63) , such as inFig. 3, the escape rate is of the form of Eq.(38), and is given by the famous Kramers formula[27] (see Appendix A for a derivation): r = (cid:112) V (cid:48)(cid:48) ( y (cid:63) ) | V (cid:48)(cid:48) ( y (cid:63) )) | π exp (cid:20) − σ ( V ( y (cid:63) )) − V ( y (cid:63) )) (cid:21) . (40)17he Kramers escape rate (40) applies as long as the barrier height E b ≡ V ( y (cid:63) )) − V ( y (cid:63) ) (cid:29) σ .As in Eq.(39), the escape rate r is exponentially small in the height E b .While the Kramers relation (40) is obtained with classical stochastic dynamics, the sameexponentially suppressed probabilities of under-barrier transitions are obtained in quantum me-chanics [34] within the semi-classical (WKB) approximation. This is of course not incidental,and is due to the fact that the Langevin dynamics can be mapped onto quantum mechanics inEuclidean time [19, 57].Such mapping of classical stochastic dynamics onto Euclidean quantum dynamics is veryuseful as it allows one to utilize the power of quantum mechanics and quantum field theory(QFT) for solving problems of classical stochastic dynamics that are conventionally ad” using theLangevin or Fokker-Planck equation. In particular, these methods enable efficient computationalmethods for tunneling in multivariate (and non-linear) setting in dimensions D > , wherethe Kramers relation (corresponding to D = 1) needs modifications. We will now consider amodern alternative QFT-based approach to computing such hopping (tunneling) probabilitiesthat is called the instanton approach. Let’s start with the Langevin equation (32). Following [37], we use the path integral formulationof the Langevin dynamics. It is given by the following generating functional (see Eq.(B.31) inAppendix B): Z [ h ] = (cid:90) Dy D ˆ p J ( y ) exp (cid:18) − σ A ( y, ˆ p ) + (cid:90) h ( t ) y ( t ) dt (cid:19) , (41)where (cid:82) Dy D ˆ p stands for integration over trajectories, ˆ p is the so-called response (Martin-Siggia-Rose) dynamic field, A ( y, ˆ p ) is the Euclidean action of fields y, ˆ p with the Lagrangian L : A ( y, ˆ p ) = (cid:90) L ( y, ˙ y, ˆ p ) dt, L ( y, ˙ y, ˆ p ) = i ˆ pσ (cid:18) dydt + dVdy (cid:19) + σ p . (42) h ( t ) is an external source field, and J ( y ) is a Jacobian J ( y ) = det (cid:18) ddt + d V ( y ) dy (cid:19) . (43)In the weak-noise limit σ → T as T = σ / J ( y ) and the source field h can beneglected [37] (see Eqs.(B.29) and (B.30)). The generating functional is dominated in this limitby solutions of classical Lagrange equations [33] (here ˙ y = dy ( t ) /dt ) ddt ∂ L ∂ ˙ y − ∂ L ∂y = 0 , ddt ∂ L ∂ ˙ˆ p − ∂ L ∂ ˆ p = 0 , (44)that are obtained by varying the action A ( y, ˆ p ) with respect to y and ˆ p , respectively. For theQED Lagrangian (42), this yields d ˆ pdt − d Vdy ˆ p = 0 , dydt + dVdy = iσ ˆ p. (45)Note that the second equation has the same form as the original Langevin equation (32) wherethe role of noise is played by the rescaled response field i ˆ p . In the financial context,
D > D different stocks. dydt = − dVdy , ˆ p = 0 , A ( y, ˆ p ) = 0 . (46)This is a “normal” solution, where the response field ˆ p vanishes, and a particle goes “downhill”,against the gradient of the potential. This solution has zero action and corresponds to thenoiseless (“classical”) limit of the dissipative Langevin dynamics (32).There is however another solution dydt = dVdy , i ˆ p = 2 σ dydt = 2 σ dVdy . (47)It is easy to see that as long as Eqs. (47) hold, the first of Eqs.(45) is obtained from the secondequation by differentiating both sides with respect to time.The second solution (47) is very different from the first, normal solution (46). As suggestedby the second equation in (47), the response field ˆ p for this solution is purely imaginary. The firstequation shows that this solution describes a noiseless limit of the Langevin equation with an inverted potential V ( y ), or equivalently under time reversal t → − t . Such inversion of a potentialwhen describing tunneling also occurs in quantum mechanics (see below), and defines the essenceof the instanton approach: classically forbidden transitions become classically allowed with aninverted potential.The solution (47) is a vacuum ( zero energy ) solution, as the Hamiltonian (B.34) vanishes forthis solution: H = − σ i ˆ p ) + iσ ˆ p dVdy = − (cid:18) dVdy (cid:19) + 2 (cid:18) dVdy (cid:19) = 0 . (48)On the other hand, the action for this solution for a transition from y R at t i to y at t f isnon-zero: A ( y, ˆ p ) = 2 σ (cid:90) t f t i (cid:18) dydt (cid:19) dt = 2 σ (cid:90) y y R dydt dy = 2 σ (cid:90) y y R dVdy dy = 2 σ ( V ( y ) − V ( y R )) ≡ E b T , (49)where we replaced σ by the effective temperature T at the last step. The probability of suchtransition is therefore P (escape) ∼ exp ( −A ( y, ˆ p )) = exp (cid:18) − E b T (cid:19) , (50)which coincides, up to a pre-exponential factor, with Eq.(39), as well as with the conventionalWKB approximation of quantum mechanics for under-barrier tunneling [34].It turns out that in quantum mechanics (QM), tunneling can be described using a similarapproach. Classical Euclidean equations of motion arise in the QM semi-classical limit (cid:126) → imaginary time τ = it (see Appendix B for a crashintroduction). Solutions of classical Euclidean (imaginary time) equations of motion describingtunneling phenomena are known in quantum field theory and statistical physics as instantons ,see e.g. [57], [55], or [45]. A pre-exponential factor Γ can be computed from the analysis offluctuations around instanton solutions.The instanton solution is illustrated for a metastable potential on the left in Fig. 4. Aninstanton starts at time t = − T (where T → ∞ ) at the point y R of the right local minimum(or equivalently of the top of the inverted potential on the right graph). At time t = T , itarrives at the return point y L . Note that y L is a reflection point rather than a local minimum19igure 4: A metastable potential in the log-space y = log x for the same choices of parametersin (9) as used in Fig. 1. The graph on the right shows the inverted potential.of a potential. Therefore, our instantons are different from instantons that arise in theorieswith vacuum tunneling between degenerate potentials. In these models, initial and terminalconditions for an instanton correspond to degenerate vacua connected by the instanton.In the present problem, we deal with a different case of a decay of a metastable state.Corresponding classical solutions for such scenarios are called bounces (see [9] or [10], Chapt.7). Unlike instantons, bounces are period solutions, where the final state coincides with theinitial one, which in its turn is chosen to be a particle located at or near the bottom of thelocal metastable minimum of the potential. As we will show below, the bounce is made of aninstanton-anti-instanton pair.The instanton solution (47) on the left of Fig. 4 describes a particle that starts at the point y R at the bottom of right-most well of the original potential V ( y ) at time t = − T , and thenclimbs the hill at y , following Eq.(47). After that, the first solution (46) kicks in, and describesthe classical downhill move from y to y L , achieving the point y L at time T . An anti-instantonfollows the same trajectory reversely in time. By stacking together the instanton and anti-instanton, we obtain a bounce that starts at y R at t = − T , and ends at the same point y R attime t = T . Note that the parts of this trajectory that go from y R to y and then from y L to y are classically forbidden as they go uphill the classical potential V ( y ).On the other hand, on the right graph in Fig. 4, the same solution is interpreted as a purelyclassical motion in the inverted potential − V ( y ) (alternatively, one can reverse time t → − t ).In this case, the particle starts on the top of the hill at y R , then relaxes to y and climbs to y L ,after which it “bounce” (hence justifying its name) back to y R by reverting the trajectory. Theexplicit form of the instanton solution will be given below in Sect. 5.4.To summarize, the instanton-anti-instanton pairs (bounces) produce exponentially smallprobabilities (50) suppressed by the height of a potential barrier, as in the Kramers escape rate(40). The pre-exponential factor can be obtained by computing fluctuations around the bounce[9, 10]. Instead of completing this calculation (that produces again the Kramers relation), inthe next section we present an equivalent approach to quantum mechanical tunneling based onthe Fokker-Planck equation. 20 Defaults and the Fokker-Planck dynamics
Let p ( x, t ) be a probability density of the non-linear diffusion of Eq.(28). Its behavior dependson rules for stochastic integration, namely the Stratonovich vs Itˆo prescriptions, see e.g. [25].The Fokker-Planck equation (FPE) reads ∂ t p ( x, t ) = − ∂ x [ f ( x ) p ( x, t )] + σ ∂ xx [ g ( x ) p ( x, t )] , (51)where f ( x ) = (cid:18) θx − κx − gx + (2 − ν ) σ x (cid:19) , g ( x ) = x , (52)and ν = 2 or ν = 1 for the Itˆo or Stratonovich interpretation of the SDE (28), respectively.Similar FPEs with polynomial drifts and a multiplicative noise (52) were studied in the physicsliterature, see e.g. [46], [53]. The steady-state solution of this equation is p s ( x ) = 1 Z g ( x ) exp (cid:40)(cid:90) x f ( y ) σ g ( y ) dy (cid:41) = 1 Z exp [ − U ( x )] , (53)where Z is a normalization factor (also known as a partition funciton), and U ( x ) is the effectivepotential: U ( x ) = − σ (cid:90) x f ( y ) g ( y ) dy + log g ( x ) = 2 σ (cid:20) κx + g x − (cid:18) θ − σ − ν ) (cid:19) log x (cid:21) . (54)This is a potential energy of a “particle” with mass g in an external field given by a combinationof a quadratic and logarithmic potentials. This potential leads to metastability that was alreadymentioned above, for certain values of model parameters, as illustrated in Fig. 5.Figure 5: Effective potentials with metastable states for the QED model. The same values ofparameters as in Fig. 1 are used. The figure on the right shows extrema of the effective potentialand a potential barrier between the local minimum and an essential singularity at zero.21he extremal points of the effective potential (54) are (we set ν = 2 here)¯ x , = − κ ± (cid:112) κ + 4 g ( θ − σ )2 g . (55)In addition, the effective potential (54) has an essential singularity at x = 0. Comparing Eq.(55)with Eq.(11), we find that that only effect of noise on the extrema ¯ x , of the classical potentialis a shift of parameter θ by an amount ∼ σ .On the other hand, the trivial vacuum x = 0 is impacted by noise in a more drastic way,and becomes an essential singularity. Consider first the scenario where the coefficient in frontof the logarithmic term in Eq.(54) is positive. In this case, the state ¯ x = 0 becomes a repellingboundary and absorption at x = 0 is impossible. Such regime would be similar to a default-freeGBM world in the sense that in both cases the boundary at x = 0 becomes unattainable.On the other hand, if the coefficient in front of the logarithm in Eq.(54) is negative, than aparticle can “fall to the origin”, x = 0, in a potential such as shown in Fig. 5. If a particle isinitially placed to the right of a local maximum of the effective potential, it will be in a meta-stable state that will decay into the absorbing state in a finite time due to thermal fluctuations.A “fall to the origin” (a corporate default) from a meta-stable initial state may occur when thecoefficient of the logarithm in Eq.(54) is negative.To summarize, adding white noise induces a new “effective potential” (54) that leads toconsiderable changes of the analysis of the phase structure of the classical model (8) withoutnoise. For certain values of parameters, there is a non-zero probability of a “collapse to theorigin” at X = 0, that can describe a corporate default. Such events would be examples of theso-called noise-induced phase transitions [25, 53]. Let’s consider the FPE (51) where we introduce a new independent variable y = log x , with aninitial condition y ( t ) = y = log x for some time t . Let ˜ p ( y, t | y ) is be a probability densityfor y . For clarity, we display the initial point y here, or its respective value x in the x -spacein this section. The two densities p ( x, t | x ) and ˜ p ( y, t ) are related as follows: p ( x, t | x ) dx = ˜ p ( y, t | y ) dy ⇔ p ( x ( y ) , t | x ) = e − y ˜ p ( y, t | y ) (cid:18) y = log x, ∂∂x = e − y ∂∂y (cid:19) . (56)This substitution transforms the FPE (51) into a new FPE for the density ˜ p ( y, t ) that containsan additive rather than a multiplicative noise: ∂ ˜ p ( y, t | y ) ∂t = − ∂∂y (cid:104) ˜ f ( y )˜ p ( y, t | y ) (cid:105) + σ ∂ ∂y ˜ p ( y, t | y ) , (57)where ˜ f ( y ) = θ − ( ν − σ − κe y − ge y ≡ − ∂V ( y ) ∂y . (58)The steady state solution for this equation reads (compare with Eq.(53))˜ p s ( y ) = 1˜ Z e − σ V ( y ) , V ( y ) ≡ − (cid:18) θ − ( ν − σ (cid:19) y + κe y + 12 ge y , (59)where ˜ Z is a normalization constant. This solution exists for parameter values that give rise to anormalizable steady state solution. For parameters that produce metastability, the normalizationconstant Z in Eq.(59) diverges, and therefore for such parameters Eq.(59) ceases to be a validsolution. For mathematical details, see Appendix C.22 .3 Schr¨odinger equation In what follows, we set ν = 2, i.e. choose the Itˆo interpretation of the FPE (51), and work withthe FPE (57) in the log-space. We use the following ansatz to solve Eq.(57) (see e.g. [31] or[54]): ˜ p ( y, t | y ) = e − σ ( V ( y ) − V ( y )) K − ( y, t | y ) . (60)Using this in Eq.(57) produces an imaginary time Schr¨odinger equation (SE) for the forwardFeynman propagator K − ( y, t | y ). As discussed in Appendix A, it is useful to analyze simulta-neously both the forward and backward propagators, resp. K − ( y, t | y ) and K + ( y, t | y ), where K + ( y, t | y ) describes the dynamics with a flipped potential (or equivalently the dynamics undertime-reversal). A pair of SEs for K ± ( y, t | y ) reads − σ ∂K ± ( y, t | y ) ∂t = H ± K ± ( y, t | y ) , (61)where H ± are the Hamiltonians H ± = − σ ∂ ∂y + 12 (cid:18) ∂V∂y (cid:19) ± σ ∂ V∂y ≡ − σ ∂ ∂y + U ± ( y ) . (62)The solution of the Schr¨odinger equation can be formally represented by the heat kernel K ± ( y, t | y ) = (cid:104) y | e − σ t H ± | y (cid:105) . (63)Let { Ψ − n } be a complete set of eigenstates of the Hamiltonian H − with eigenvalues E − n : H − Ψ − n = E − n Ψ − n , (cid:88) n Ψ − n ( x )Ψ − n ( y ) = δ ( x − y ) . (64)We assume here a discrete spectrum corresponding to a motion in a bounded domain, so that theset of values E ( − ) n is enumerable by integer values n = 0 , , . . . . Substituting this into Eqs.(63)and (60), we obtain the spectral decomposition of the original FPE (see e.g. [54]):˜ p ( y, t | y ) = e − σ ( V ( y ) − V ( y )) ∞ (cid:88) n =0 e − σ E − n t Ψ − n ( y )Ψ − n (0) . (65)When t → ∞ , only one term with the lowest energy survives in the sum in Eq.(65). If suchlowest energy E ( − )1 is larger than zero for a particle located in a potential well, such as shownin Fig. 4, this means that such an initial state y is metastable with the decay rate E ( − )1 /σ .Therefore, in the quantum mechanical (QM) approach to barrier transitions in the Langevindynamics computing rate transitions amounts to computing the eigenvalue spectra of the QMHamiltonians H ± . The transform (60) and the resulting first factor in Eq.(65) take care of thenon-equilibrium component of the dynamics, and reduce the problem of metastability in theLangevin dynamics to a stationary or quasi-stationary quantum mechanical problem [19].Using function V ( y ) from Eq.(59), we obtain (recall that ¯ θ = θ − σ / (cid:18) ∂V∂y (cid:19) = ¯ θ − κ ¯ θe y + (cid:18) κ − g ¯ θ (cid:19) e y + κge y + g e y ,σ ∂ V∂y = κ σ e y + gσ e y . (66)23he QM potentials U ± ( y ) are therefore as follows: U ± ( y ) = ¯ θ − κ (cid:18) ¯ θ ∓ σ (cid:19) e y + (cid:18) κ − g ¯ θ ± gσ (cid:19) e y + κge y + g e y . (67)Therefore, both potentials U ± ( y ) are quartic polynomials in x = e y , and moreover differ fromone another only in coefficients in front of x and x . As y → −∞ , both potentials approach thesame constant level ¯ θ /
2, see Fig. 6.Figure 6: Quantum mechanical potentials U ± ( y ) for the QED model.Extrema of U ± ( y ) are determined by the equation (here x = e y ) x (cid:0) x + ax + b ± x + c ± (cid:1) = 0 , (68)where a = 3 κ g , b ± = 1 g (cid:18) κ − g ¯ θ ± gσ (cid:19) , c ± = − κ g (cid:18) ¯ θ ∓ σ (cid:19) . (69)Eq.(68) has a trivial root x = 0 ( y = −∞ ) corresponding to a particle absorbed at the origin X = 0. In addition, the cubic polynomial arising in Eq.(68) can have either one real root and twocomplex roots, or three real-valued roots. The roots are found by noticing that by a substitution x = u − a/
3, the cubic equation x + ax + b ± x + c ± = 0 reduces to the standard Cardano form u + p ± u + q ± = 0 , p ± = − a b ± , q ± = 2 (cid:16) a (cid:17) − ab ± c ± . (70)This equation has three roots, that we write here in terms of the original variable x : x = A ± + B ± − a , x , = − A ± + B ± ± i √ A ± − B ± − a , (71)where A ± = (cid:114) − q ± (cid:112) Q ± , B ± = (cid:114) − q ± − (cid:112) Q ± , Q ± = (cid:16) p ± (cid:17) + (cid:16) q ± (cid:17) . (72)If the discriminant Q ± in (72) is positive, Eq.(70) has one real root and two complex roots. If Q ± = 0, then there are tree roots such that two of them are equal each other. Finally, when Q ± <
0, there are three distinct real roots. 24ig. 6 shows that while the potential U − has two local minima and a local maximum (so that Q − < U + has only one global real-valued minimum (corresponding to Q + < H + than of H − . It turns out that the spectra (and eigenfunctions) of the Hamiltonians H ± areelated by a symmetry known in physics as supersymmetry . Analysis of supersymmetry of theFPE and a supersymmetric derivation of the Kramers relation are presented in Appendix D. While reproducing the Kramers relation along with calculable corrections, the quantum-mechanicalSUSY-based calculation of the decay rate given in in Appendix D does not provide an explicitlink to noise-induced barrier transitions that we described by Langevin instantons in Sect. 4.3.Here we show that the same instantons arise in the Schr¨odinger equation (61).To this end, consider the stationary Schr¨odinger equation for the lowest energy E , which isobtained from the time-dependent Schr¨odinger equation (61) by the substitution K − ( y, t | y ) → e − Et Ψ( y ): H − Ψ = E Ψ , H − = − σ ∂ ∂y + 12 (cid:18) ∂V∂y (cid:19) − σ ∂ V∂y . (73)We look for a solution of the SE (73) in the formΨ( y ) = exp (cid:20) − σ S ( y ) − S ( y ) (cid:21) , (74)which replaces one wave function Ψ by two unknown functions S ( y ) and S ( y ). The changeof variables (74) is the same as in the conventional WKB approximation of quantum mechanics(also known as the Eikonal approximation, see e.g. [34]). The QM WKB approximation com-putes functions S , S (and possibly higher-order corrections in σ ) by assuming a perturbativeexpansion for the energy E of the ground state as follows: E = E (0) + σ E (1) + σ E (2) + . . . . (75)The standard WKB approximation for functions S , S is obtained by substituting Eqs.(74) and(75) into the SE (73), and matching coefficients in front of different powers of (cid:126) = σ .Instead of the conventional WKB method, we use a modified WKB method in which E = 0,so that any difference of the ground state energy from zero is a pure quantum effect (i.e. itvanishes in the limit σ →
0) [43, 44]. While the modified WKB approach was developed in[43, 44] in the context on non-supersymmetric quantum mechanics, here we apply it to SUSYQM obtained from our initial Langevin formulation of classical stochastic dynamics. Note thatthe classical vacuum energy in SUSY QM is zero as can be seen from (73). Therefore, setting E (0) = 0 is perfectly justified in our case.Substituting (74) and (75) into Eq.(73) and equating expressions in front of like powers of σ , we obtain, to the zeroth order, the following equation: − (cid:18) ∂S ∂y (cid:19) + 12 (cid:18) ∂V∂y (cid:19) = 0 . (76)The first-order equation reads 12 ∂ S ∂y − ∂S ∂y ∂S ∂y − ∂ V∂y = E (1) . (77)25he first equation (76) is the Euclidean Hamilton-Jacobi (HJ) equation for the shortened (orMaupertuis) action S = S ( y | y ) as a function of a final position y starting with an initialposition y [33], with zero energy E = 0. Its difference from the real-time HJ equation is in theflipped sign of the kinetic energy. Replacing the partial derivative ∂S /∂y with the canonicalmomentum p = ∂S /∂y = ˙ y , Eq.(76) is solved by a solution of one of the following two equations: dydt = ± ∂V∂y ⇔ S ( y | y ) = ± ( V ( y ) − V ( y )) (78)As the action should be non-negative and we assume that V ( y ) > V ( y ), we should select theplus sign. This is the same instanton solution that was obtained in Eq.(47) in the path integralLangevin dynamics in Sect. 4.3. Choosing the minus sign produces anti-instanton that can beinterpreted as an instanton that propagates backward in time. The action for a bounce is twicethe instanton action (78).Therefore, we see that Langevin instantons re-appear as quantum-mechanical instantons inthe equivalent QM formulation. We view three different derivations (FPE, SUSY, and QMinstantons) of the same Kramers escape rate (40) presented here not as a useless mathemati-cal exercise but rather as a demonstration that powerful methods of quantum mechanics andquantum field theory can be applied to stochastic nonlinear dynamics of stock prices both for asingle stock (as shown in this paper) and in a multivariate setting (which is left here for futureresearch).The inverse of the instanton solution can be obtained by integrating Eq.(78): t − t = ± (cid:90) yy dy (cid:48) ∂V /∂y (cid:48) = ± g (cid:90) xx dx (cid:48) x (cid:48) ( x (cid:48) − x )( x (cid:48) − x )= ± g x − x (cid:18) x log | x − x || x − x | − x log | x − x || x − x | (cid:19) ± g x x log xx , (79)where plus or minus signs correspond to the instanton and anti-instanton, respectively, and x = e y with x , = e y , where y , are defined in Eq.(33). Given the inverse of the instanton (oranti-instanton) solution, the instanton itself can be obtained by a numerical inversion of Eq.(79).The instanton solution starts at x at time t = − T (where T → ∞ ), and arrives at point x attime t = T . The anti-instanton solution is the same in the reverse order: it starts at time t = − T at the reflection point x , and arrives at point x at t = T . A bounce is a periodic solutionwith period 2 T obtained with an instanton followed by a time-shifted anti-instanton that startsat time t = 0, instead of t = − T , see Fig. 7. Note that the instanton and anti-instanton arewell localized in time, and are different from a constant solution only for a relatively short timeinterval. Therefore, in the limit T → ∞ typically assumed in instanton-based calculations, abounce can be thought of as a pair of well-separated instanton and anti-instanton. This implies,in particular, that the action for a bounce is twice the instanton action .Note that in our problem, the instanton action can be directly integrated as in Eq.(78)because the quantum mechanical potential in the SE (73) is W ( y ) ≡ / ∂V /∂y ) (plus thequantum correction proportional to ∂ V /∂y .) Therefore, Eq.(78) coincides with the regularWBK expression for the instanton action, as an integral of the momentum corresponding to thezero total energy E in the potential W ( y ) and the canonical momentum p ≡ (cid:112) W ( y ) = ∂V /∂y As was mentioned above, only periodic bounce solutions that start and end at same local minimum can beclassical extrema for metastable potentials [9, 10] such as obtained in the QED model for ¯ θ <
0. We representbounces as instanton-anti-instanton pairs, but our instantons are not transitions between classical local minima. S = (cid:90) yy p ( y (cid:48) ) dy (cid:48) = (cid:90) yy (cid:112) W ( y (cid:48) ) dy (cid:48) = (cid:90) yy ∂V∂y (cid:48) dy (cid:48) = V ( y ) − V ( y ) . (80)Therefore, the Euclidean HJ equation (76) in our setting is the same as the leading-order HJequation in the (non-supersymmetric) modified WKB approach of [43, 44] provided we use it forthe QM potential W ( y ) ≡ / ∂V /∂y ) . This particular form of the QM potential allows oneto compute the instanton action explicitly as in Eq.(78). The bounce action exactly reproducesthe exponential factor in the Kramers relation (40).The pre-exponential factor can be obtained from the second equation (77). Note the differ-ence of this equation relatively to its regular QM version in [43, 44]: the instanton solution ofEq.(77) leads to a cancellation of the first and third terms in Eq.(77). Rearranging terms, weobtain ∂S ∂y = − E (1) ∂S /∂y (81)that can be easily integrated: S ( y | y ) = − E (1) (cid:90) yy dy (cid:48) ∂S /∂y (cid:48) = − E (1) (cid:90) yy dy (cid:48) ∂V /∂y (cid:48) = − E (1) ( t − t ) , (82)where t = t ( y ) as determined by its dependence on x = e y in Eq.(79). Further approximationsof this expression reproduce the pre-exponential factor in the Kramers relation (40). Assuming that data correspond to a stable (quasi-stationary) market regime, the FPE equationin the y -variable (57) can be used to estimate the model parameter using maximum likelihoodestimation (additional constraints will be introduced below). To this end, note that the FPE(57) corresponds to the following diffusion law in the y -space (where we re-install the drivers z t ): dy t = − ∂V ( y ) ∂y dt + σdW t , V ( y ) ≡ − (cid:18) θ − σ wz t (cid:19) y + κe y + 12 ge y , (83)27here W t is a standard Brownian motion. In terms of variables y = log x , the negative log-likelihood of data is therefore (assuming a small time step ∆ t → LL M (Θ) = − log T − (cid:89) t =0 √ πσ ∆ t exp (cid:40) − σ ∆ t (cid:18) y t +∆ t − y t ∆ t + ∂V ( y ) ∂y (cid:19) (cid:41) , (84)where y t = log x t now stands for observed values of log-cap. Note that because the model isMarkov, the product over t = 0 , . . . , T − σ , θ , κ , g and w .To guide the numerical optimization, one typically imposes regularization. In our case, weuse a regularization that ensures the presence of a barrier between a “diffusive” and “default”regions of stock log-cap value y t (or equivalently of market prices). These regions would beconcentrated around the metastable minimum of the effective potential and the reflection point,respectively. In such a regime, defaults via tunneling become possible, and therefore we canapproximate model-implied hazard rates by the Kramers escape rate formula (40) .By fitting to CDS data referencing stocks in our dataset, we get approximate calibration totwo sorts of data: stock returns and CDS spreads. Note that from the point of view of calibrationto equity returns, calibration to the Kramers formula with a fit to CDS spreads amounts to whatcan be called the “Kramers regularization”. With this regularization, we simply compute theKramers rate as a function of model parameters, and then add a penalty to the negative log-likelihood that is proportional to the squared difference of this quantity and a CDS-impliedhazard rate. The total penalty (regularization) is made of this contribution plus the penaltyensuring the presence of a barrier.We used 7 years of daily returns and daily CDS spreads for 16 stocks that make just overa half of all names in the Dow Jones Industrial index. All results are reported for the simplestcase with no signals (so that z t = 0 or equivalently w = 0).We modify the loss function to fit the QED model to the observed CDS spreads. For eachyear of daily spread history, we estimate the sample mean ¯ r obs and then add a penalty term ofthe form λ (¯ r obs − r ) , where r is the Kramers rate.Furthermore, we have to enforce the condition that all transitions included in the likelihoodare within the diffusive region. It is convenient to add this as a soft penalty for the initialvalue y t for each transition to be to the right of the maximum y of the potential. This canbe enforced e.g. by adding a penalty λ max ( y − y t , λ = 1 × .To ensure positivity of the discriminant defining the roots y , (and hence that they are bothreal-valued) and the presence of a barrier, we use the following simple numerical trick. At eachiteration step, for any given previous value of κ , we define a new variable κ (cid:48) = − (2 (cid:112) g | ¯ θ | + | κ | ),which is negative by construction, as well as it automatically satisfies the condition ( κ (cid:48) ) ≥ . g ¯ θ .Using such recomputed value κ → κ (cid:48) to reinforce negativity of κ at each iteration step ofcalibration, we eventually converge to a stable solution with both a negative κ and a barrier.We use this trick instead of a more brute-force approach that would enforce the real-valued rootand a barrier at a cost of two additional Lagrange multipliers.Figure 8 compares the annual mean spread (bps) of the AXP CDS spread with the Kramershazard rate. Alternatively, we could perform unconstrained optimization with model parameters that do not necessarilyensure the existence of a potential barrier. This may provide a better in-sample fit if we are only interestedin fit to equity returns but not to credit spreads. Out-of-sample performance may however be worse with suchapproach, presumably depending on a fraction of large market moves in an out-of-sample dataset. λ . The maximum loglikelihoods are observed to be consistently higher under the QED model (without constraints)than the GBM model. With increasing values of λ , we observe a general trend of decreasinglog likelihood values, due to greater emphasis placed on the CDS spread history.For completeness, the calibrated model parameters, using both prices and CDS spreads, arereported for all symbols in Tables 4 to 7 using λ = 10. Additionally, the calibrated Kramersescape rate and annual CDS spread history is given in Tables 2 and 3 respectively. All numericalexperiments are performed using TensorFlow version 1.3.Figure 8: A comparison of the observed annual mean spread (bps) of the AXP CDS (black solid)with the calibrated Kramers spread: hazard × (1 − R ). The calibrated spreads are shown forthree different values of λ corresponding to Table 1. When λ = 10 (green dashed), the modeland observed annual mean CDS spreads are practically identical. Model 2010 2011 2012 2013 2014 2015 2016 2017GBM 878.739 895.622 991.374 983.511 1007.410 946.753 1011.842 1008.518QED (unconstrained) 882.301 900.213 997.0149 984.307 1008.076 947.164 1012.872 1010.485QED (constrained, λ = 0 .
1) 875.025 891.0852 989.552 977.831 1008.032 946.543 994.030 1005.150QED (constrained, λ = 1 .
0) 866.31 879.7991 980.2564 961.6697 1008.0321 946.7632 1010.5832 989.9231QED (constrained, λ = 10 .
0) 838.477 832.561 850.736 938.622 1005.481 946.821 994.871 1004.654
Table 1: Comparison of the maximum log likelihood function under GBM and QED models forAXP price history between 2010 to 2017. Note that the second row shows the values of themaximum log likelihood function for the QED model calibrated without constraints to pricehistory only. The bottom three rows show the maximum log likelihood functions for the QEDmodel calibrated, with constraints, to price and CDS spread history, for different values of theregularization parameter. 29
010 2011 2012 2013 2014 2015 2016 2017AXP 93.844 95.2148 85.4901 53.4911 41.0538 43.8309 45.9352 26.8229BA 152.9697 240.5589 232.8387 109.406 70.0338 72.3545 88.2742 54.3214CAT 77.0086 85.8795 95.7828 71.005 43.4454 61.1637 79.0823 41.6516CSCO 57.6397 80.3585 79.1721 43.3382 36.8152 30.7575 34.0111 27.4797DIS 48.913 37.9151 29.388 22.2677 19.3259 18.4489 25.1561 30.2328GS 144.1252 193.7553 241.3186 125.7899 84.7 89.1107 101.7708 71.5498HD 67.5283 63.3773 49.3592 35.1921 25.4508 21.9486 26.1475 25.2557IBM 39.3108 41.4169 36.4958 34.752 41.2068 48.8432 57.4105 37.2366JNJ 42.302 40.0684 36.3307 23.9545 14.4517 15.2582 17.4625 17.8959JPM 87.7228 104.4257 119.2014 83.2466 59.6081 71.0212 68.7511 49.8118MCD 44.9489 34.8205 25.3095 18.6306 21.2441 39.4674 34.741 26.4765PFE 54.2214 67.408 62.2827 35.3363 23.3582 21.4857 26.4003 28.567PG 43.9813 44.7094 50.3334 34.7521 26.4204 18.1977 20.7467 20.9639UNH 132.8105 100.0228 93.4989 50.3178 37.7885 28.2273 34.7879 24.5506VZ 76.6715 67.0173 60.6482 65.1896 52.368 64.9342 62.631 71.1288WMT 44.973 48.2212 42.4751 30.3341 17.5377 19.7762 39.3315 36.0448
Table 2: The calibrated Kramers hazard (default) rate r without a signal. Note that the hazardrate has been scaled here by the factor (1 − R ), where R = 0 . Table 3: Observed annual mean CDS spreads (in bps).
This paper presents the “Quantum Equilibrium Disequilibrium” (QED) model of a financialmarket, which is inspired by reinforcement learning and physics. The initial formulation ofthe QED model for a special case g = 0 was proposed in [26] based on inverse reinforcement30earning (IRL) applied to the modeling of investment portfolios and a market as a whole. Ageneralization of the resulting dynamics to the case g > θ, σ, κ and g (plus the weights w , if any).For practical applications with more stringent requirements for quality of fit to data, onemight include time-dependent predictors (alpha-signals) z t . In addition to producing calibratedweights w , calibrated model parameters θ, σ, κ will also be different in the presence of signals.We focused in our examples on the basic benchmark case with no signals.Calibration of our model to CDS data can also be viewed as a proxy to calibration to(unavailable) large market moves data. Unlike the GBM model and other models of “small”marker fluctuations, our non-linear model is both capable of modeling large market moves, andoffers an approximate way to calibrate probabilities of such events to available CDS marketdata.With strong practical motivation, we showed in this paper non-linearities of resulting pricedynamics are non-perturbative (non-analytic) in friction parameters. Consequently, the GBMmodel turns out to be only a formal mathematical limit of the QED model, or any other non-linear model where defaults arise as noise-induced transitions into an absorbing state.In reality, this limit is non-analytic , which shows up, depending on a parameter regime, as adivergence of perturbation theory, or as a non-equilibrium and non-perturbative noise-activatedphase transition into an absorbing state, associated in our model with a corporate default. As aresult, the QED and GBM model are not related by a physically-meaningful smooth transition.In the language of physics, they cannot be in the same universality class - which is easy tounderstand as the GBM model altogether misses defaults, in the first place.The formal limit of zero market frictions thus leads to both divergence of a formal pertur-bation theory around a zero-friction limit, and missing defaults. Such rare events become amathematical impossibility in the GBM and other linear models, simply because they cease toexists in this formal limit. Similarly to tunneling phenomena in statistical physics, quantummechanics and quantum field theory, such exponentially rare events cannot be caught at anyfinite order of perturbation theory around a zero-friction limit. Therefore such events are notattainable either in the GBM model or any other model that treats market frictions as a pertur-bation around a “friction-free” market. On the other hand, such market does not exist anyway,at least not as any meaningful physical limit.To describe defaults and other large market moves, we need different tools that do not rely on perturbative expansions around such non-existing “friction-free” markets. Fortunately,such tools are readily available in physics. In this paper, we presented three inter-related, non-perturbative, physics-based approaches.The first approach is based on the analysis of the Fokker-Planck equation. In a one-dimensional setting, this leads to the celebrated Kramers escape rate formula as a theoreticalprediction for hazard (default) rates in the QED model. Such analysis of defaults as tunnelingphenomena is simple in 1D, but in higher dimensions the Kramers relation requires modifica-tions.Such modifications for D > z t into the dynamics. Thisrequires modifications to calculations we presented above for the hazard rates. For example,if signals z t are modeled as colored (correlated) noise, this gives rise to a fluctuating barrierseparating the diffusive and default regions. Such analysis will be presented elsewhere.In our approach, both large market moves and defaults are described by instanton solutionswhose distinguished property is their locality in time: the instanton is well approximated bya smoothed step function, see Fig. 7. The instanton transition happens very fast in time.Therefore, even though the noise term in a simplest version of our model can be just whitenoise, nonlinearity and instantons provide a mechanism explaining large market jumps withoutthe need for introducing jump-diffusion or stochastic time changes that are sometimes introducedin mathematical finance models to include jumps in prices.Our model does not claim to provide an accurate time-dependent (inter-day, within eachyear) calibration to daily CDS spreads. This is of course because without time-dependent sig-nals z t , the model is stationary and does not exhibit any explicit time-dependence. In addition,incorporating non-zero trading signals z t produces fluctuating barriers. Consequently, predic-tions for CDS spreads based on the Kramers relation will be affected by time-varying correctionsthat might be considerably sizable.Another interesting problem is to model CDS spreads at different tenors. Instanton-based orSUSY QM-based approaches are only able to model the simplest term structure of CDS spreads.Namely, only a simplest CDS curve with a constant hazard rate (equal to the Kramers escaperate) can be modeled based on instantons or SUSY QM for the ground state. This is becauseboth these methods are asymptotic in the transition time T → ∞ . Producing non-flat CDScurves requires computing corrections to this asymptotic behavior, which is another problemthat we leave for future research. Appendix A: Escape from a metastable minimum
Here we provide a brief summary of the derivation of the probability to escape from a metastableminimum using the Fokker-Planck equation. For more details, see e.g. [21], Chapter 9.Consider the FPE equation ∂ t p ( y, t ) = ∂ t (cid:2) V (cid:48) ( y ) p ( y, t ) (cid:3) + D∂ y p ( y, t ) ≡ − ∂ y J ( y ) , D ≡ σ , (A.1)where a potential V ( y ) has a metastable minimum at point a , a stable minimum at point c , anda local maximum between these points at point b , and J ( y ) is the probability current: J = − ∂V∂y P ( y, t ) − σ ∂P ( y, t ) ∂y = − σ e − σ V ∂∂y (cid:16) e σ V P ( y, t ) (cid:17) . (A.2)In equilibrium with a stationary distribution P ( x ), the left hand side of the FPE vanishes. Theright hand side trivially vanishes as well, if the probability current is zero. This can be achieved ifthe product e σ V P ( x ) is independent of x . This immediately produces the equilibrium density: P ( y ) = 1 Z e − σ V ( y ) , (A.3)32here Z is a normalization factor. When the potential V ( y ) is metastable, Eq.(A.3) leads toa diverging normalization constant Z and hence to non-existence of the stationary solutionEq.(A.3).A proper treatment for a metastable scenario is obtained as follows. First, we assume aconstant non-vanishing current J throughout the system, so that we have − σ e − σ V ∂∂y (cid:16) e σ V P (cid:17) = J ⇔ ∂∂y (cid:16) e σ V P (cid:17) = − J σ e σ V . (A.4)Let’s integrate this equation between points a and c : e σ V P (cid:12)(cid:12)(cid:12) ca = − Jσ (cid:90) ca e σ V ( y (cid:48) ) dy (cid:48) . (A.5)Neglecting the term proportional to P ( c ) in comparison to P ( a ), we obtain e σ V ( a ) P ( y = a ) = − Jσ (cid:90) ca e σ V ( y (cid:48) ) dy (cid:48) ⇒ J = σ P ( a ) exp (cid:2) V ( a ) /σ (cid:3)(cid:82) ca e σ V ( y (cid:48) ) dy (cid:48) . (A.6)The quantity J gives the total probability of a particle to leave the metastable well given thatit is initially placed in this well. We can therefore write J = pr where p is the probability to bein the well, and r is the escape rate. For the former, we can use use the fact that probabilitiesat y and y = a are approximately related by the equilibrium relation P ( y ) = P ( a ) exp (cid:20) − V ( y ) − V ( a ) T (cid:21) , T ≡ σ . (A.7)Integrating this relation in the limits [ a − ∆ , a +∆] where ∆ is the size of the well, and expandingthe integrand on the right hand side around the point y = a to perform a Gaussian integration,we obtain p = (cid:90) a +∆ a − ∆ P ( y ) dy = P ( a ) (cid:90) a +∆ a − ∆ exp (cid:20) − V (cid:48)(cid:48) ( a )( y − a ) T (cid:21) = P ( a ) (cid:115) πT | V (cid:48)(cid:48) ( a ) | . (A.8)For the denominator in Eq.(A.6), we can expand around the peak of the potential at y = b toobtain (cid:90) ca e σ V ( y (cid:48) ) dy (cid:48) = (cid:115) πT | V (cid:48)(cid:48) ( b ) | e σ V ( b ) . (A.9)Using this together with the relation r = J/p and Eq.(A.8), we finally obtain the Kramers escaperate formula: r = (cid:112) V (cid:48)(cid:48) ( a ) | V (cid:48)(cid:48) ( b ) | π exp (cid:20) − σ ( V ( b ) − V ( a )) (cid:21) . (A.10)This formula applies as long as the barrier height ∆ E ≡ ( V ( b ) − V ( a ) (cid:29) σ . Appendix B: Path integrals for the Langevin dynamics
Here we describe steps needed to re-formulate the Langevin dynamics in terms of path integrals.For more details, see e.g. [57], [31], or [22]. 33he method of path integrals for the Langevin dynamics is based on using the followingidentity valid for an arbitrary function f ( x ) acting in a real-valued space R n : (cid:90) dxδ ( f ( x )) (cid:12)(cid:12)(cid:12)(cid:12) det ∂f∂x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:90) dx (cid:12)(cid:12)(cid:12) det ∂f∂x (cid:12)(cid:12)(cid:12) δ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) det ∂f∂x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:90) dxδ ( x ) = 1 . (B.1)In quantum mechanics and quantum field theory, a similar identity is used for infinite-dimensionalpath integrals.We introduce two conjugated (“Nicolai”) maps defined by the following pair of relations fortwo independent Gaussian white noise processes ξ ± t : ξ ± t = dydt ∓ ∂V∂y . (B.2)If we take the minus sign in this relation, we recover the Langevin equation (32) which we repeathere for convenience: dydt = − ∂V∂y + ξ − t (cid:104) ξ − t ξ − t (cid:105) = σ δ ( t − t ) . (B.3)On the other hand, picking the plus sign in (B.2), we obtain a similar Langevin equation to(B.3) with the independent noise ξ + t , and with an inverted potential, or equivalently written inthe backward time t → − t . As we will see shortly, we need both the forward and backwardsolutions if we want to describe periodic solutions that propagate in both directions.Now consider a transition probability m ± t ( y, y ) to some state y given an initial state y attime t under the influence of noise ξ ± t . It can be represented as an integral of a delta-functionwith respect to all realizations of noise ξ ± t , where Dξ ± stands for integration over values of ξ ± t for all times t , see e.g. [57] or [31]: m ± t ( y, y ) = (cid:104) δ ( y ( t ) − y ) (cid:105) ± = (cid:90) Dξ ± exp (cid:18) − σ (cid:90) tt (cid:0) ξ ± t (cid:1) dt (cid:19) δ ( y ( t ) − y ) . (B.4)This is a Gaussian path integral for the noise ξ ± ( t ). We can convert it into a Wiener pathintegral for y t using the Nicolai maps Eqs.(B.2) as the change of measure ξ ± t → y ( t ). Note thatthis involves the Jacobian of this transformation J ≡ (cid:12)(cid:12)(cid:12)(cid:12) det (cid:20) δξ ± ( t ) δy ( t (cid:48) ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) det (cid:20)(cid:18) ∂∂t ∓ ∂V∂y (cid:19) δ ( t − t (cid:48) ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) . (B.5)As was shown in [17], [31] (see also below), the value of the determinant depends on the definitionof the stochastic calculus. For the Itˆo and Stratonovich prescriptions, we obtain, respectively J = (cid:40) const × , Itˆo , const × exp (cid:110) ∓ (cid:82) t dτ ∂ V ( y ( τ )) ∂y (cid:111) , Stratonovich . (B.6)The fact that with the Itˆo prescription, the Jacobian is equal to an unessential constant factorcan be seen directly in our initial discrete-time equations (2). When viewed “backwards” as aparticular discretization of a continuous-time diffusion, Eq.(2) correspond to the most conven-tional forward Euler discretization of an Itˆo’s SDE. The Jacobian of transition from the noise ξ t to the next-value price x t +1 is clearly a constant factor here. Such constant can be ignoredas it does not impact the dynamics, and is eventually re-absorbed in a denominator of the pathintegral. 34ow we replace the path integration wrt ξ ± ( t ) in the Gaussian path integral (B.4) by inte-gration with respect to paths of the state variable y t using the Nicolai maps (B.2) (with eitherthe plus or minus sign) as a definition of the mapping, and keeping the Jacobian (B.5). Thisyields m ± t ( y, y ) = (cid:90) Dy J exp (cid:32) − σ (cid:90) tt (cid:18) dydu ∓ ∂V∂y (cid:19) du (cid:33) δ ( y ( t ) − y )= (cid:90) y t y Dy J exp (cid:32) − σ (cid:90) tt (cid:18) dydu ∓ ∂V∂y (cid:19) du (cid:33) (B.7)= (cid:90) y t y Dy J exp (cid:34) − σ (cid:90) tt (cid:32)(cid:18) dydu (cid:19) + (cid:18) ∂V∂y (cid:19) (cid:33) du ± σ (cid:90) tt ∂V∂y dydu du (cid:35) . The last integral entering this expression requires a proper interpretation, as its value dependson whether we use Itˆo’s or Stratonovich’s calculus [31]: (cid:90) tt ∂V∂y dydu du = (cid:40) [ V ( y t ) − V ( y )] − σ (cid:82) t dτ ∂ V∂y , Itˆo , [ V ( y t ) − V ( y )] , Stratonovich . (B.8)If we now substitute this expression along with the Jacobian (B.6) into Eq.(B.7), we obtain afinal Lagrangian path integral representation for the transition probability. Importantly, thisfinal representation is independent of the rule of stochastic calculus: m ± t ( y, y ) = e ± V ( yt ) − V ( y σ × (cid:90) y t y Dy e − σ (cid:82) tt L ± du = e ± V ( yt ) − V ( y σ × (cid:90) y t y Dy e − S ± σ , (B.9)where S ± = (cid:82) L ± dt is the Euclidean action with the Euclidean Lagrangian L ± = 12 (cid:18) dydt (cid:19) + 12 (cid:18) ∂V∂y (cid:19) ± σ ∂ V∂y . (B.10)Therefore, once we obtained the path integral (B.9) using Itˆo’s calculus, from that point on wecan assume the Stratonovich rules in for integrals appearing in what follows within the pathintegral formulation of stochastic dynamics. This is convenient because it allows one to useconventional rules of calculus such as e.g. integration by parts.The path integral entering Eq.(B.9) is equivalent to an Euclidean quantum mechanical pathintegral with the Planck constant σ . We recall that in quantum mechanics, an EuclideanLagrangian in imaginary time τ is obtained by making the so-called Wick rotation from thephysical time t to the imaginary time τ ≡ it in the action: i Sσ = i σ (cid:90) t du (cid:34) (cid:18) dydu (cid:19) − U ( y ) (cid:35) ⇒ − σ (cid:90) it dτ (cid:34) (cid:18) dydτ (cid:19) + U ( y ) (cid:35) . (B.11)As a result, the Euclidean Lagrangian is given by the kinetic energy plus the potential energy,see e.g. [55] or [57]. In our case, a pair of Euclidean Lagrangians corresponding, respectively, tothe transition probabilities m ± t ( y, y ), are given by Eq.(B.10).We can also construct a corresponding Euclidean Hamiltonian from the Euclidean Lagrangian(B.9) by making the Legendre transform from the velocity ˙ y to the canonical momentum p : H ± = − p dydt + L , p ≡ ∂ L ∂ ˙ y . (B.12)35or the Lagrangian (B.9), this produces H ± = − p + 12 (cid:18) ∂V∂y (cid:19) ± σ ∂ V∂y . (B.13)Note the peculiar properties of these Euclidean Hamiltonians. While the last two terms in H ± can be identified with a potential energy term U ± , the first term is a kinetic energy. In real-time classical mechanics, the kinetic energy of a particle of mass m and velocity v = ˙ y is mv ,However, in the Euclidean Hamiltonians H ± , the coefficient in from of this term is negative ratherthan positive. This has important implications, in particular, it lets us to describe noise-inducedtunneling as a classical motion in Euclidean time with zero energy.The “quantum-mechanical” Fokker-Planck Hamiltonians are obtained from these classicalHamiltonians by the regular quantum mechanical replacement of the canonical momentum bya derivative p → − σ ∂/∂y . This gives a pair of FP Hamiltonians H ( F P ) ± = − σ ∂ ∂y + 12 (cid:18) ∂V∂y (cid:19) ± σ ∂ V∂y . (B.14)The Fokker-Planck Hamiltonians (B.14) can also be obtained directly from the Fokker-Plankequations (FPE) for the transition probabilities m ± t ( y, y ) [57, 31]: ∂∂t m ± t ( y, y ) = ∓ ∂∂y (cid:20) ∂V∂y m ± t ( y, y ) (cid:21) + 12 σ ∂ ∂y m ± t ( y, y ) . (B.15)Note that the conventional FPE corresponds to the equation for m − t ( y, y ). The second equationfor m + t ( y, y ) corresponds to the forward dynamics in an inverted potential, or equivalently thebackward in time dynamics with the original potential. Define the change of the dependentvariable in Eq.(B.15) as follows: m ± t ( y, y ) = e ± V ( yt ) − V ( y σ K ± ( y, t | y ) . (B.16)Substituting this ansatz in the FPE (B.15), the latter is transformed into the Schr¨odinger equa-tion in imaginary time with the Planck constant (cid:126) = σ and the Hamiltonian H ( F P ) ± defined inEq.(B.14): − σ ∂∂t K ± ( y, t | y ) = H ( F P ) ± K ± ( y, t | y ) , K ± ( y, t | y ) = δ ( y − y ) . (B.17)Using position eigenstates of the Hamiltonian H ( F P ) − that we denote as | y t (cid:105) , the formal solutionof this imaginary-time Schr¨odinger equation is given by the heat kernel K ± ( y, t | y ) = (cid:104) y | exp (cid:18) − σ t H ( F P ) ± /σ (cid:19) | y (cid:105) . (B.18)Combining Eqs.(B.18) and (B.9) which produces the Euclidean path integral representation oftransition probability (cid:104) y | exp (cid:16) − t H ( F P ) ± /σ (cid:17) | y (cid:105) = (cid:90) y t y Dy e − σ (cid:82) L ± dt , (B.19)where the Euclidean Lagrangians L ± are defined in Eq.(B.10).36ransition probabilities such as (B.19) are conventionally obtained in the path integralmethod by adding a source term h ( t ) in the exponent, and defining the generating functional Z [ h ] as follows: Z [ h ] = (cid:90) Dy exp (cid:18) − σ (cid:90) L ± dt + (cid:90) h ( t ) y ( t ) dt (cid:19) . (B.20)In particular, the propagator (cid:104) y ( t ) y ( t ) (cid:105) can be obtained by twice differentiating Z [ h ] withrespect to h ( t ) and h ( t ), and then setting h ( t ) = 0 in the final expression, see e.g. [57]. B.1 Tunneling and instantons in quantum mechanics
A convenient formulation of quantum mechanics in general, and tunneling phenomena in par-ticular, is provided by the Feynman path integral method [23]. Feynman’s path integral versionof quantum mechanics is formulated in terms of classical paths. Weights of such paths arecomplex-valued, and are given by exp( iS/ (cid:126) ) where i is the square root of − (cid:126) is the Planckconstant, and S is the action on a path S = (cid:90) y f y L ( y, ˙ y ) dτ = (cid:90) y f y (cid:34) (cid:18) dydτ (cid:19) − V ( y ) (cid:35) dτ. (B.21)Consider the classical action S multiplied by i starting from the time τ and making theEuclidean rotation (also known as the Wick rotation) iS = i (cid:90) τ τ (cid:34) (cid:18) dydτ (cid:19) − V − ( y ) (cid:35) dτ −−−−→ τ = − it − (cid:90) t t (cid:34) (cid:18) dydt (cid:19) + V − ( y ) (cid:35) dt ≡ − S E . (B.22)Minimization of the Euclidean action S E is done by solving the classical Lagrange equation ddτ ∂ L ∂ ˙ y + ∂ L ∂y = 0 ⇒ d ydτ = ∂V∂y . (B.23)The key observation is that the potential V ( y ) got flipped as a result of the Euclidean (Wick)rotation [55]. What was a classically forbidden region in the original time τ becomes a “classi-cally” feasible region in the Euclidean time t , as now the potential is turned upside down, andwhat was a forbidden region now becomes a “classically” allowed region. B.2 Path integral with the response field
Instead of the Euclidean Lagrangian path integral, we can introduce an equivalent phase-spaceform by introducing the so-called Habbard-Stratonovich transform that expresses the quadraticterm in the action in the path integral (B.7) as an integral over an auxiliary (real-valued) fieldˆ p also known as a response or Martin-Siggia-Rose (MSR) field (we omit an unessential constantfactor in front of this integral, as it will be re-absorbed into the overall normalization factor N of a path integral, see below): m ± t ( y, y ) = (cid:90) y t y Dy J exp (cid:32) − σ (cid:90) tt (cid:18) dydu ∓ ∂V∂y (cid:19) du (cid:33) = (cid:90) y t y Dy D ˆ p J exp (cid:20) − (cid:90) tt (cid:18) σ p + i ˆ p (cid:18) dydu ∓ ∂V∂y (cid:19)(cid:19) du (cid:21) . (B.24)37n this formulation, we use the Stratonovich rules of stochastic calculus. The Jacobian J isgiven by the determinant of differential operator M obtained by differentiating the Langevinequation with respect to y ( t (cid:48) ): J = det M ≡ det (cid:20)(cid:26) ∂∂t + ∂ V∂y ( t ) ∂y ( t (cid:48) ) (cid:27) δ ( t − t (cid:48) ) (cid:21) = exp (cid:20) Tr log (cid:20)(cid:26) ∂∂t + ∂ V∂y ( t ) ∂y ( t (cid:48) ) (cid:27) δ ( t − t (cid:48) ) (cid:21)(cid:21) = exp (cid:20) Tr log (cid:20) ∂ t (cid:26) δ ( t − t (cid:48) ) + ∂ − t ∂ V∂y ( t ) ∂y ( t (cid:48) ) (cid:27)(cid:21)(cid:21) . (B.25)Here we used the identity log Det A = Tr log A , and ∂ − t stands for the Green’s function G ( t − t (cid:48) )that satisfies ∂ t G ( t − t (cid:48) ) = δ ( t − t (cid:48) ) . (B.26)The solutions of this equation are G ( t − t (cid:48) ) = θ ( t − t (cid:48) ) if we choose propagation forward in time,or G ( t − t (cid:48) ) = − θ ( t (cid:48) − t ) for propagation backward in time [22]. If we choose propagation forwardin time, we obtain J = exp (cid:20) Tr (cid:26) log ∂ t + log (cid:20) δ ( t − t (cid:48) ) + θ ( t − t (cid:48) ) ∂ V∂y ( t ) ∂y ( t (cid:48) ) (cid:21)(cid:27)(cid:21) = exp [Tr log ∂ t ] exp (cid:20) Tr log (cid:26) log (cid:20) δ ( t − t (cid:48) ) + θ ( t − t (cid:48) ) ∂ V∂y ( t ) ∂y ( t (cid:48) ) (cid:21)(cid:27)(cid:21) . (B.27)Here the first term can be dropped, as it cancels out a similar term in an overall normalizationfactor in Eq.(B.24). On the other hand, the second term can be evaluated using the Taylorexpansion of the logarithm: J = exp (cid:20) Tr (cid:26) θ ( t − t (cid:48) ) ∂ V∂y ( t ) ∂y ( t (cid:48) ) + θ ( t − t (cid:48) ) θ ( t (cid:48) − t ) ∂ V∂y ( t ) ∂y ( t (cid:48) ) ∂ V∂y ( t (cid:48) ) ∂y ( t ) + · · · (cid:27)(cid:21) = exp (cid:20)(cid:90) dtθ (0) ∂ V∂y ( t ) + (cid:90) dt (cid:48) θ ( t − t (cid:48) ) θ ( t (cid:48) − t ) ∂ V∂y ( t ) ∂y ( t (cid:48) ) ∂ V∂y ( t (cid:48) ) ∂y ( t ) + · · · (cid:21) . (B.28)The second term and all the subsequent terms here vanish because θ ( t − t (cid:48) ) θ ( t (cid:48) − t ) = 0, thereforeonly the first term survives. Choosing θ (0) = , one has [22] (see also Sect.4.8.2 in [57]) J = det M = exp (cid:18) (cid:90) ∂ V∂y dt (cid:19) . (B.29)On the other hand, if we choose backward propagation in Eq.(B.26), the Jacobian (B.29) getsa negative sign in the exponent. Respectively, with either of this choices, we obtain a pair of aforward and backward Euclidean Lagrangians L and L B with actions A ( y, ˆ p ) and A B ( y, ˆ p ): A ( B ) ( y, ˆ p ) = 1 σ (cid:90) L ( B ) dt, L ( B ) = σ p + i ˆ pσ (cid:18) dydt ∓ dVdy (cid:19) ∓ σ ∂ V∂y . (B.30)To the leading order for the forward propagator m − t ( y, y ), the last term in Eq.(B.30) originatingfrom the Jacobian can be neglected, producing the following Lagrangian: L = σ p + i ˆ pσ (cid:18) dydt + dVdy (cid:19) . (B.31)38s an alternative to the Lagrangian mechanics, one can change variables from y, dydt to Hamil-tonian mechanics using the Legendre transform from the velocity ˙ y = dy/dt to the momentum p = ∂ L ∂ ˙ y = iσ ˆ p. (B.32)This defines the classical Euclidean Hamiltonian H = − p ˙ y + L ( y, ˙ y, ˆ p ) . (B.33)In our case with the specific Lagrangian (42), we obtain the Hamiltonian H = σ p + iσ ˆ p dVdy = − p + p dVdy . (B.34)The Hamilton form of dynamics is given by two first-order PDEs for y and p , instead of onesecond-order PDE of the Lagrangian formulation (44) [33]:˙ y = ∂ H ∂p , ˙ p = − ∂ H ∂y . (B.35)This of course produces the same equations of motion (45) if we substitute here the QEDHamiltonian (B.34) with the canonical momentum (B.32). B.3 SUSY path integral
As was discussed above, while the path integral representation (B.7) is independent of the rulesof stochastic calculus (see Eq.(B.9)), our derivation above relied on the Itˆo calculus, underwhich the Jacobian J is trivial, while the last term in Eq.(B.7) is given by the first expressionin Eq.(B.8).As the whole expression is independent of the stochastic calculus rules, we can insteaduse the Stratonovich rule, under which the Jacobian J is non-trivial. We can now expressthis determinant as an integral over dynamic Grassmann (fermion) variables ψ t , ¯ ψ t using theGrassmann integration rules (B.36).Unlike ordinary (boson) variables that commute, i.e. [ y , y ] ≡ y y − y y = 0, Grassmann(fermion) variables ψ, ¯ ψ anti-commute , i.e. { ψ , ψ } ≡ ψ ψ + ψ ψ = 0 (which means, inparticular that they are nilpotent , i.e. ψ = 0). Grassmann variables satisfy the followingintegration rules: (cid:90) dψ = 0 , (cid:90) dψ ψ = 1 , (cid:90) dψ d ¯ ψ . . . dψ n d ¯ ψ n exp n (cid:88) i,j =1 ¯ ψ i A ij ψ j = det [ A ij ] . (B.36)These rules can be used in order to express the Jacobian J in terms of an integral over Grassmannvariables. To this end, we introduce two dynamic Grassmann fields ψ and ¯ ψ , that satisfy thefollowing anti-commuting relations: { ψ ( t ) , ψ ( t ) } = { ¯ ψ ( t ) , ¯ ψ ( t ) } = 0 , { ψ ( t ) , ¯ ψ ( t ) } = 1 . (B.37)This produces an equivalent representation of transition probabilities (B.7) in terms of a pathintegral that involves both the original (boson) variables y t and Grassmann variables ψ, ¯ ψ : m ± t ( y, y ) = e ± V ( yt ) − V ( y σ × (cid:90) y t y DyDψD ¯ ψ e − σ (cid:82) tt L ± du . (B.38)39here L ± = 12 (cid:18) dydt (cid:19) + 12 (cid:18) ∂V∂y (cid:19) − σ ψ T (cid:18) ddt ∓ ∂ V∂y (cid:19) ψ. (B.39)This Lagrangian is invariant under the SUSY transformations δy = ¯ εψ − ¯ ψε, δψ = ε (cid:18) ˙ y ± ∂V∂y (cid:19) , δ ¯ ψ = (cid:18) ˙ y ∓ ∂V∂y (cid:19) ¯ ε. (B.40)where ε, ¯ ε are infinitesimal Grassmann parameters of the SUSY transformation.We can relate the SUSY Lagrangians (B.39) to the Fokker-Planck Hamiltonians (B.14) byintroducing a matrix-valued representation of anti-commutation relations (B.37). This can bedone by the following choice for ψ, ¯ ψ : ψ = (cid:20) (cid:21) ≡ σ − , ¯ ψ = (cid:20) (cid:21) ≡ σ + (B.41)Using this in Eq.(B.39) and making the Legendre transform to proceed from the Lagrangian toa Hamiltonian, we obtain the SUSY Hamiltonian H : H = − σ ∂ ∂y + 12 (cid:18) ∂V∂y (cid:19) + σ σ ∂ V∂y = (cid:20) H + H − (cid:21) , σ = (cid:20) − (cid:21) , where. (B.42) σ is the Pauli matrix. This is the Hamiltonian of the Euclidean supersymmetric (SUSY)quantum mechanics of Witten [56], see also below in Appendix D. The two FP Hamiltonians(B.14) describing the forward and backward dynamics are therefore combined together withinthe SUSY approach. Fore more details on supersymmetric path integrals, see e.g. [31]. Appendix C: The Fokker-Planck equation and non-analyticity
The partition function Z for the FPE equation is given by the normalization factor in the steadystate solution (53): Z = (cid:90) ∞ exp [ − U ( x )] dx = (cid:90) ∞ x θσ − ν exp (cid:26) − σ (cid:104) κx + g x (cid:105)(cid:27) dx. (C.1)When Re (cid:0) θσ + 1 − ν (cid:1) > (cid:0) g/σ (cid:1) >
0, this integral is known analytically, see [24],Eq.(3.462)): Z = (cid:18) gσ (cid:19) − z Γ ( z ) e κ gσ D − z (cid:32) κσ (cid:115) σ g (cid:33) , z ≡ θσ + 1 − ν, (C.2)where D − z ( · ) stands for a parabolic cylinder function ([24], Eq.(9.240)) D − z ( x ) = 2 − z W − z , − (cid:18) x (cid:19) = 2 − z e − x (cid:34) √ π Γ (cid:0) z (cid:1) Φ (cid:18) z ,
12 ; x (cid:19) − √ πx Γ (cid:0) z (cid:1) Φ (cid:18) z ,
32 ; x (cid:19)(cid:35) . (C.3)Here W p,q ( x ) is a Whittaker function, and Φ( α, γ, x ) is a confluent hypergeometric function, alsoknown as the Kummer function , which is defined as a sum of the following infinite series:Φ( α, γ ; x ) = 1 + αγ x
1! + α ( α + 1) γ ( γ + 1) x
2! + α ( α + 1)( α + 2) γ ( γ + 1)( γ + 2) x
3! + . . . . (C.4) Other notations for this function are M ( a, b, x ) and F ( a ; b ; x ), see [24]. α, γ, x ) is an analytic (holomorphic) function of x , there-fore the only singularities of the partition function (C.2) in a complex plane of θ are due tosingularities of gamma functions entering Eqs.(C.2) and (C.3).It is known that for real values of z , the parabolic cylinder function D z ( x ) has [ z + 1] real-valued roots, where [ z + 1] stands for a largest natural number that is less than z + 1 if it exists,and zero if it does not [3]. According to the Lee-Yang theory of phase transitions, physicallyobserved phase transitions correspond to accumulation of non-analytic points of the free energy F ≡ log Z on a real positive axis for parameter z used as an external control parameter drivingthe phase transition (see e.g. [30], Sect. 3.2).These non-analytic points of the free energy F correspond to zeros of the partition function Z . The latter can have zeros on a real positive axis of parameter φ due to the presence of theparabolic cylinder function D − z ( x ) in Eq.(C.2). It should have [ − z +1] real valued zero, thereforethe first zero arises when − z + 1 ≥
1, or z ≤
0. Recalling the definition of z = θσ + 1 − ν inEq.(C.2), we conclude that a phase transition according to the Lee-Yang theory should happenfor the following values of parameter θ = r f − c + φ :2 θσ ≤ ν − ⇔ φ ≤ ( ν − σ c − r f . (C.5) Appendix D: Supersymmetry in the Fokker-Planck dynamics
D.4 Supersymmetry (SUSY)
The Schr¨odinger equation (61) for the FPE possesses hidden supersymmetry (SUSY) [7], [38],[54] that makes it mathematically identical to supersymmetric quantum mechanics (SUSY QM)of Witten [56].Supersymmetry of the problem is rooted in the fact that the Hamiltonians (62) can befactorized into two first-order operators as follows: H − = A + A , H + = AA + (D.1)where A = 1 √ (cid:20) σ ∂∂y + ∂V∂y (cid:21) , A + = 1 √ (cid:20) − σ ∂∂y + ∂V∂y (cid:21) . (D.2)Note that the Hamiltonian H − transforms into the partner Hamiltonian H + if we flip the sign ofthe potential V ( y ) → − V ( y ). They can be paired in the following matrix-valued Hamiltonian: H = (cid:20) H + H − (cid:21) = (cid:20) AA + A + A (cid:21) . (D.3)This is the Hamiltonian of the Euclidean supersymmetric quantum mechanics of Witten [56].It can also be represented in a form that involves fermion (anti-commuting) fields ψ t , ψ + t , inaddition to the conventional boson (i.e., commuting) field y t , see Appendix B.3. Alternatively,two purely boson Hamiltonians H − = A + A and H + = AA + can be thought of as representingtwo different fermion sectors of the model. The potential V ( y ) is referred to in the contextof SUSY models as the superpotential. For a brief review of SUSY quantum mechanics, seeAppendix B.3, while a more complete treatment can be found e.g. in [31].Instead of representation in terms of operators A , A + , we can equivalently express the Hamil-tonian (D.3) in terms of supercharges Q = 1 √ (cid:20) AA + (cid:21) , Q = i √ (cid:20) −AA + (cid:21) (D.4)41his gives H = 2 Q = 2 Q = Q + Q . (D.5)This means that the Hamiltonian H commutes with both supercharges, i.e. [ H , Q ] ≡ HQ −Q H = 0, and [ H , Q ] = 0, therefore the supercharges Q , Q are constants in time. Further-more, Eq.(D.5) shows that eigenvalues of both Hamiltonians H ± are non-negative, with zerobeing the lowest possible eigenvalue.Due to the factorization property (D.1), if Ψ − n is an eigenvector of H − with an eigenvalue E − n > n = 1 , , . . . ), than the state Ψ + n ≡ ( E − n ) − / A Ψ − n will be an eigenstate of theSUSY partner Hamiltonian H + with the same eigenvalue (energy) E − n (the factor ( E − n ) − / isintroduced here for a correct normalization.) This is seen from the following transformation H + Ψ + n = (cid:0) E − n (cid:1) − / AA + A Ψ − n = (cid:0) E − n (cid:1) − / AH − Ψ − n = (cid:0) E − n (cid:1) − / A E − n Ψ − n = E − n Ψ + n , n = 1 , , . . . , which means that all eigenstates of spectra of H , except possibly for a ’vacuum’ state with energy E − = 0, should be degenerate in energy with eigenstates of the SUSY partner Hamiltonian H + [56]. Such zero-energy ground state would be unpaired, while all higher states would be doublydegenerate between the SUSY partner Hamiltonians H ± : H − Ψ − = A Ψ − = 0 , E − = 0Ψ − n +1 = (cid:0) E + n (cid:1) − / A + Ψ + n , Ψ + n = (cid:0) E − n +1 (cid:1) − / A Ψ − n +1 , n = 0 , , . . . (D.6) E − n +1 = E + n , n = 0 , , . . . For more details on supersymmetry in quantum mechanics, see e.g. [55], [31], [45].The existence or non-existence of a zero-energy ground state E − = 0 has to do with su-persymmetry being unbroken or spontaneously broken. In scenarios with spontaneous breakingof SUSY, supersymmetry is a symmetry of a Hamiltonian but not of a ground state of thatHamiltonian. On the other hand, an unbroken SUSY is characterized by the existence of a nor-malizable ground state Ψ with strictly zero energy E = 0, while for a spontaneously brokenSUSY the energy of the ground state is larger than zero [56]:Unbroken SUSY: A Ψ = 0 · Ψ = 0 ( E = 0)Spontaneously broken SUSY: A Ψ = E Ψ , E > . (D.7)For SUSY to be unbroken, the derivative of the superpotential V (cid:48) ( y ) = ∂V /∂y should havedifferent signs at y = ±∞ , which means that that it should have an odd number of zeros at realvalues of y . In our case, the superpotential is V ( y ) = − ¯ θy + κe y + 12 ge y . (D.8)For y → ∞ , we have V (cid:48) ( y ) → ∞ , while the behavior at y → −∞ depends on the sign of ¯ θ . If¯ θ >
0, we have lim y →−∞ V (cid:48) ( y ) <
0, therefore SUSY is unbroken.On the other hand, if ¯ θ <
0, SUSY would be spontaneously broken. Note that this is thesame condition under which defaults through barrier penetration in the Langevin dynamicsbecome possible. This is not coincidental, and is due to the fact that the ground state of H − with zero energy turns out to be non-normalizable due to a decay via tunneling.Indeed, due to the factorization property (D.1), the formal zero-energy solution of the SE H − Ψ ( − )0 = E ( − )0 Ψ ( − )0 with zero energy E ( − )0 = 0 is given by a solution of a simpler equation A Ψ = 0. The solution of the latter equation isΨ − = Ce − σ V ( y ) , (D.9)42here C is a normalization constant. Note that the square of this solution coincides with theequilibrium FPE density (59). Therefore, it will be square-integrable for the same choice ofparameters for which the FPE density (59) is integrable. In particular, as tunneling becomespossible for ¯ θ <
0, the zero-energy ground state (D.9) ceases to be a normalizable state. Thisleads to a spontaneous breaking of SUSY.We therefore obtain the following conditions for unbroken and broken SUSY in our setting:Unbroken SUSY: ¯ θ > , E = 0Spontaneously broken SUSY: ¯ θ < , E > . (D.10)For certain types of potentials, it is often the case that while the potential U − is bistable,the potential V + has a single minimum. In this case, SUSY can help to solve the problem oftunneling for potential U − by a simpler problem of finding a ground state for the SUSY partnerpotential U + [7]. As suggested by Fig. 6, our case is exactly of this sort. D.5 SUSY at work: the decay rate calculation
As discussed above, defaults in the Langevin dynamics correspond to a noise-induced tunnelingin the Langevin drift potential V ( y ), which becomes possible under the same constraint ¯ θ < C for such a solution.In the quantum mechanical SUSY approach, this means that the zero-energy state of QMHamiltonian H − does not exist, and the spectrum starts with the first eigenvalue (energy) E − >
0. However, supersymmetry relations (D.6) still apply for higher states n >
0. They canbe used to replace a hard problem of computing the first level E − > H − by a simplerproblem of computing the ground state energy E +0 of the SUSY partner Hamiltonian H − [7].As indicated in Eq.(65), the decay rate of a metastable state in the original problem is given by E − /σ .We therefore want to compute the lowest eigenvalue E − > H − when parameters are such that ¯ θ <
0, SUSY is spontaneously broken, and a normalizable groundstate with E − = 0 does not exists.The candidate ground-state solution of H + is easy to compute from the equation A + Ψ +0 = 0,whose solution can be obtained by flipping the sign of V ( y ) in Eq.(D.9):Ψ +0 ( y ) ∼ − ( y ) ∼ exp (cid:20) σ V ( y ) (cid:21) . (D.11)However, this cannot be a right zero energy solution because it is not normalizable due todivergence at y → ∞ . Following [32, 20] ((see also [31] for a similar derivation) consider thefollowing ansatz: Ψ( y ) = − ( y ) (cid:82) ∞ y dz (cid:2) Ψ − ( z ) (cid:3) for y > − ( − y ) (cid:82) ∞− y dz (cid:2) Ψ − ( z ) (cid:3) for y < − ( y ) is assumed to be normalized, while Ψ( y ) is not normalized. One can easily checkthat Ψ( y ) is continuous at y = 0 with Ψ(0) = − (0) , and that H + Ψ( y ) = 0 for y (cid:54) = 0.However, Ψ( y ) defined in Eq.(D.12) is not an eigenstate of H + because its derivative has adiscontinuity at y = 0:lim ε → d Ψ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ε − d Ψ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) − ε = lim ε → ε d Ψ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) y =0 = − − (0) . (D.13)43nstead, Ψ( y ) can be seen as a ground state wave function of the singular Hamiltonian H s givenby H s = H + − σ (cid:2) Ψ − (0) (cid:3) δ ( y ) ≡ H + − δ H , (D.14)that differs from H + by the δ -function term δ H . Indeed, with this Hamiltonian we obtain (cid:90) dy Ψ( y ) [ H s − E s ] Ψ( y ) = (cid:90) dy Ψ( y ) H s Ψ( y ) = 0 , (D.15)as a result of the fact that H s Ψ( y ) = 0 for y (cid:54) = 0, while finite contributions coming from thediscontinuity of the first derivative (D.13) and the additional δ -function term cancel each other: − σ (cid:90) ε − ε Ψ( y ) d Ψ( y ) dy dy − (cid:90) ε − ε [Ψ( y )] δ H dy = σ − σ . (D.16)Equivalently, we can express H + in terms of H s : H + = H s + 2 σ (cid:2) Ψ − ( y (cid:63) ) (cid:3) δ ( y ) = H s + δ H . (D.17)Therefore, the ground-state eigenvalue and the eigenfunction of H + can be found using quantummechanical perturbation theory (see e.g. [34]). To this end, we treat the additional term δ H as a perturbation around the exactly solvable singular Hamiltonian H s . To the first order inperturbation δ H , the change of energy E +0 = E − is E +0 = (cid:82) ∞−∞ Ψ ( y ) δ H dy (cid:82) ∞−∞ Ψ ( y ) dy = 2 σ [Ψ(0)] (cid:2) Ψ − (0) (cid:3) (cid:82) ∞−∞ Ψ ( y ) dy (cid:39) σ (cid:82) ∞−∞ e − Vσ dy (cid:82) ∞−∞ e Vσ dy . (D.18)Here in the last equality we assumed that the barrier is high, and therefore the ground statewave function is concentrated around the minimum of V ( y ) where the exact form of V ( y )is replaced by its approximation around this minimum. In other words, tunneling is neglectedwhen we compute the energy splitting formula (D.18), similarly to how tunneling is computed inquantum mechanics ([34], Sect. 50). The constant C is then determined from the normalizationcondition (under the same approximation for V ( y )) C (cid:90) e − V ( y ) σ dy = 1 , while Ψ( y ) can be approximated as follows: Ψ( y ) (cid:39) / Ψ − ( y ) = 1 /C exp (cid:104) V ( y ) σ (cid:105) While the main contribution to the first integral in the denominator of Eq.(D.18) comesfrom the region around the minimum y (cid:63) of V ( y ), the second integral is determined by a vicinityof the maximum y (cid:63) of V ( y ) . Therefore, we expand V ( y ) for the first and second integrals asfollows: V ( y ) = V ( y (cid:63) ) + 12 V (cid:48)(cid:48) ( y (cid:63) )( y − y (cid:63) ) + . . .V ( y ) = V ( y (cid:63) ) − (cid:12)(cid:12) V (cid:48)(cid:48) ( y (cid:63) ) (cid:12)(cid:12) ( y − y (cid:63) ) + . . . (D.19) See Sect. 5.10.1 in [47] for a different, FPE-based derivation of the the Kramers relation that leads to thesame expression as in the last form in Eq.(D.18). r = E +0 /σ : r = (cid:112) V (cid:48)(cid:48) ( y (cid:63) ) | V (cid:48)(cid:48) ( y (cid:63) ) | π exp (cid:20) − σ ( V ( y (cid:63) ) − V ( y (cid:63) )) (cid:21) . (D.20)Therefore, we demonstrated that the classical Kramers escape rate relation can be obtained usingmethods of SUSY [7, 32, 20, 31]. While the SUSY-based approach may appear more involvedthan the FPE-based derivation given in Appendix A, its merit is that it can also be relativelyeasily extended to more difficult multivariate settings, unlike a 1D FPE-based derivation. TheSUSY-based approach can also be used to compute corrections to the Kramers relation byincluding higher-order terms in perturbation theory with the singular potential (D.17). Appendix E: Numerical Experiments
The calibrated model parameters used for the results presented in Table 1 and Figure 8 arelisted in Tables 4 to 7 below. The regularization parameter λ = 10. Table 4: Calibrated θ without a signal.45
010 2011 2012 2013 2014 2015 2016 2017AXP 0.0318 0.0343 0.0446 0.0634 0.014 0.0271 0.0315 0.0227BA 0.1344 0.0705 0.0387 0.2705 0.0221 0.027 0.0476 0.1582CAT 0.1196 0.0449 0.0347 0.0147 0.02 0.0311 0.0635 0.1079CSCO 0.0274 0.0345 0.026 0.0285 0.0361 0.0165 0.0225 0.0263DIS 0.0261 0.024 0.0374 0.0291 0.019 0.0154 0.0115 0.0128GS 0.0391 0.1093 0.1144 0.0311 0.0302 0.0374 0.1406 0.0205HD 0.0389 0.0263 0.0641 0.0224 0.0235 0.0135 0.0099 0.0246IBM 0.0127 0.0296 0.0114 0.0138 0.0174 0.0229 0.0337 0.0129JNJ 0.0093 0.0158 0.0138 0.0261 0.0129 0.0114 0.0119 0.0122JPM 0.0258 0.0537 0.0677 0.0401 0.0146 0.029 0.0909 0.0399MCD 0.0321 0.0354 0.0095 0.008 0.0104 0.0115 0.0102 0.0317PFE 0.0188 0.0293 0.0297 0.0122 0.0106 0.0131 0.016 0.0108PG 0.0087 0.0103 0.0165 0.0131 0.014 0.0111 0.0092 0.009UNH 0.0419 0.08 0.0328 0.0494 0.049 0.0165 0.0346 0.038VZ 0.0353 0.0162 0.044 0.0346 0.0168 0.0149 0.0132 0.0176WMT 0.0104 0.0128 0.0293 0.0104 0.0126 0.0147 0.0121 0.0445
Table 5: Calibrated σ without a signal. Table 6: Calibrated κ without a signal.46
010 2011 2012 2013 2014 2015 2016 2017AXP 3.7041 2.8461 1.6995 0.4664 0.6134 0.6051 0.9075 0.3931BA 4.5167 8.9223 12.337 0.9479 1.2692 1.0384 1.0297 0.1441CAT 1.11 1.0411 1.252 2.312 0.7425 1.3847 1.2797 0.2278CSCO 0.8851 2.0819 2.6494 0.8562 0.5472 0.5526 0.4801 0.2242DIS 3.3125 1.6859 0.8832 0.3762 0.2222 0.156 0.4095 0.4265GS 2.1306 1.7729 4.4156 3.033 1.66 1.24 0.7853 1.212HD 5.1744 4.73 1.0941 0.8118 0.3607 0.2811 0.3797 0.1402IBM 1.2355 0.5482 0.7554 0.5961 0.8786 1.2554 1.3665 1.5904JNJ 3.6373 2.054 1.8562 0.415 0.2547 0.2969 0.2659 0.2117JPM 3.1696 2.2257 2.9556 1.6542 1.8757 1.2114 0.4552 0.2777MCD 1.2216 0.6152 0.8064 0.6305 0.6084 1.3799 0.9683 0.2409PFE 2.2225 1.8951 1.2921 0.8726 0.5384 0.375 0.4962 0.7079PG 2.5674 2.2259 1.7419 0.9365 0.5376 0.3889 0.5218 0.5266UNH 14.5879 4.0245 5.1527 1.2451 0.5163 0.4409 0.2286 0.0741VZ 4.7781 5.1793 1.723 1.6505 1.0168 1.4157 1.2974 1.3782WMT 2.201 2.0996 0.7268 0.8806 0.3209 0.3327 1.3503 0.3602
Table 7: The calibrated g without a signal.47 eferences [1] Y. Amihud, H. Mendelson, and L.H. Pedersen, ”Liquidity and Asset Prices”, Foundationsand Trends in Finance , 2005, vol. 1, no. 4, pp. 269-364.[2] 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).[3] H. Bateman and A. Erd´elyi,
Higher Transcendental Functions: Volume 2 , McGrau-HillBook Company (1953).[4] F. Black and M. Scholes, ”The Pricing of Options and Corporate Liabilities”, Journal ofPolitical Economy, Vol. 81(3), 637-654, 1973.[5] J.P. Bouchaud and R. Cont, ”A Langevin Approach To Stock Market”,
The EuropeanPhysical Journal B , (4), 543-550 (1998).[6] J.P. Bouchaud and M. Potters, Theory of Financial Risk and Derivative Pricing , secondedition, Cambridge University Press (2004).[7] M. Bernstein and L.S. Brown, ”Supersymmetry and the Bistable Fokker-Planck Equation”,
Physical Review Letters , (22), 1933-1935 (1984).[8] S. Boyd, E, Busetti, S. Diamond, R.N. Kahn, K. Koh, P. Nystrup, and J. Speth, ”Multi-Period Trading via Convex Optimization”, Foundations and Trends in Optimization . Vol.XX, no. XX, 1-74 (2017).[9] C.G. Callan and S. Coleman, ”Fate of the False Vacuum.II. First Quantum Corrections”,
Phys. Rev. D , (6), 1762-1768 (1977).[10] S. Coleman, Aspects of Symmetry. Selected Erice Lectures , Cambridge University Press(1988).[11] J. Dash,
Quantitative Finance and Risk Management: a Physicist’s Approach , World Sci-entific, (2004).[12] A. Dixit and R. Pindyck,
Investment Under Uncertainty , Princeton University Press,Princeton NJ (1994).[13] D. Duffie, ”Black, Scholes and Merton - Their Central Contributions to Economics” (1997).[14] D. Duffie and K.J. Singleton,
Credit Risk , Princeton Series in Finance (2003).[15] F.J. Dyson, ”Divergence of Perturbation Theory in Quantum Electrodynamics”,
PhysicalReview , , 631 (1952).[16] C.O. Ewald and Z. Yang, ”Geometric Mean Reversion: Formulas for the Equilibrium Den-sity and Analytic Moment Matching”, (2007).[17] H. Ezawa and J.R. Klauder, ”Fermions without Fermions”, Progress of Theoretical Physics , (4), 904-915 (1985). 4818] Eugene F. Fama and Kenneth R. French. Size, value, and momentum in international stockreturns. Journal of Financial Economics , 105(3):457–472, 2012.[19] M.V. Feigelman and A.M. Tsvelik, ”Hidden Supersymmetry of Stochastic Dissipative Dy-namics”, Sov. Phys. JEPT, (4), 823-830 (1982).[20] A. Gangopadhyaya, P. Panigrahi, and U. Sukhatme, ”Supersymmetry and Tunneling in anAsymmetric Double Well”, Physical Review A , (4), 2720-2724 (1993).[21] Gardiner, Handbook of Stochastic Methods (1996).[22] E. Gozzi, ”Functional-Integral Approach to Parisi-Wu Stochastic Quantization: Scalar The-ory”,
Phys. Rev. D , (8), 1922-1929 (1983).[23] R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals , Dover (2010).[24] I.S. Gradshtein and I.M. Ryzhik,
Table of Integrals, Series, and Products , Fifth Edition,Academic Press (1994).[25] W. Horsthemke and R. Lefever,
Noise-Induced Transitions: Theory and Applications inPhysics, Chemistry and Biology , Springer (1984).[26] I. Halperin and I. Feldshteyn, ”Market Self-Learning of Signals, Im-pact and Optimal Trading: Invisible Hand Inference with Free Energy.(or, How We Learned to Stop Worrying and Love Bounded Rationality)”,https://papers.ssrn.com/sol3/papers.cfm?abstract id=3174498.[27] P. Hanggi, ”Escape from a Metastable State”,
Journal of Statistical Physics , (1/2)105-148 (1986).[28] S. L. Heston. A closed-form solution for options with stochastic volatility with applicationsto bond and currency options. Review of Financial Studies , 6:327–343, 1993.[29] H. Hinrichsen, ” Nonequilibrium Critical Phenomena and Phase Transitions into AbsorbingStates”,
Advances in Physics , (7) (2000).[30] C. Itzykson and J.M. Druffe, Statistical Field Theory. Volume I: From Brownian Motionto Renormalization and Lattice Gauge Theory , Cambridge University Press (1989).[31] G. Junker,
Supersymmetric Methods in Quantum and Statistical Physics , Springer 1996.[32] W.Y. Keung, E. Kovacs, and U. Sukhatme, ”Supersymmetry and Double Well Potentials”,
Phys. Rev. Lett. , , p. 41 (1988).[33] L.D. Landau and E.M. Lifschitz, Mechanics , Elsevier (1980).[34] L.D. Landau and E.M. Lifschitz,
Quantum Mechanics , Elsevier (1980).[35] L.D. Landau and E.M. Lifschitz,
Statistical Physics , Elsevier (1980).[36] P. Langevin, ”Sur la Th´eorie du Mouvement Brownien”,
Comps Rendus Acad. Sci. (Paris)146, 530-533 (1908).[37] A.V. Lopatin and L.B. Ioffe, ”Instantons in the Langevin Dynamics: an Application to SpinGlasses”,
Phys. Rev. B (9), 6412 (1999).4938] F. Marchesoni, P. Sodano, and M. Zannetti, ”Supersymmetry and Bistable Soft Potentials”, Physical Review Letters , (10), 1143-1146 (1988).[39] J. Mathews and R.L. Walker, Mathematical Methods of Physics , Second Edition, Addison-Wesley Publishing (1969). 1964.[40] R. Merton, ”On the Pricing of Corporate Debt: the Risk Structure of Interest Rates”,
Journal of Finance , , 449-470 (1974).[41] R. Merton, ”Theory of Rational Option Pricing”, Bell Journal of Economics and Manage-ment Science, Vol.4(1), 141-183, 1974.[42] R. Merton, ”An Asymptotic Theory of Growth Under Uncertainty”, Review of EconomicStudies , (3), 375-393 (1975).[43] G. Mil’nikov and H. Nakamura, ”Tunneling Splitting and Decay of Metastable States inPolyatomic Molecules: Invariant Instanton Theory”, Phys. Chem. Chem. Phys. , 1374-1393 (2008).[44] H. Nakamura and G. Mil’nikov, Quantum Mechanical Tunneling in Chemical Physics , CRCPress (2013).[45] M. Morano,
Instantons and Large N : An Introduction to Non-Perturbative Methods inQFT , Cambridge University Press (2015).[46] M.A. Munoz, ”Nature of Different Types of Absorbing States”, Physical Review E , (2),1377 (1998).[47] H. Risken, The Fokker-Plank Equation , Springer (1989).[48] P. Samuelson, ”Rational theory of warrant pricing”,
Industrial Management Review , (Spring), 13-32 (1965).[49] W.F. Sharpe, ”Capital asset prices: A theory of market equilibrium under conditions ofrisk”, Journal of Finance , (3), 425?442 (1964).[50] B. Schmittmann and R.K.P. Zia, Statistical Mechanics of Driven Diffusiive Systems: Vol17: Phase Transitions and Critical Phenomena , Ed. C. Domb and J.L. Lebowitz, AcademicPress (1995).[51] D. Sornette, ”Stock Market Speculations: Spontaneous Symmetry Breaking of EconomicValuation”,
Physica A , (1-4), 355-375 (2000).[52] D. Sornette, Why Stock Markets Crash , Princeton University Press (2003).[53] C. Van den Broeck, J.M.R. Parrondo, R. Toral, and R. Kawai, ”Nonequilibrium PhaseTransitions Induced by Multiplicative Noise”,
Phys. Rev. E , (4), 4084-4094 (1997).[54] N.G. Van Kampen, Stochastic Processes in Physics and Chemistry , North-Holland (1981).[55] E. Weinberg,
Classical Solutions in Quantum Field Theory: Solitons and Instantons inHigh Energy Physics , Cambridge University Press (2012).[56] E. Witten, ”Dynamical Breaking of Supersymmetry”,
Nuclear Physics B (3-5), 513-554(1981). 5057] C. Zinn-Justin,