Igor Halperin, VP, AI Asset Management, Fidelity Investments
Financial markets are highly complex systems whose dynamics are driven by interactions of a large (or very large) number of market participants that pursue different objectives, as well as operate at different time scales. Furthermore, they are examples of open, rather than closed, systems. In particular, money is not a conserved quantity in finance because of new cash pumped into the market by retail investors, mostly via investment into pension funds or individual brokerage accounts.
As recent Robinhood meme stocks stories demonstrated, when ejected in large volumes over a short period of time, new cash inflows can produce very strong (and nonlinear) distortions of equity prices away from their fundamental values. Strong effects of new cash infusions as an origin of stock price fluctuations were proposed in a recent influential paper by X. Gabaix and R.S.J. Koijen [1], and links of such price impact mechanics with the market microstructure dynamics were proposed by J.F. Bouchaud [2] (see also the related Bloomberg story [3]).
This article deals with further effects caused by inflows of new money in the market. More specifically, I will argue that they result in non-linear market dynamics.
Non-linearity of dynamics implies, among other things, that small changes of the state of the system, due to either internal fluctuations or external control, can produce large changes in the state of the system. The meme stock craze can be viewed as an indication of an important role of non-linearity of market dynamics ensuing in the regime of large cash inflows. These events clearly demonstrated that the market often behaves in a highly non-equilibrium and non-linear regime.
On the other hand, most models employed by practitioners, such as factor models (CAMP, APT, etc.) for portfolio management, or derivatives pricing models that grew out of the celebrated Geometric Brownian Motion (GBM) model of Samuelson, are linear models. For example, the GBM model postulates that both the drift and diffusion functions of the price process are linear functions of the asset price. It can be considered as a particular specification of a stochastic process called Ito’s diffusion, with linear drift and diffusion functions. Its later extensions such as local or stochastic volatility, jump-diffusion, Levy models etc. enrich the ensuing price dynamics by considering progressively more complex models of the diffusion term, for example by making it dependent on additional risk factors, but they do not modify the linear state dependence of a drift term. Equivalently, a linear drift term for prices can be expressed as a constant drift for log-returns.
Of course, both the academics and practitioners are well aware that non-linear effects often arise in various financial settings, e.g. in models of market impact and transaction costs, as well as in certain models of incomplete markets and XVA models. On the other hand, it is also commonly believed that non-linear effects can usually be tackled using the means of perturbation theory methods, where nonlinear terms are considered as small perturbations around a linear model. The latter linear model (such as the GBM model) would then serve as a ‘zero-order’ approximation to the actual dynamics. Effects due to financial frictions and/or new money inflows would then be treated as corrections that could be obtained using methods of perturbation theory.
However, such an approach has a problem, as becomes apparent if we analyze the GBM model (or its later extensions) using insights provided by physics. More specifically, notice that the GBM model and its extensions can be interpreted as the so-called overdamped Langevin equations known in physics since the 1908 work of the French physicist Paul Langevin. Langevin extended the free Brownian motion theory of Einstein to the case of Brownian particles experiencing diffusion in an external potential field (which can be caused by other heavy molecules, electric or magnetic field, etc.). One of the simplest specifications of a potential in physics is a quadratic function of the particle position, which is known as a harmonic oscillator potential. Non-linear interactions in physical systems amenable to modeling within the Langevin approach typically give rise to higher order non-linearities, producing cubic or quartic potentials, or even non-polynomial potentials. Potentials of this sort commonly appear in many important problems in statistical and quantum physics. In particular, processes of a thermally-induced or a quantum mechanical escape (tunneling) from a potential well are often described using a quartic potential with two minima, called a double well potential in physics.
Back to finance, the Langevin dynamics of a particle in a potential is mathematically equivalent to Ito’s diffusion, but it gives a physics-provided interpretation to the drift term. More specifically, the drift term in the mathematical construction of Ito’s diffusion is interpreted as a negative gradient of the Langevin potential function. While this simple observation is based on comparison of two very famous equations, apparently it did not attract the due attention of the wider mathematical finance community. In the meantime, it produces a critically important observation, namely that the linear drift term of the classical GBM model and its offspring is equivalent to an inverted harmonic potential, or equivalently a harmonic potential with a negative mass!
The reason I believe that this observation is critically important is that an inverted harmonic potential describes a globally unstable system, whose (unstable) dynamics continue indefinitely. On the other hand, no systems in physics ever produce such globally unstable dynamics. Unstable systems in physics exist only for relatively short times, and arise in some models of the early universe, or in models of lasers, for example. On longer time horizons, non-linear effects in physical systems stabilize dynamics around some globally stable or metastable (i.e. very long-lived) states corresponding, respectively, to global or local minima of the potential.
What happens if we follow the logic of perturbation theory methods, and try to treat all possible non-linear effects in the price dynamics as small perturbations around the linear ‘GBM limit’ corresponding to a linear drift, or equivalently an inverted harmonic potential?
The problem is that such a ‘zero-order’ limit produces highly problematic behaviour in both the long time and small price limits. In the long-time limit, it implies an unlimited exponential (or average) growth of the stock price, while in the small price limit it implies a presence of a totally fictitious and counter-intuitive force that somehow ‘saves’ the firm from attaining the zero price – thus preventing the possibility to model corporate defaults within the same diffusion-based approach. While such linear dynamics can approximate the true non-linear dynamics over short time scales, the very range of time scales where such approximations are reasonably accurate should come from the underlying non-linear model itself. The main problem of linear models such as the GBM model is that when viewed as ‘fundamental’ models, they give no indication of their suggested range of applicability. In a way, this is analogous to how the equations of Newtonian mechanics give no indication that they have to be replaced by equations of quantum mechanics for very small distances, or equations of the special relativity theory for very high velocities.
The undesirable behaviour at very small or very large prices could be cured if the stock drift corresponded to a non-linear Langevin potential with global and local minima. But what physical mechanism could produce such non-linear potentials? A possible solution was proposed in Halperin & Dixon 2020 [4] (see also [5] for a non-technical presentation) as a composition of two effects. First, a new cash infusion produces a price impact in the spirit of the ‘dumb money’ effect of Frazzini and Lamont [6], and second, the amount of this new cash is itself driven by the current stock price (or return), among other driving factors. Together, this produces the Langevin potential for the stock price as a quartic polynomial in price. In [7], I proposed a similar but more tractable approach in the log-price space, with a non-linear potential that can smoothly vary from a single-well to a double well potential, and showed how parameters of this potential can be calibrated to available option quotes.
This produces a model that is capable of producing both small and large fluctuations of returns. In particular, in certain market regimes the resulting potential can be of a double well form. For such scenarios, the critical role in dynamics of large fluctuations is played by non-perturbative solutions which are named so because they cannot be established using a perturbation theory around a zero-friction limit. Such non-perturbative solutions arise in many problems in quantum and statistical physics, where they are known as instantons. In particular, instantons arising in statistical physics models are responsible for non-equilibrium dynamics of escape from a local potential minimum towards a global minimum. While non-perturbative solutions such as instantons are frequently encountered in physics, the framework developed in [4,5,7] based on the analysis of money flows and price impact suggests that similar mechanisms may be theoretically important in finance, and lead to tractable non-linear models that are suitable for practitioners. Extending this approach to multi-asset market models is a logical next step in this program, which is left here for future research.
