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What are nonlinear characteristics? Phases of MANES: Multi-Asset Non-Equilibrium Skew Model of a Strongly Non-Linear Market with Phase Transitions

What are nonlinear characteristics? Phases of MANES: Multi-Asset Non-Equilibrium Skew Model of a Strongly Non-Linear Market with Phase Transitions


Keywords: Non-equilibrium market dynamics, Langevin dynamics, mean field approximation, statistical mechanics, McKean-Vlasov equation

Let’s take a look at Dr. Igor Halperin’s work!

Part 2 Nonlinear stochastic dynamics of market returns

The paper first introduced a potential function that models ensembles of interacting stocks with individual log-returns.

This function can become decomposed to a self-interaction potential for an individual stock and a pairwise interaction potential, and both of them are explained in terms of the correlations between log-returns of individual stocks originate from the dependence on money flow into a certain stock on the average previous performance of all other stocks. By applying the Langevin equation to a homogeneous setting, the interacting and self-interacting oscillators are non-linear (quadratic and quadric polynomials respectively). 

Part 3 Markets as interacting nonlinear oscillators

Then, in order to approximate the return of market index in a heterogeneous market, the NES potential is introduced as a tractable approximation to a non-linear self-interaction potential. Along with the quadratic Curie-Weiss interaction potential, this results in the multi-asset NES model (MANES). Where the non-linear self-interaction potential is given by the logarithm of two-component Gaussian Mixture.

The Curie-Weiss interaction potential is a theoretical model used to describe the behavior of ferromagnetic materials. The model was developed by the physicists Pierre Curie and Pierre Weiss in the late 19th century.

Pierre Curie by Dujardin c1906.jpg
Pierre Curie

The Curie-Weiss model assumes that the magnetic moments of the individual atoms in a ferromagnetic material are aligned in a certain direction, creating a net magnetic field. This alignment is due to the interaction between the neighboring atoms, which results in a potential energy that can be described by the Curie-Weiss interaction potential.

The potential energy is given by the formula U = -C/(T – θ), where U is the potential energy, C is the Curie constant (a measure of the strength of the interaction), T is the temperature, and θ is the Curie temperature (the temperature above which the material loses its ferromagnetic properties).

At temperatures below the Curie temperature, the potential energy is negative.

Pierre and Marie Skłodowska-Curie, 1895

As a result, which means that the interaction between neighboring atoms is attractive. This leads to the alignment of the magnetic moments and the creation of a net magnetic field. Moreover, at temperatures above the Curie temperature, the potential energy becomes positive, which means that the interaction becomes repulsive and the material loses its ferromagnetic properties.

A Gaussian mixture is a statistical model that represents a probability distribution as a weighted sum of Gaussian (normal) distributions. The idea behind a Gaussian mixture becomes approximating a complex probability distribution with a simpler model that can become more easily analyzed and manipulated.

In a Gaussian mixture model, each Gaussian distribution becomes called a component. And the weights assigned to each component represent the probability of belonging to that component. The Gaussian components may have different means, variances, and weights, which allows for greater flexibility in modeling complex probability distributions.

Gaussian mixtures often find themselves used in machine learning and data analysis, particularly in unsupervised learning tasks such as clustering and density estimation. Clustering involves grouping data points together based on their similarity. And Gaussian mixture models can identify groups of data points that are most likely to belong to the same cluster. Density estimation involves estimating the probability density function of a set of data points. And Gaussian mixtures can model this density as a weighted sum of Gaussian distributions.

Gaussian mixture models have several advantages over other probability distribution models, including their ability to model complex data distributions with fewer parameters and their ability to capture multimodal data distributions. However, they can also be more difficult to estimate and optimize compared to simpler models, and the choice of the number of components can be a challenging problem.

The further processes mainly use the McKean-Vlasov equation.

Which is the non-linear extension of classical Fokker-Planck equation, for the mean-field dynamics, and then renormalize the parameters of the single-stock NES potential by interaction so that the dynamics of market return can be modeled as the dynamic of the fictitious single ‘representative’ stock. The McKean-Vlasov equation is a partial differential equation that describes the evolution of a probability distribution in a dynamical system with many interacting particles.

The McKean-Vlasov equation is unique in that it takes into account the effect of the particles on the distribution itself, through a self-consistency condition.

Mathematically, the McKean-Vlasov equation can become written as:

∂ρ/∂t + div(ρf) = div(ρD(ρ)gradρ)

where ρ(x,t) is the probability density function of the system at position x and time t, f(x,t) is the drift velocity of the particles, and D(ρ) is the diffusion coefficient. The self-consistency condition is expressed through the dependence of the diffusion coefficient on the probability density ρ.

The McKean-Vlasov equation arises in many fields of physics, including plasma physics, astrophysics, and statistical mechanics. It is also used in various applications in finance, such as modeling the behavior of stock prices and interest rates.

The equation is difficult to solve analytically, and numerical methods are often used to approximate the solution. Furthermore, there has been significant research on the properties and applications of the McKean-Vlasov equation, including its stability properties, its relation to game theory and optimization problems, and its use in machine learning and artificial intelligence.

Part 4 Phase transitions of MANES

In the next part, the MANES model also admits regimes of large fluctuations involving both first and second order phase transitions, due to the non-linearity of McKean-Vlasov equation. The self-consistency equation is derived from the NES potential within the McKean-Vlasov equation, and then the first-order phase transition of asymmetric potential and the second-order phase transition of symmetric potential are explored in different parameter regimes. Finally the critical exponents alpha and beta are computed.

Part 5 Fitting model parameters using option data

In the last, for practical use, the MANES model is calibrated to market quotes on options of market indexes, given the dynamics of the market index can be represented as a single-stock dynamics with renormalized parameters given in part 3. As a result, in order to obtain the equilibrium market log-return, a closed-form explicit solutions of the self-consistency condition in terms of renormalized parameters calibrating to option prices is derived.

Furthermore, the model also demonstrates that a single volatility parameter is sufficient enough to accurately match both benign and distressed market conditions, as the implied potentials derived from the parameters of calibrating the single-stock potentials to market index options can replace the implied volatility smile when fitting the market data.

Full paper:

Written by Shangxian Liu

What are nonlinear characteristics? Phases of MANES: Multi-Asset Non-Equilibrium Skew Model of a Strongly Non-Linear Market with Phase Transitions