Newton-Raphson, sometimes just Newton or Newton-Fourier, method is an approximate method in mathematical analysis for finding local roots of very complex functions (such as polynomials with large powers). Recall that root of the function is defined as a solution of \( f(z) = 0 \). The essence of this method is to linearize function at the guessing point. The point where linearized function passes the abscissa axis is assumed to be a more precise estimate of the actual root. Mathematically:

Previously we have discussed ant colony model [1, 2] (see Kirman's agent based and stochastic model of ant colony), which is an interesting example of applying knowledge obtained from one field to another. Human (ex., trader in the financial markets) crowd behavior is ideologically quite similar to the behavior in ant colonies, thus the success and relevancy of the aforementioned model were to be expected. Though the key to success lies in the description of large number of entities.

Interestingly enough one can also create, and thus provide additional backing for the argument above, a successful model for human crowd behavior using classical models of statistical physics as an inspiration. In this text we will discuss agent-based spin model of the financial markets proposed by Bornholdt [3, 4], which is based on widely known Ising model. Despite the fact that Ising model models inanimate system, natural interactions are introduced by assuming two different types of interactions between the agents - local herding (local feromagnetic interaction) and global minority game (coupling with total magnetic field generated by whole lattice).

Bornholdt's model is also interesting as recently there were some attempts to propose macro treatment of the original agent-based model [5]. Previously similar thing was done with Kirman's model [2].

From the practical point of view price is the most interesting observable of the financial markets. Though modeling and analysis of price fluctuations are hindered by the fact that price itself is non-stationary process - mean price and market volatility constantly change. While price changes, at least at short time scales, behave as stationary process - mean price change is equal, or at least approximately equal, to zero. Thus it is convenient to introduce variable related to the relative price changes, which is known as return

Ising model is a generalized mathematical model of feromagnetism in statistical physics. In this model particles having magnetic spin are put inside vertices of graph. In general case structure of graph can vary a lot, but in the usual case selection is limited to the lattices of different dimensions. Behavior of such system is observed at different temperature in the quest to find critical temperature at which phase shift occurs. If you want to familiarize yourself more with Ising model and its various interpretations, you should read [1, 2] works, because in this text we will only consider one possible, numerical, algorithm - heat bath algorithm.

Previously on Physics of Risk website we have presented Kirman's ant colony agent-based model [1], where each ant was represented as an agent. In this article we will move from the agent-based model framework to the stochastic differential equation framework. Thus showing that in case of simple agent-based models full transition to stochastic framework is possible. This transition is very important as stochastic framework is very popular and well developed in quantitative finance. The problem is that stochastic framework mainly gives only a macroscopic insight into the modeled system, while microscopic behavior currently is also of big interest.