Kirman's ant colony model, previously presented on our website as agent based (based on [1]) and stochastic (based on [2, 3]) model, has become classical example of herding modeling. Application of this model towards economic, financial or other social scenarios might seem doubtful as human society is far more complex than ant colony, but methodologically it is more useful to start from very simple and stylized model and later add complexity on top of it. Furthermore we have already shown that Kirman's herding dynamics could be applicable in agent-based marketing (see comparison of Kirman's and Bass diffusion model). In this text we will consider financial market scenario and obtain stochastic differential equations similar to the existing stochastic models considered in [4, 5].

Working on Physics of Risk is very interesting and useful experience. This experience provides valuable insights into the mechanics behind various complex systems, well modeled by macroscopic models. Using our experience we are able to obtain qualitative and quantitative agreements between varying models. In our newest publication [1] we have used one-step formalism [2] to obtain macroscopic treatments of Kirman model [3].

Bass diffusion model [1] is widely known and very important model in marketing science. This model predicts diffusion, sales, of new successful products inside the market. While previously discussed model, Kirman model [2], has wide range of possible applications - from biology to finance. In [2] it is noted that behavior observed by entomologists is similar to one observed in economic scenarios (ex. popularity of books and restaurants). Thus one might expect that Kirman's model might be modified to work in the marketing scenario.

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].