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We give a new way to price American options by using Samuelson’s formula. We first obtain the option price corresponding to a European option at time t, weighing it by the probability that the underlying asset takes the value S at time t. We then use Samuelson’s formula with this factor which is given by the solution of the Fokker-Planck (Kolmogorov) equation for the transition probability density. The main advantage of this approach is that we can systematically introduce the effect of macroeconomic factors. If a macroeconomic framework is given by a dynamical system in the form of a set of ordinary differential equations we only have to solve a partial differential equation for the transition probability density. In this context, we verify, for the sake of consistency, that this formula coincides with the Black-Scholes model and compare several numerical implementations.
The ultimate goal of environmental impact assessment is to guarantee
that benefits generated by a development project will not cause highly negative
effects on the environment or public health. The fulfillment of this goal
depends on the willingness of proponents and society to cooperate. The
information management, its accessibility to community and the educational
level of participants are of great relevancy too. Cooperation is not always
attainable due to conflicts between individual and community interests. Conflict
leads to a variety of cooperative and non-cooperative responses, depending on
the information available to the actors. In order to capture the tendency in
which a community perceives the proposals, we introduced an information index.
We prove that computer models have a direct impact on this information index.
This computer-based approach, leads the EIA to the paradigm of adaptive
environmental assessment and management. To implement this, a system based on
artificial intelligence and game theory was used to resolve a study case of
conflict in groundwater management.
A new analysis of a previously studied traveling agent model, showed that there is a relation between the degree of homogeneity of the medium where the agents move, agent motion patterns, and the noise generated from their displacements. We proved that for a particular value of homogeneity, the system self organizes in a state where the agents carry out Lévy walks and the displacement signal corresponds to 1/f noise. Using probabilistic arguments, we conjectured that 1/f noise is a fingerprint of a statistical phase transition, from randomness (disorder) to predictability (order), and that it emerges from the contextuality nature of the system which generates it.