| Popis: |
In mathematical finance, economies are often presented with the specification of a probability space equipped with a filtration that encodes information flow. The information-based framework of Brody, Hughston and Macrina (BHM) emphasises the role of market information in deriving asset price dynamics, instead of assuming price behaviour from the start. We extend the BHM framework by (i) modelling the nature of access to information through information blockages and activations of new information sources, and (ii) introducing a new class of multivariate Markov processes that we call Generalised Liouville Processes (GLPs) which can model the flow of information about vectors of assets. The analysis of access to information allows us to derive price dynamics with jumps. It additionally enables us to develop an information-switching framework, and price derivatives under regime-switching economies. We also indicate some geometrical aspects of appearances of new information sources. We represent information jumps on the unit sphere in the Hilbert space of square-integrable functions, and on hyperbolic spaces. We use differential geometry, information theory and what we call n-order piecewise enlargements of filtrations to dynamically quantify the impact of sudden changes in the sources of information. This helps us to model the stochastic evolution of what may be viewed as information asymmetry. In related work, we construct GLPs on finite time horizons by splitting so-called Levy random bridges into non-overlapping subprocesses. The terminal values of GLPs have generalised multivariate Liouville distributions, and GLPs can model a wide spectrum of information-driven dependence structures between assets. The law of an n-dimensional GLP under an equivalent measure is that of an n-vector of independent Levy processes. We focus on a special type of GLPs that we call Archimedean Survival Processes (ASPs). The terminal value of an ASP has an [Symbol appears here. To view, please open pdf attachment] 1-norm symmetric distribution, and hence, an Archimedean survival copula. |
