Crofton Risk and Relative Transactional Entropy

We develop a geometric approach to financial risk based on Crofton's idea and the tools of the Radon transform. The trajectory of a financial instrument is defined with respect to a frame of reference (money, benchmark). A central role is played by simple instruments, inspired by the annual percentage rate (APR) concept, whose graphs in a fixed reference frame are line segments. Risk is interpreted transactionally as the density of exchange dilemmas that arise when the instrument's trajectory intersects the trajectories of simple instruments. This perspective leads to a risk measure given by the trajectory length in the Crofton-Steinhaus sense. We also introduce new notions, such as geometric volatility, transactional entropy, and trajectory temperature, associated with the distribution of the number of intersections, enabling thermodynamic analogies to be incorporated into the description of risk and market complexity.