We propose a real time benchmark for bitcoin called QRTR whose calculation is much simpler than that of BRTI. Then, we introduce two types of reference rates that are usable for daily quotes of the currency, based on QRTR. The methodologies we propose in this paper are applicable not only for...

We introduce a category that represents varying risk as well as uncertainty, and give a generalized conditional expectation as a contravariant functor on the category. Then, we reformulate dynamic monetary value measures as a contravariant functor on the category. We show some axioms of dynamic...

We introduce a notion of market times that are stochastic processes in order to represent information delay in structural credit risk models. The market times are extensions of the time change process introduced by Guo, Jarrow and Zeng in the sense that each component of the market time is not...