Continuous trading models have been the central tools in modern financial theory ever since they were used to establish the Black-Scholes formula for correctly pricing options. Unfortunately, the continuous trading models rest on economic foundations that can be questioned. The object of this project is to put continuous trading models on a sounder economic footing. The most fundamental difficulty addressed is the meaning of continuous trading. The `obvious` interpretation of continuous trading is as a limit of models in which trading occurs at a finite number of discrete times, but the interval between trading times becomes very short. It can be argued on the basis of counter examples, that there is a serious economic difficulty with this interpretation. The aim of this research is to seek a more satisfactory interpretation. The second set of difficulties addressed concerns the nature and existence of equilibrium. Most of the analysis of continuous trading models is non-equilibrium in nature; for example, the Black-Scholes analysis takes the process describing security prices as given and solves for option prices by arbitrage arguments. Other analyses simply assume the existence of a suitable equilibrium. From an economic point of view, assuming the existence of such an equilibrium is generally unsatisfactory. It is particularly unsatisfactory in the continuous time setting because the existence of equilibrium is very much in doubt in many cases of interest, and because even when the existence of an equilibrium satisfying regularity conditions is guaranteed, there may be many other equilibria. This research will seek to establish the existence of equilibrium and to rule out equilibria not satisfying regularity conditions.
