Collaborative Research: Economics of Continuous Trading

自从连续交易模型被用来建立Black-Scholes公式，为期权正确定价以来，它一直是现代金融理论的核心工具。 不幸的是，连续交易模式建立在可以质疑的经济基础上。 该项目的目标是将连续交易模式置于更坚实的经济基础之上。 解决的最根本的困难是连续交易的含义。 对连续交易的“显而易见”的解释是作为一种模型的限制，在这种模型中，交易发生在有限数量的离散时间，但交易时间之间的间隔变得非常短。 根据反例，可以认为这种解释存在严重的经济困难。 本研究的目的是寻求一个更令人满意的解释。 第二类困难涉及均衡的性质和存在。 大多数连续交易模型的分析本质上是非均衡的;例如，布莱克-斯科尔斯分析将描述证券价格的过程视为给定的，并通过套利参数求解期权价格。 其他分析只是假设存在一个合适的均衡。 从经济学的观点来看，假设存在这样一种均衡通常是不能令人满意的。 在连续时间的情况下，它尤其不能令人满意，因为在许多有趣的情况下，均衡的存在性是非常值得怀疑的，而且即使保证了满足正则性条件的均衡的存在，也可能存在许多其他均衡。 本研究将寻求建立平衡的存在性，并排除不满足正则性条件的平衡。

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.