Lindgren Jussi
Department of Mathematics and Systems Analysis, Aalto University, 02150 Espoo, Finland.
Entropy (Basel). 2020 Nov 12;22(11):1283. doi: 10.3390/e22111283.
This research article shows how the pricing of derivative securities can be seen from the context of stochastic optimal control theory and information theory. The financial market is seen as an information processing system, which optimizes an information functional. An optimization problem is constructed, for which the linearized Hamilton-Jacobi-Bellman equation is the Black-Scholes pricing equation for financial derivatives. The model suggests that one can define a reasonable Hamiltonian for the financial market, which results in an optimal transport equation for the market drift. It is shown that in such a framework, which supports Black-Scholes pricing, the market drift obeys a backwards Burgers equation and that the market reaches a thermodynamical equilibrium, which minimizes the free energy and maximizes entropy.
这篇研究文章展示了如何从随机最优控制理论和信息理论的背景来理解衍生证券的定价。金融市场被视为一个信息处理系统，它优化一个信息泛函。构建了一个优化问题，其线性化的哈密顿 - 雅可比 - 贝尔曼方程就是金融衍生品的布莱克 - 斯科尔斯定价方程。该模型表明，可以为金融市场定义一个合理的哈密顿量，这会导致市场漂移的最优传输方程。结果表明，在这样一个支持布莱克 - 斯科尔斯定价的框架中，市场漂移服从一个反向的伯格斯方程，并且市场达到一个热力学平衡，该平衡使自由能最小化并使熵最大化。
