期望与价格的相对熵
The Relative Entropy of Expectation and Price
AI总结：
本文提出一种不完备市场中的衍生品定价原则，用对数鞅条件替代完备市场中的鞅条件，通过最小相对熵价格测度与投资者风险厌恶参数来量化市场不完备性和资本耗尽导致的违约风险，并应用于模型风险分析、深度对冲和量子信息。
摘要：
作为作用于衍生证券未定最终结算的算子，期望是线性的，但价格是非线性的。当标的证券市场不完备时，非线性源于中间价周围的买卖价差，该价差反映了最优融资与对冲策略的残余风险。在极端情形下，非线性还来自资本上嵌入的期权，这些期权在违约时被执行。本文在一个熵风险度量中对这些凸性进行量化，该度量评估策略性风险，并通过引入双边保证金实现为一种成本。随后，价格针对市场不完备性以及因资本耗尽而导致的违约风险进行调整。在完备市场理论中，价格由鞅条件导出。在本文所提出的不完备市场理论中，价格改由对数鞅条件导出：\begin{equation} p=-\frac{1}{α}\log\mathbb{E}\exp[-αP] \notag \end{equation}，其中 $p$ 和 $P$ 分别为已融资并对冲的衍生证券的价格与收益，价格测度 $\mathbb{E}$ 相对于经济期望具有最小熵，参数 $α$ 与投资者的风险厌恶程度相匹配。这一价格原则易于应用于市场演化的标准模型，本文考虑其在模型风险分析、深度对冲和量子信息中的应用。
英文摘要：
As operators acting on the undetermined final settlement of a derivative security, expectation is linear but price is non-linear. When the market of underlying securities is incomplete, non-linearity emerges from the bid-offer around the mid price that accounts for the residual risks of the optimal funding and hedging strategy. At the extremes, non-linearity also arises from the embedded options on capital that are exercised upon default. In this essay, these convexities are quantified in an entropic risk metric that evaluates the strategic risks, which is realised as a cost with the introduction of bilateral margin. Price is then adjusted for market incompleteness and the risk of default caused by the exhaustion of capital. In the complete market theory, price is derived from a martingale condition. In the incomplete market theory presented here, price is instead derived from a log-martingale condition: \begin{equation} p=-\frac{1}α\log\mathbb{E}\exp[-αP] \notag \end{equation} for the price $p$ and payoff $P$ of a funded and hedged derivative security, where the price measure $\mathbb{E}$ has minimum entropy relative to economic expectations, and the parameter $α$ matches the risk aversion of the investor. This price principle is easily applied to standard models for market evolution, with applications considered here in model risk analysis, deep hedging and quantum information.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
