arXiv 2209.042642022-09-09q-fin.PRmath.PRq-fin.MF

概率的生物几何表示及其在期权定价中的应用

Biology-inspired geometric representation of probability and applications to completion and options' pricing

Felix Polyakov

备注：: 31 pages, 19 figures

摘要：

几何构成了存在于所有人类中的一组核心直觉，无论他们的语言或学校[1]。大脑内建的几何信息处理机制能否参与不确定性表征？几十年来，交易员一直在引用基于不确定性的外汇期权合约的价格，而不是用货币单位定价，这是一个与标准差相关的虚拟变量，即隐含波动率。这项工作介绍了一种方法，几何表示的概率方面的隐含波动率，并试图找到方法来近似某些概率分布使用直观的几何对称性。特别是，它示出了任何概率分布支持$\mathbb{R}_{+}$和具有有限的期望可以表示与一个平面曲线，其几何特征可以进一步分析。对数正态分布用以原点为中心的圆表示。某些具有钟形密度分布的非对数正态分布由曲线表示，这些曲线可以用圆心远离原点的圆近似。定义一个圆只需要三个点，而它代表了沿着整个$\mathbb{R}_{+}$近似分布的候选概率密度。只有三个数字：沿$x$和$y$轴的缩放和平移将一个圆映射到另一个圆。可以引入等价类，其成员分布可以通过几何变换在任何相应表示上的传递作用来获得。近似完成的概率与非圆形形状和情况下，概率支持外$\mathbb{R}_{+}$也被认为是。建议完成隐含波动率进行比较的瓦纳-伏尔加法。

英文摘要：

Geometry constitutes a core set of intuitions present in all humans, regardless of their language or schooling [1]. Could brain's built in machinery for processing geometric information take part in uncertainty representation? For decades already traders have been citing the price of uncertainty based FX optional contracts in terms of implied volatility, a dummy variable related to the standard deviation, instead of pricing with units of money. This work introduces a methodology for geometric representation of probability in terms of implied volatility and attempts to find ways to approximate certain probability distributions using intuitive geometric symmetry. In particular, it is shown how any probability distribution supported on $\mathbb{R}_{+}$ and having finite expectation may be represented with a planar curve whose geometric characteristics can be further analyzed. Log-normal distributions are represented with circles centered at the origin. Certain non-log-normal distributions with bell-shaped density profiles are represented by curves that can be closely approximated with circles whose centers are translated away from the origin. Only three points are needed to define a circle while it represents the candidate probability density approximating the distribution along the entire $\mathbb{R}_{+}$. Just three numbers: scaling and translations along the $x$ and $y$ axes map one circle to another. It is possible to introduce equivalence classes whose member distributions can be obtained by transitive actions of geometric transformations on any of corresponding representations. Approximate completion of probability with non-circular shapes and cases when probability is supported outside of $\mathbb{R}_{+}$ are considered too. Proposed completion of implied volatility is compared to the vanna-volga method.

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。

摘要：

英文摘要：

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