Representation of Probability Distributions With Implied Volatility and Biological Rationale

F. Polyakov
{"title":"Representation of Probability Distributions With Implied Volatility and Biological Rationale","authors":"F. Polyakov","doi":"10.2139/ssrn.3213650","DOIUrl":null,"url":null,"abstract":"Economic and financial theories and practice essentially deal with uncertain future. Humans encounter uncertainty in different kinds of activity, from sensory-motor control to dynamics in financial markets, what has been subject of extensive studies. Representation of uncertainty with normal or lognormal distribution is a common feature of many of those studies. For example, proposed Bayessian integration of Gaussian multisensory input in the brain or log-normal distribution of future asset price in renowned Black-Scholes-Merton (BSM) model for pricing contingent claims.<br><br>Standard deviation of log(future asset price) scaled by square root of time in the BSM model is called implied volatility. Actually, log(future asset price) is not normally distributed and traders account for that to avoid losses. Nevertheless the BSM formula derived under the assumption of constant volatility remains a major uniform framework for pricing options in financial markets. I propose that one of the reasons for such a high popularity of the BSM formula could be its ability to translate uncertainty measured with implied volatility into price in a way that is compatible with human intuition for measuring uncertainty.<br><br>The present study deals with mathematical relationship between uncertainty and the BSM implied volatility. Examples for a number of common probability distributions are presented. Overall, this work proposes that representation of various probability distributions in terms of the BSM implied volatility profile may be meaningful in both biological and financial worlds. Necessary background from financial mathematics is provided in the text.","PeriodicalId":269529,"journal":{"name":"Swiss Finance Institute Research Paper Series","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2018-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Swiss Finance Institute Research Paper Series","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2139/ssrn.3213650","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2

Abstract

Economic and financial theories and practice essentially deal with uncertain future. Humans encounter uncertainty in different kinds of activity, from sensory-motor control to dynamics in financial markets, what has been subject of extensive studies. Representation of uncertainty with normal or lognormal distribution is a common feature of many of those studies. For example, proposed Bayessian integration of Gaussian multisensory input in the brain or log-normal distribution of future asset price in renowned Black-Scholes-Merton (BSM) model for pricing contingent claims.

Standard deviation of log(future asset price) scaled by square root of time in the BSM model is called implied volatility. Actually, log(future asset price) is not normally distributed and traders account for that to avoid losses. Nevertheless the BSM formula derived under the assumption of constant volatility remains a major uniform framework for pricing options in financial markets. I propose that one of the reasons for such a high popularity of the BSM formula could be its ability to translate uncertainty measured with implied volatility into price in a way that is compatible with human intuition for measuring uncertainty.

The present study deals with mathematical relationship between uncertainty and the BSM implied volatility. Examples for a number of common probability distributions are presented. Overall, this work proposes that representation of various probability distributions in terms of the BSM implied volatility profile may be meaningful in both biological and financial worlds. Necessary background from financial mathematics is provided in the text.
用隐含波动率表示概率分布和生物学原理
经济金融理论和实践本质上是处理不确定的未来。人类在各种各样的活动中都会遇到不确定性,从感觉-运动控制到金融市场的动态,这些都是广泛研究的主题。用正态或对数正态分布表示不确定性是许多研究的共同特征。例如,提出了大脑中高斯多感官输入的贝叶斯积分,或著名的Black-Scholes-Merton (BSM)模型中未来资产价格的对数正态分布,用于对或有债权进行定价。在BSM模型中,对数(未来资产价格)按时间的平方根进行缩放的标准差称为隐含波动率。实际上,log(未来资产价格)不是正态分布的,交易者考虑这一点是为了避免损失。然而,在恒定波动假设下导出的BSM公式仍然是金融市场期权定价的主要统一框架。我认为,BSM公式如此受欢迎的原因之一可能是它能够将隐含波动率测量的不确定性转化为价格,这种方式与人类测量不确定性的直觉相兼容。本文研究了不确定性与BSM隐含波动率之间的数学关系。给出了一些常见概率分布的例子。总的来说,这项工作提出,根据BSM隐含波动率剖面的各种概率分布的表示可能在生物和金融领域都有意义。本文从金融数学的角度提供了必要的背景。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信