英文标题：
《Probability-free models in option pricing: statistically
  indistinguishable dynamics and historical vs implied volatility》
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作者：
Damiano Brigo
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最新提交年份：
2021
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英文摘要：
  We investigate whether it is possible to formulate option pricing and hedging models without using probability. We present a model that is consistent with two notions of volatility: a historical volatility consistent with statistical analysis, and an implied volatility consistent with options priced with the model. The latter will be also the quadratic variation of the model, a pathwise property. This first result, originally presented in Brigo and Mercurio (1998, 2000), is then connected with the recent work of Armstrong et al (2018, 2021), where using rough paths theory it is shown that implied volatility is associated with a purely pathwise lift of the stock dynamics involving no probability and no semimartingale theory in particular, leading to option models without probability. Finally, an intermediate result by Bender et al. (2008) is recalled. Using semimartingale theory, Bender et al. showed that one could obtain option prices based only on the semimartingale quadratic variation of the model, a pathwise property, and highlighted the difference between historical and implied volatility. All three works confirm the idea that while historical volatility is a statistical quantity, implied volatility is a pathwise one. This leads to a 20 years mini-anniversary of pathwise pricing through 1998, 2008 and 2018, which is rather fitting for a talk presented at the conference for the 45 years of the Black, Scholes and Merton option pricing paradigm.
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中文摘要：
我们研究是否有可能在不使用概率的情况下建立期权定价和套期保值模型。我们提出了一个与波动率的两个概念一致的模型：与统计分析一致的历史波动率和与模型定价的期权一致的隐含波动率。后者也是模型的二次变化，是一种路径特性。这第一个结果最初发表在Brigo和Mercurio（1998，2000）中，然后与Armstrong等人（2018，2021）最近的工作相联系，其中使用粗糙路径理论表明，隐含波动率与股票动力学的纯粹路径提升相关联，尤其是不涉及概率和半鞅理论，导致无概率的期权模型。最后，回顾了Bender等人（2008）的一个中间结果。Bender等人利用半鞅理论表明，仅基于模型的半鞅二次变化（一种路径性质）就可以获得期权价格，并强调了历史波动率和隐含波动率之间的差异。这三部著作都证实了一个观点，即历史波动率是一个统计量，而隐含波动率是一个路径量。这导致pathwise定价在1998年、2008年和2018年迎来了20年的迷你周年纪念日，这非常适合在Black、Scholes和Merton期权定价范式45年会议上发表的演讲。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Pricing of Securities        证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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