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摘要翻译：
众所周知，Black-Scholes公式是在股票波动不变的假设下推导出来的。尽管有证据表明这个参数不是常数，但这个公式被金融市场广泛使用。本文讨论了是否存在Black-Scholes或类似公式所适用的股票价格模型的问题。由于没有对模型的动力学进行假设，无论是基础价格过程、波动过程，还是它们之间的关系，本文得到的结果都是非常一般的。我们证明，如果公式对于连续的罢工和三个终端时间成立，那么波动率一定是常数。然而，当波动率只在有限的几个罢工和三个或更多的到期日成立时，我们得到了波动率变化的一个普遍的界。当罢工序列增加到覆盖整个半线时，这一界限使隐含波动率保持不变。这可以通过不同的方法恢复连续罢工的结果。
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英文标题：
《Volatility in options formulae for general stochastic dynamics》
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作者：
Kais Hamza, Fima Klebaner and Olivia Mah
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最新提交年份：
2013
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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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英文摘要：
It is well-known that the Black-Scholes formula has been derived under the assumption of constant volatility in stocks. In spite of evidence that this parameter is not constant, this formula is widely used by financial markets. This paper addresses the question of whether an alternative model for stock price exists for which the Black-Scholes or similar formulae hold. The results obtained in this paper are very general as no assumptions are made on the dynamics of the model, whether it be the underlying price process, the volatility process or how they relate to each other. We show that if the formula holds for a continuum of strikes and three terminal times, then the volatility must be constant. However, when it only holds for finitely many strikes, and three or more maturity times, we obtain a universal bound on the variation of the volatility. This bound yields that the implied volatility is constant when the sequence of strikes increases to cover the entire half-line. This recovers the result for a continuum of strikes by a different approach.
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PDF链接：
https://arxiv.org/pdf/1306.0980
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