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摘要翻译：
在Neri和Schneider(2012)中，我们提出了一种方法来恢复由n个罢工集合上的看涨期权和数字期权的价格推断出的最大熵密度(MED)。为了找到MED，我们需要数值反演n个值的一维函数，并提出了Newton-Raphson方法。在本注记中，我们重新讨论了这个反演问题，并证明它可以用朗之万函数重写，对于朗之万函数，它的逆的数值逼近是已知的。这种方法与Buchen和Kelly(BK)的方法非常相似，不同之处在于BK只要求看涨期权价格。接着，在我们第一篇论文的基础上，我们提出了另一种方法，它只使用呼叫价格，并且恢复了与BK相同的密度，具有一些优点，特别是数值稳定性。第二篇文章详细分析了迭代算法的收敛性，特别是给出了迭代算法离解多远（在不同意义上）的各种估计。这些估计依赖于常数m>0。m越大，估计值就越好。在第二篇论文中提出了m的具体值，这篇注释提供了一个更尖锐的值。
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英文标题：
《A Note on "A Family of Maximum Entropy Densities Matching Call Option
Prices"》
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作者：
Cassio Neri, Lorenz Schneider
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最新提交年份：
2012
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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：
In Neri and Schneider (2012) we presented a method to recover the Maximum Entropy Density (MED) inferred from prices of call and digital options on a set of n strikes. To find the MED we need to numerically invert a one-dimensional function for n values and a Newton-Raphson method is suggested. In this note we revisit this inversion problem and show that it can be rewritten in terms of the Langevin function for which numerical approximations of its inverse are known. The approach is very similar to that of Buchen and Kelly (BK) with the difference that BK only requires call option prices. Then, in continuation of our first paper, we presented another approach which uses call prices only and recovers the same density as BK with a few advantages, notably, numerical stability. This second paper provides a detailed analysis of convergence and, in particular, gives various estimates of how far (in different senses) the iterative algorithm is from the solution. These estimates rely on a constant m > 0. The larger m is the better the estimates will be. A concrete value of m is suggested in the second paper, and this note provides a sharper value.
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PDF链接：
https://arxiv.org/pdf/1212.4279
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