摘要翻译：
我们考虑了一个线性市场影响模型，并在此框架中解决了未定权益的复制问题。我们导出了一个非线性的Black-Scholes方程，它提供了一个精确的复制策略。该方程是完全非线性的奇异方程，但我们证明了它是适定的，并证明了一大类最终收益的光滑解的存在性，包括常数波动性和局部波动性。为了获得解的正则性，我们提出了一种基于勒让德变换的新方法。讨论了Cheridito、Soner和Touzi等人研究的带有gamma约束的套期保值问题和带有流动性成本的套期保值问题之间的密切联系。我们还导出了适用于渐近小冲击参数的修正Black-Scholes公式，并给出了数值模拟的例子。
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英文标题：
《Option pricing with linear market impact and non-linear Black and
  Scholes equations》
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作者：
Gregoire Loeper
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最新提交年份：
2016
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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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一级分类：Mathematics        数学
二级分类：Analysis of PDEs        偏微分方程分析
分类描述：Existence and uniqueness, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDE's, conservation laws, qualitative dynamics
存在唯一性，边界条件，线性和非线性算子，稳定性，孤子理论，可积偏微分方程，守恒律，定性动力学
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英文摘要：
  We consider a model of linear market impact, and address the problem of replicating a contingent claim in this framework. We derive a non-linear Black-Scholes Equation that provides an exact replication strategy.   This equation is fully non-linear and singular, but we show that it is well posed, and we prove existence of smooth solutions for a large class of final payoffs, both for constant and local volatility. To obtain regularity of the solutions, we develop an original method based on Legendre transforms.   The close connections with the problem of hedging with it gamma constraints studied by Cheridito, Soner and Touzi and with the problem of hedging under it liquidity costs are discussed.   We also derive a modified Black-Scholes formula valid for asymptotically small impact parameter, and finally provide numerical simulations as an illustration.
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PDF链接：
https://arxiv.org/pdf/1301.6252