英文标题：
《Lie Symmetry Analysis of the Black-Scholes-Merton Model for European
Options with Stochastic Volatility》
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作者：
A. Paliathanasis, K. Krishnakumar, K.M. Tamizhmani and P.G.L. Leach
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最新提交年份：
2016
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英文摘要：
We perform a classification of the Lie point symmetries for the Black--Scholes--Merton Model for European options with stochastic volatility, $\\sigma$, in which the last is defined by a stochastic differential equation with an Orstein--Uhlenbeck term. In this model, the value of the option is given by a linear (1 + 2) evolution partial differential equation in which the price of the option depends upon two independent variables, the value of the underlying asset, $S$, and a new variable, $y$. We find that for arbitrary functional form of the volatility, $\\sigma(y)$, the (1 + 2) evolution equation always admits two Lie point symmetries in addition to the automatic linear symmetry and the infinite number of solution symmetries. However, when $\\sigma(y)=\\sigma_{0}$ and as the price of the option depends upon the second Brownian motion in which the volatility is defined, the (1 + 2) evolution is not reduced to the Black--Scholes--Merton Equation, the model admits five Lie point symmetries in addition to the linear symmetry and the infinite number of solution symmetries. We apply the zeroth-order invariants of the Lie symmetries and we reduce the (1 + 2) evolution equation to a linear second-order ordinary differential equation. Finally, we study two models of special interest, the Heston model and the Stein--Stein model.
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中文摘要：
我们对随机波动率为$\\ sigma$的欧式期权的Black-Scholes-Merton模型的Lie点对称性进行了分类，其中最后一个由一个带有Orstein-Uhlenbeck项的随机微分方程定义。在该模型中，期权的价值由一个线性（1+2）演化偏微分方程给出，其中期权的价格取决于两个自变量，即标的资产价值S$和一个新变量y$。我们发现，对于任意函数形式的波动率，$\\σ（y）$，除了自动线性对称和无穷多个解对称外，（1+2）演化方程总是允许两个Lie点对称。然而，当$\\ sigma（y）=\\ sigma\\u{0}$且期权价格取决于定义波动率的第二个布朗运动时，（1+2）演化并没有简化为Black-Scholes-Merton方程，该模型除了线性对称和无穷多个解对称外，还允许五个Lie点对称。我们应用李对称的零阶不变量，将（1+2）发展方程化为线性二阶常微分方程。最后，我们研究了两个特别有趣的模型，赫斯顿模型和斯坦-斯坦模型。
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分类信息：

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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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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