英文标题：
《On the Solution of the Multi-asset Black-Scholes model: Correlations,
Eigenvalues and Geometry》
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作者：
Mauricio Contreras, Alejandro Llanquihu\\\'en and Marcelo Villena
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最新提交年份：
2015
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英文摘要：
In this paper, we study the multi-asset Black-Scholes model in terms of the importance that the correlation parameter space (equivalent to an $N$ dimensional hypercube) has in the solution of the pricing problem. We show that inside of this hypercube there is a surface, called the Kummer surface $\\Sigma_K$, where the determinant of the correlation matrix $\\rho$ is zero, so the usual formula for the propagator of the $N$ asset Black-Scholes equation is no longer valid. Worse than that, in some regions outside this surface, the determinant of $\\rho$ becomes negative, so the usual propagator becomes complex and divergent. Thus the option pricing model is not well defined for these regions outside $\\Sigma_K$. On the Kummer surface instead, the rank of the $\\rho$ matrix is a variable number. By using the Wei-Norman theorem, we compute the propagator over the variable rank surface $\\Sigma_K$ for the general $N$ asset case. We also study in detail the three assets case and its implied geometry along the Kummer surface.
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中文摘要：
本文从相关参数空间（相当于一个N维超立方体）在定价问题求解中的重要性出发，研究了多资产Black-Scholes模型。我们证明，在这个超立方体内部有一个曲面，叫做Kummer曲面$\\Sigma_K$，其中相关矩阵$\\rho$的行列式为零，因此，$N$asset Black-Scholes方程的传播子的常用公式不再有效。更糟糕的是，在这个表面之外的一些区域，$\\rho$的行列式变为负，所以通常的传播子变得复杂和发散。因此，在$\\Sigma_K$之外的这些地区，期权定价模型没有得到很好的定义。相反，在Kummer曲面上，$\\rho$矩阵的秩是一个可变数。利用Wei-Norman定理，我们计算了一般$N$资产情况下变秩曲面$\\Sigma_K$上的传播子。我们还详细研究了三资产情况及其沿Kummer曲面的隐含几何。
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分类信息：

一级分类：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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