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On the Solution of the Multi-asset Black-Scholes model: Correlations, Eigenvalues and Geometry

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  • Mauricio Contreras
  • Alejandro Llanquihu'en
  • Marcelo Villena

Abstract

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correlation matrix $\rho$ is zero, so the usual formula for the propagator of the $N$ asset Black-Scholes equation is no longer valid.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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  • Mauricio Contreras & Alejandro Llanquihu'en & Marcelo Villena, 2015. "On the Solution of the Multi-asset Black-Scholes model: Correlations, Eigenvalues and Geometry," Papers 1510.02768, arXiv.org.
  • Handle: RePEc:arx:papers:1510.02768
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    References listed on IDEAS

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    Cited by:

    1. Bustamante, M. & Contreras, M., 2016. "Multi-asset Black–Scholes model as a variable second class constrained dynamical system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 457(C), pages 540-572.
    2. Panumart Sawangtong & Kamonchat Trachoo & Wannika Sawangtong & Benchawan Wiwattanapataphee, 2018. "The Analytical Solution for the Black-Scholes Equation with Two Assets in the Liouville-Caputo Fractional Derivative Sense," Mathematics, MDPI, vol. 6(8), pages 1-14, July.
    3. Tristan Guillaume, 2019. "On the multidimensional Black–Scholes partial differential equation," Annals of Operations Research, Springer, vol. 281(1), pages 229-251, October.

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