Projection of the two-dimensional Black-Scholes equation for options with underlying stock and strike prices in two different currencies
DOI:
https://doi.org/10.31349/RevMexFis.68.011401Keywords:
Diffusion in channels, entropic potential, black-scholes equation, option-pricing modelAbstract
The two-variable Black-Scholes equation is used to study the option exercise price of two different currencies. Due to the complexity of dealing with several variables, reduction methods have been implemented to deal with these problems. This paper proposes an alternative reduction by using the so-called Zwanzig projection method to one-dimension, successfully developed to study the diffusion in confined systems. In this case, the option price depends on the stock price and the exchange rate between currencies. We assume that the exchange rate between currencies will depend on the stock price through some model that bounds such dependence, which somehow influences the final option price.
As a result, we find a projected one-dimensional Black-Scholes equation similar to the so-called Fick-Jacobs equation for diffusion on channels. This equation is an effective Black-Scholes equation with two different interest rates, whose solution gives rise to a modified Black-Scholes formula. The properties of this solution are shown and were graphically compared with previously found solutions, showing that the corresponding difference is bounded.
References
L. Jiang, Mathematical Modeling and Methods of Option Pricing (World Scientific, Singapore, 2005), https://doi.org/10.1142/5855.
S. Benninga, T. Bj¨ork, and Zvi Wiener, On the Use of Numeraires in Option Pricing, J. Deriv. 10 (2002) 43, https://doi.org/10.3905/jod.2002.319195.
H.-C. O, Y.-H. Ro, and N. Wan, The Use of Numeraires in Multi-dimensional Black-Scholes Partial Differential Equations, arXiv:1310.8296.
M. Jeanblanc, M. Yor, and M. Chesney, Mathematical Methods for Financial Markets (Springer-Verlag, London, 2009), https://doi.org/10.1007/978-1-84628-737-4.
H.-C. O, Y.-H. Ro and N. Wan, Matematika, 30 (2014) 1, (arXiv:1406.2053). https://matematika.utm.my/index.php/matematika/article/view/706
R. Zwanzig, Diffusion past an entropy barrier, J. Phys. Chem. 96 (1992) 3926, https://doi.org/10.1021/j100189a004.
P. Kalinay and J. K. Percus, Projection of two-dimensional diffusion in a narrow channel onto the longitudinal dimension, J.
Chem. Phys. 122 (2005) 204701, https://doi.org/10.1063/1.1899150.
P. Kalinay and J. K. Percus, Corrections to the Fick-Jacobs equation, Phys. Rev. E 74 (2006) 041203, https://doi.org/10.1103/PhysRevE.74.041203.
narrow varying width channel onto the longitudinal dimension, J. Chem. Phys. 137 (2012) 024107, https://doi.org/10.1063/1.4733394.
Y. Ch´avez, M.-V. V´azquez, and L. Dagdug, Unbiased Diffusion through a Linear Porous Media with Periodic Entropy Barriers:
A Tube Formed by Contacting Ellipses, J. Chem. 2015 (2015) 986402, https://doi.org/10.1155/2015/986402.
M. Bohner and Y. Zheng, On analytical solutions of the Black-Scholes equation, Appl. Math. Lett. 22 (2009) 309, https://doi.org/10.1016/j.aml.2008.04.002.
D. Jeong et al., Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations,
Discret. Dyn. Nat. Soc. 2015 (2015) 359028, https://doi.org/10.1155/2015/359028.
M. B. Garman and S: W. Kohlhagen, Foreign currency option values, J. Int. Money Finance 2 (1983) 231, https://doi.org/10.1016/S0261-5606(83)80001-1.
J. Michael Steele, Stochastic Calculus and Financial Applications (Springer-Verlag, New York, 2001), https://doi.org/10.1007/978-1-4684-9305-4.
M. B. Garman, A general theory of asset valuation under diffusion state processes (University of California, Berkeley, 1977).
A. Bemporad and M. Morari, Robust model predictive control: A Survey, in Robustness in identification and control, edited by
A. Garulli and A. Tesi (Springer, London, 1999), pp. 207-226, https://doi.org/10.1007/BFb0109870.
W. Langson, I. Chryssochoos, S. V. Rakovi´c, and D. Q. Mayne, Robust model predictive control using tubes, Automatica 40 (2004) 125, https://doi.org/10.1016/j.automatica.2003.08.009.
T. Guillaume, On the multidimensional Black-Scholes partial differential equation, Ann. Oper. Res. 281 (2019) 229, https://doi.org/10.1007/s10479-018-3001-1.
A. Berezhkovskii and A. Szabo, Time scale separation leads to position-dependent diffusion along a slow coordinate, J. Chem.
Phys. 135 (2011) 074108, https://doi.org/10.1063/1.3626215.
L. Dagdug, A. A. Garc´ıa-Chung, and G. Chac´on-Acosta, On the description of Brownian particles in confinement on a non-Cartesian coordinates basis, J. Chem. Phys. 145 (2016) 074105, https://doi.org/10.1063/1.4960652.
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