# American Institute of Mathematical Sciences

February  2020, 19(2): 699-714. doi: 10.3934/cpaa.2020032

## (1+2)-dimensional Black-Scholes equations with mixed boundary conditions

 1 Department of Applied Mathematics & Institute of Natural Science, Kyung Hee University, Yongin, 17104, Republic of Korea 2 Department of Mathematics, Kyungpook National University, Daegu, 41566, Republic of Korea

* Corresponding author

Received  September 2018 Revised  April 2019 Published  October 2019

In this paper, we investigate (1+2)-dimensional Black-Scholes partial differential equations(PDE) with mixed boundary conditions. The main idea of our method is to transform the given PDE into the relatively simple ordinary differential equations(ODE) using double Mellin transforms. By using inverse double Mellin transforms, we derive the analytic representation of the solutions for the (1+2)-dimensional Black-Scholes equation with a mixed boundary condition. Moreover, we apply our method to European maximum-quanto lookback options and derive the pricing formula of this options.

Citation: Junkee Jeon, Jehan Oh. (1+2)-dimensional Black-Scholes equations with mixed boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (2) : 699-714. doi: 10.3934/cpaa.2020032
##### References:
 [1] J. Bertrand, P. Bertrand and J. P. Ovarlez, The Mellin transform, in The Transforms and Applications Handbook (ed. A.D. Poularikas), IEEE Press, New York, (1996), 829–885. [2] M. Dai, H. Y. Wong and Y. K. Kwok, Quanto lookback options, Math. Finance, 14 (2004), 445-467.  doi: 10.1111/j.0960-1627.2004.00199.x. [3] A. Dravid, M. Richardson and T. S Sun, Pricing foreign index contingent claims: An application to Nikkei index warrants, Journal of Derivatives, 1 (1993), 33-51. [4] H. Eltayeb and A. Kılıçman, A note on Mellin transform and partial differential equations, Int. J. Pure Appl. Math., 34 (2007), 457-467. [5] J. Jeon, H. Han, H. Kim and M. Kang, An integral equation representation approach for valuing Russian options with a finite time horizon, Commun. Nonlinear Sci. Numer. Simul., 36 (2016), 496-516.  doi: 10.1016/j.cnsns.2015.12.019. [6] I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York., 1972. [7] J. H. Yoon and J. H. Kim, The pricing of vulnerable options with double Mellin transforms, J. Math. Anal. Appl., 422 (2015), 838-857.  doi: 10.1016/j.jmaa.2014.09.015.

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##### References:
 [1] J. Bertrand, P. Bertrand and J. P. Ovarlez, The Mellin transform, in The Transforms and Applications Handbook (ed. A.D. Poularikas), IEEE Press, New York, (1996), 829–885. [2] M. Dai, H. Y. Wong and Y. K. Kwok, Quanto lookback options, Math. Finance, 14 (2004), 445-467.  doi: 10.1111/j.0960-1627.2004.00199.x. [3] A. Dravid, M. Richardson and T. S Sun, Pricing foreign index contingent claims: An application to Nikkei index warrants, Journal of Derivatives, 1 (1993), 33-51. [4] H. Eltayeb and A. Kılıçman, A note on Mellin transform and partial differential equations, Int. J. Pure Appl. Math., 34 (2007), 457-467. [5] J. Jeon, H. Han, H. Kim and M. Kang, An integral equation representation approach for valuing Russian options with a finite time horizon, Commun. Nonlinear Sci. Numer. Simul., 36 (2016), 496-516.  doi: 10.1016/j.cnsns.2015.12.019. [6] I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York., 1972. [7] J. H. Yoon and J. H. Kim, The pricing of vulnerable options with double Mellin transforms, J. Math. Anal. Appl., 422 (2015), 838-857.  doi: 10.1016/j.jmaa.2014.09.015.
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