Computational Economics

, Volume 53, Issue 1, pp 111–124

# A Hybrid Monte Carlo and Finite Difference Method for Option Pricing

• Darae Jeong
• Minhyun Yoo
• Changwoo Yoo
• Junseok Kim
Article

## Abstract

We propose an accurate, efficient, and robust hybrid finite difference method, with a Monte Carlo boundary condition, for solving the Black–Scholes equations. The proposed method uses a far-field boundary value obtained from a Monte Carlo simulation, and can be applied to problems with non-linear payoffs at the boundary location. Numerical tests on power, powered, and two-asset European call option pricing problems are presented. Through these numerical simulations, we show that the proposed boundary treatment yields better accuracy and robustness than the most commonly used linear boundary condition. Furthermore, the proposed hybrid method is general, which means it can be applied to other types of option pricing problems. In particular, the proposed Monte Carlo boundary condition algorithm can be implemented easily in the code of the existing finite difference method, with a small modification.

## Keywords

Black–Scholes equation Finite difference method Option pricing Boundary condition Monte Carlo simulation

## Notes

### Acknowledgements

The author (D. Jeong) was supported by a Korea University Grant. The corresponding author (J. S. Kim) was supported by the Korea Institute for Advanced Study (KIAS) for supporting research on the financial pricing model using artificial intelligence. The authors are grateful to the reviewers whose valuable suggestions and comments significantly improved the quality of this paper.

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## Authors and Affiliations

• Darae Jeong
• 1
• Minhyun Yoo
• 2
• Changwoo Yoo
• 2
• Junseok Kim
• 1
Email author
1. 1.Department of MathematicsKorea UniversitySeoulRepublic of Korea
2. 2.Department of Financial EngineeringKorea UniversitySeoulRepublic of Korea