The numerical approximation of nonlinear Black–Scholes model for exotic path-dependent American options with transaction cost

Abstract

In this paper, a new second-order exponential time differencing (ETD) method based on the Cox and Matthews approach is developed and applied for pricing American options with transaction cost. The method is seen to be strongly stable and highly efficient for solving the nonlinear Black–Scholes model. Furthermore, it does not incur unwanted oscillations unlike the ETD–Crank–Nicolson method for exotic path-dependent American options. The computational efficiency and reliability of the new method are demonstrated by numerical examples and by comparing it with the existing methods.

Keywords:

• exponential time differencing
• transaction cost
• butterfly spread
• discrete barrier option
• nonlinear Black–Scholes model

2000 AMS Subject Classifications :

• 65M12
• 65M15
• 65Y05
• 65Y20

Acknowledgements

The work of M. Yousuf was supported by the Fast Track Project # FT100003, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. The third author and consultant of the project A.Q.M. Khaliq thanks KFUPM for the hospitality during his visit. The authors are grateful to the anonymous referees for their constructive comments and valuable suggestions, which have improved the quality of this paper.