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29 pages, 477 KiB  
Article
An Analytically Modified Finite Difference Scheme for Pricing Discretely Monitored Options
by Guo Luo and Min Huang
Mathematics 2025, 13(2), 241; https://doi.org/10.3390/math13020241 - 13 Jan 2025
Viewed by 755
Abstract
Finite difference methods are commonly used in the pricing of discretely monitored exotic options in the Black–Scholes framework, but they tend to converge slowly due to discontinuities contained in terminal conditions. We present an effective analytical modification to existing finite difference methods that [...] Read more.
Finite difference methods are commonly used in the pricing of discretely monitored exotic options in the Black–Scholes framework, but they tend to converge slowly due to discontinuities contained in terminal conditions. We present an effective analytical modification to existing finite difference methods that greatly enhances their performance on discretely monitored options with non-smooth terminal conditions. We apply this modification to the popular Crank–Nicolson method and obtain highly accurate option pricing results with significantly reduced CPU cost. We also introduce an adaptive mesh refinement technique that further improves the computational speed of the modified finite difference method. The proposed method is especially useful for options with high monitoring frequencies, which are difficult to price using other existing methods. Full article
(This article belongs to the Section E5: Financial Mathematics)
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