| Issue |
ESAIM: M2AN
Volume 59, Number 2, March-April 2025
|
|
|---|---|---|
| Page(s) | 643 - 670 | |
| DOI | https://doi.org/10.1051/m2an/2025003 | |
| Published online | 11 February 2025 | |
High order Semi-IMEX BDF schemes for nonlinear partial integro-differential equations arising in finance
1
Department of Mathematics, Shanghai Normal University, Shanghai 200234, P.R. China
2
School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia
* Corresponding author: w.s.wang@163.com
Received:
1
August
2024
Accepted:
12
January
2025
In this paper, semi-implicit–explicit (Semi-IMEX) and semi-implicit multistep methods are proposed to solve nonlinear partial integro-differential equations (PIDEs), which describe the option pricing models with transaction costs or illiquid markets under Merton’s jump-diffusion process. After spatial differential operators are treated by using finite difference methods and the jump integral is computed by using the composite trapezoidal rule, the consistency error and global error bounds for the semi-IMEX and semi-implicit multistep methods for abstract PIDEs are provided when the nonlinear operator satisfies the boundedness and coercivity conditions. A numerical study is carried out for different option pricing models based on the convergence properties of the schemes and the comparison of the different Greek values. Numerical experiments verify our theoretical results and illustrate the intrinsic nature of the proposed option pricing models under jump-diffusion models with transaction costs or illiquid markets.
Mathematics Subject Classification: 65M06 / 65M15 / 65L06 / 91B25 / 91G60 / 65J10
Key words: Nonlinear partial integro-differential equations / jump-diffusion models / implicit–explicit BDF methods / finite difference method / error estimates / transaction costs / illiquid markets
© The authors. Published by EDP Sciences, SMAI 2025
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