Abstract
The aim of this paper is to develop an implicit–explicit (IMEX) scheme for solving the 2-dimensional (2-D) partial integro-differential equations with spatial delays arising in option pricing under the hard-to-borrow jump-diffusion models. First, a new second-order accurate numerical integration scheme that combines a mesh-dependent expansion and the trapezoidal rule is proposed to handle the integral-delayed term. Then, the IMEX scheme discretizes the integral-delayed term explicitly and the other terms implicitly. The second-order convergence rates for space and time are proved. Numerical examples are consistent with the theoretical results.
Similar content being viewed by others
References
Avellaneda M, Lipkin M (2009) A dynamic model for hard-to-borrow stocks. Risk 6:92–97
Bank RE, Santos RF (1993) Analysis of some moving space-time finite element methods. SIAM J Numer Anal 30:1–18
Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81:637–654
Cai N, Kou SG (2011) Option pricing under a mixed-exponential jump diffusion model. Manag Sci 57:2067–2081
Chen Y, Ma JT (2018) Numerical methods for a partial differential equation with spatial delay arising in option pricing under hard-to-borrow model. Comput Math Appl 76:2129–2140
Chen Y, Xiao A, Wang W (2019) An IMEX-BDF2 compact scheme for pricing options under regime-switching jump-diffusion models. Math Methods Appl Sci 42:2646–2663
Kadalbajoo MK, Tripathi LP, Kumar K (2015) Second order accurate IMEX methods for option pricing under Merton and Kou jump-diffusion model. J Sci Comput 65:979–1024
Kadalbajoo MK, Tripathi LP, Kumar K (2017) An error analysis of a finite element method with IMEX-time semidiscretizations for some partial integro-differential inequalities arising in the pricing of American options. SIAM J Numer Anal 55:869–891
Kazmi K (2019) An IMEX predictor-corrector method for pricing options under regime-switching jump-diffusion models. Int J Comput Math 96:1137–1157
Kwon Y, Lee Y (2011) A second-order finite difference method for option pricing under jump-diffusion models. SIAM J Numer Anal 49:2598–2617
Ma JT, Chen Y (2020) Convergence rates of the numerical methods for the delayed PDEs from option pricing under regime-switching hard-to-borrow models. Int J Comput Math 97:2210–2232
Ma JT, Zhou Z (2016) Convergence rates of moving mesh Rannacher methods for PDEs of Asian options pricing. J Comput Math 34:265–286
Ma GY, Zhu SP (2018) Pricing American call options under a hard-to-borrow stock model. Eur J Appl Math 29:494–514
Ma GY, Zhu SP, Chen WT (2019) Pricing European call options under a hard-to-borrow stock model. Appl Math Comput 357:243–257
Morton KW, Mayers DF (2005) Numerical solution of partial differential equations. Cambridge University Press, Cambridge
Salmi S, Toivanen J (2014) IMEX schemes for pricing options under jump-diffusion models. Appl Numer Math 84:33–45
Salmi S, Toivanen J, Von Sydow L (2014) An IMEX-scheme for pricing options under stochastic volatility models with jumps. SIAM J Sci Comput 36:B817–B834
Shreve SE (2004) Stochastic calculus for finance II: continuous-time models. Springer, Berlin
von Sydow L, Toivanen J, Zhang C (2015) Adaptive finite difference and IMEX time-stepping to price options under Bates model. Int J Comput Math 92:2515–2529
Wang W, Chen Y, Fang H (2019) On the variable two-step IMEX BDF methods for parabolic integro-differential equations with nonsmooth initial data arising in finance. SIAM J Numer Anal 57:1289–1317
Acknowledgements
The author is sincerely grateful to the editor and anonymous referees for their valuable comments that have led to a greatly improved paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Ethical statement
The authors declare that there are no following cases: conflicts of interest, research involving human participants and/or animals, and informed consent.
Additional information
Communicated by Pierre Etore.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The work was supported by Technology and Venture Finance Research Center of Sichuan Key Research Base for Social Sciences (Grant no. KJJR2019-003)
Appendix
Appendix
In this appendix, we derive the PIDE (6) with the initial and boundary conditions (7)–(11) for European call option pricing under the jump-diffusion A&L model (1)–(3).
Proof
Denote the continuous parts of the SDEs (1) and (66) as
Applying Itô lemma to \(e^{-rt}{\widehat{u}}(x,S,t)\) yields
Since \(e^{-rt}{\widehat{u}}(x,S,t)\) must be a martingale according to the risk-neutral pricing theory, we set the drift term of (69) to be zero and further obtain
with the payoff function for European call option
Just as in Shreve (2004) and Ma et al. (2019), we impose the asymptotic boundary conditions as follows:
Applying the change of the variables \(\tau =T-t\), \(y=\ln \left( \frac{S}{K}\right) \) and denote \(u(x,y,\tau ):={\widehat{u}}(x,Ke^{y},T-\tau )\), we obtain the PIDE (6) with the initial and boundary conditions (7)–(11). \(\square \)
Rights and permissions
About this article
Cite this article
Chen, Y. Second-order convergent IMEX scheme for integro-differential equations with delays arising in option pricing under hard-to-borrow jump-diffusion models. Comp. Appl. Math. 41, 75 (2022). https://doi.org/10.1007/s40314-022-01783-9
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-022-01783-9
Keywords
- PDEs with delays
- Option pricing
- Hard-to-borrow stock models
- Jump-diffusion models
- Finite-difference methods
- Convergence rates