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Second-order convergent IMEX scheme for integro-differential equations with delays arising in option pricing under hard-to-borrow jump-diffusion models

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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.

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Acknowledgements

The author is sincerely grateful to the editor and anonymous referees for their valuable comments that have led to a greatly improved paper.

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Correspondence to Yong Chen.

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Communicated by Pierre Etore.

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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

Plugging (1) into (2) gives

$$\begin{aligned} {\mathrm{d}}x_{t}=\left[ \alpha ({\overline{x}}-x_{t})+\beta r \right] {\mathrm{d}}t+\kappa {\mathrm{d}}{\widetilde{Z}}_{t}+\beta \sigma {\mathrm{d}}{\widetilde{W}}_{t}-\beta {\mathrm{d}}\left( \sum _{i=1}^{N_{\lambda _{t}}(t)} \gamma _{i}\right) . \end{aligned}$$
(66)

Denote the continuous parts of the SDEs (1) and (66) as

$$\begin{aligned} {\mathrm{d}}S^{c}_{t}&=S_{t}r {\mathrm{d}}t+S_{t}\sigma {\mathrm{d}}{\widetilde{W}}_{t}, \end{aligned}$$
(67)
$$\begin{aligned} {\mathrm{d}}x^{c}_{t}&=\left[ \alpha ({\overline{x}}-x_{t})+\beta r\right] {\mathrm{d}}t+\kappa {\mathrm{d}}{\widetilde{Z}}_{t}+\beta \sigma {\mathrm{d}}{\widetilde{W}}_{t}. \end{aligned}$$
(68)

Applying Itô lemma to \(e^{-rt}{\widehat{u}}(x,S,t)\) yields

$$\begin{aligned} d\left[ e^{-rt}{\widehat{u}}(x,S,t)\right]&= -re^{-rt}{\widehat{u}}{\mathrm{d}}t+e^{-rt}\Big \{\frac{\partial {\widehat{u}}}{\partial t}{\mathrm{d}}t+\frac{\partial {\widehat{u}}}{\partial x}{\mathrm{d}}x^{c}+\frac{\partial {\widehat{u}}}{\partial S}{\mathrm{d}}S^{c} \nonumber \\&\quad +\, \frac{1}{2}\frac{\partial ^{2}{\widehat{u}}}{\partial x^{2}}{\mathrm{d}}x^{c}{\mathrm{d}}x^{c}+\frac{1}{2}\frac{\partial ^{2}{\widehat{u}}}{\partial S^{2}}{\mathrm{d}}S^{c}{\mathrm{d}}S^{c}+\frac{\partial ^{2}{\widehat{u}}}{\partial x \partial S}{\mathrm{d}}x^{c}{\mathrm{d}}S^{c} \nonumber \\&\quad +\, \left[ {\widehat{u}}\left( x-\beta \gamma ,S(1-\gamma ),t\right) -{\widehat{u}}(x,S,t)\right] dN_{\lambda _{t}}(t)\Big \} \nonumber \\&=e^{-rt}\Big \{S\sigma \frac{\partial {\widehat{u}}}{\partial S}{\mathrm{d}}{\widetilde{W}}_{t}+\kappa \frac{\partial {\widehat{u}}}{\partial x}{\mathrm{d}}{\widetilde{Z}}_{t}+\beta \sigma \frac{\partial {\widehat{u}}}{\partial x}{\mathrm{d}}{\widetilde{W}}_{t} \nonumber \\&\quad +\, \left[ {\widehat{u}}\left( x-\beta \gamma ,S(1-\gamma ),t\right) -{\widehat{u}}(x,S,t)\right] \left[ dN_{\lambda _{t}}(t)-\lambda _{t}{\mathrm{d}}t\right] \Big \} \nonumber \\&\quad +\, e^{-rt}\Big \{\frac{\partial {\widehat{u}}}{\partial t}+\frac{\kappa ^{2}+\beta ^{2}\sigma ^{2}}{2}\frac{\partial ^{2}{\widehat{u}}}{\partial x^{2}}+\frac{\sigma ^{2}S^{2}}{2}\frac{\partial ^{2}{\widehat{u}}}{\partial S^{2}}+\beta \sigma ^{2}S\frac{\partial ^{2}{\widehat{u}}}{\partial x \partial S} \nonumber \\&\quad +\, \left[ \alpha ({\overline{x}}-x)+\beta r\right] \frac{\partial {\widehat{u}}}{\partial x}+rS\frac{\partial {\widehat{u}}}{\partial S}-r{\widehat{u}} \nonumber \\&\quad +\, \lambda _{t}\left[ {\widehat{u}}\left( x-\beta \gamma ,S(1-\gamma ),t\right) -{\widehat{u}}(x,S,t)\right] \Big \}{\mathrm{d}}t. \end{aligned}$$
(69)

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

$$\begin{aligned} -\frac{\partial {\widehat{u}}}{\partial t}&=\frac{\kappa ^{2}+\beta ^{2}\sigma ^{2}}{2}\frac{\partial ^{2}{\widehat{u}}}{\partial x^2}+\frac{1}{2}\sigma ^{2}S^{2}\frac{\partial ^{2}{\widehat{u}}}{\partial S^2}+\beta \sigma ^{2}S\frac{\partial ^{2}{\widehat{u}}}{\partial x \partial S}+\left[ \alpha {({\overline{x}}-x)}+\beta r \right] \frac{\partial {\widehat{u}}}{\partial x} \nonumber \\&\quad +\, rS\frac{\partial {\widehat{u}}}{\partial S}-(r+e^{x}) {\widehat{u}}+e^{x}\int _{0}^{\infty }{\widehat{u}}\left( x-\beta (1-e^{-v}),Se^{-v},t\right) f(v){\mathrm{d}}v, \end{aligned}$$
(70)

with the payoff function for European call option

$$\begin{aligned} {\widehat{u}}(x,S,T)=\max {(S-K,0)}. \end{aligned}$$
(71)

Just as in Shreve (2004) and Ma et al. (2019), we impose the asymptotic boundary conditions as follows:

$$\begin{aligned}&\lim _{x\rightarrow -\infty }{\widehat{u}}(x,S,t)=C^{{\mathrm{BS}}}(S,T-t,K,r,\sigma ), \end{aligned}$$
(72)
$$\begin{aligned}&\lim _{x\rightarrow +\infty }{\widehat{u}}(x,S,t)=\max {\Big (Se^{-\zeta \lambda (T-t)}-Ke^{-r(T-t)},0\Big )},\end{aligned}$$
(73)
$$\begin{aligned}&\lim _{S\rightarrow 0}{\widehat{u}}(x,S,t)=0,\end{aligned}$$
(74)
$$\begin{aligned}&\lim _{S\rightarrow +\infty }\left[ {\widehat{u}}(x,S,t)-\Big (Se^{-\zeta \lambda (T-t)}-Ke^{-r(T-t)}\Big )\right] =0. \end{aligned}$$
(75)

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). \(\square \)

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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

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  • DOI: https://doi.org/10.1007/s40314-022-01783-9

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