An implicit-explicit preconditioned direct method for pricing options under regime-switching tempered fractional partial differential models

Abstract

Recently, fractional partial differential equations have been widely applied in option pricing problems, which better explains many important empirical facts of financial markets, but rare paper considers the multi-state options pricing problem based on fractional diffusion models. Thus, multi-state European option pricing problem under regime-switching tempered fractional partial differential equation is considered in this paper. Due to the expensive computational cost caused by the implicit finite difference scheme, a novel implicit-explicit finite difference scheme has been developed with consistency, stability, and convergence guarantee. Since the resulting coefficient matrix equals to the direct sum of several Toeplitz matrices, a preconditioned direct method has been proposed with ${\mathcal O}(\bar {S}N\log N+\bar {S}^{2} N)$ operation cost on each time level with adaptability analysis, where $\bar {S}$ is the number of states and N is the number of grid points. Related numerical experiments including an empirical example have been presented to demonstrate the effectiveness and accuracy of the proposed numerical method.

Appendix

In Appendix, the steps for obtaining ${\mathcal L}_{s} F(x,t)$ in (2) are presented. By the definition of F(x, t), we have:

$$ \begin{array}{@{}rcl@{}} F(x,t)&=&\frac{V_{s}(x_{r},t)-V_{s}(x_{l},t)}{e^{x_{r}}-e^{x_{l}}}(e^{x}-e^{x_{l}})+V_{s}(x_{l},t)\\ &=&d_{s, 1} e^{x}+d_{s, 2}, \end{array} $$

where $d_{s, 1}=\frac {V_{s}(x_{r},t)-V_{s}(x_{l},t)}{e^{x_{r}}-e^{x_{l}}}$ and $d_{s, 2}=V_{s}(x_{l},t)-\frac {V_{s}(x_{r},t)-V_{s}(x_{l},t)}{e^{x_{r}}-e^{x_{l}}}e^{x_{l}}$. To obtain ${\mathcal L}_{s} F(x,t)$, the most difficult step is to calculate $D^{\xi _{s},\alpha _{s}}_{+}[d_{s, 1}e^{x}]$, $D^{\xi _{s},\alpha _{s}}_{+}[d_{s, 2}]$, $D^{\lambda _{s},\alpha _{s}}_{-}[d_{s, 1}e^{x}]$, $D^{\lambda _{s},\alpha _{s}}_{-}[d_{s, 2}]$. In fact, they can be obtained by the following deduction:

$$ \begin{array}{@{}rcl@{}} D^{\lambda_{s},\alpha_{s}}_{-}[d_{s, 1} e^{x}]&=&\frac{e^{-\lambda_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}{\int}^{x}_{-\infty}\frac{d_{s, 1} e^{\lambda_{s}\zeta}e^{\zeta}}{(x-\zeta)^{\alpha_{s}-1}}d\zeta-\lambda_{s}^{\alpha_{s}} d_{s, 1} e^{x}\\ &=&\frac{e^{-\lambda_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}} \Big[d_{s, 1}{\int}^{\infty}_{0} e^{(\lambda_{s}+1)(x-y)}y^{1-\alpha_{s}}dy\Big]-\lambda_{s}^{\alpha_{s}} d_{s, 1} e^{x} \\ &=&\frac{e^{-\lambda_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 1} e^{(\lambda_{s}+1)x} {\int}^{\infty}_{0} e^{-(\lambda_{s}+1)y}y^{1-\alpha_{s}}dy\Big]-\lambda_{s}^{\alpha_{s}} d_{s, 1} e^{x}\\ &=&\frac{e^{-\lambda_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 1} e^{(\lambda_{s}+1)x} {\int}^{\infty}_{0} e^{-\mu}\mu^{1-\alpha_{s}}(\lambda_{s}+1)^{\alpha_{s}-2}d\mu\Big]\\&&-\lambda_{s}^{\alpha_{s}} d_{s, 1} e^{x}\\ &=&\frac{e^{-\lambda_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 1} e^{(\lambda_{s}+1)x} (\lambda_{s}+1)^{\alpha_{s}-2}{\Gamma}(2-\alpha_{s})\Big]-\lambda_{s}^{\alpha_{s}} d_{s, 1} e^{x}\\ &=&d_{s, 1} (\lambda_{s}+1)^{\alpha_{s}} e^{x}-\lambda_{s}^{\alpha_{s}} d_{s, 1} e^{x}, \end{array} $$

$$ \begin{array}{@{}rcl@{}} D^{\lambda_{s},\alpha_{s}}_{-}[{d_{s, 2}}]&=&\frac{e^{-\lambda_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}{\int}^{x}_{-\infty}\frac{d_{s, 2} e^{\lambda_{s}\zeta}}{(x-\zeta)^{\alpha_{s}-1}}d\zeta-\lambda_{s}^{\alpha_{s}} d_{s, 2}\\ &=&\frac{e^{-\lambda_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}} \Big[d_{s, 2}{\int}^{\infty}_{0} e^{\lambda_{s}(x-y)}y^{1-\alpha_{s}}dy\Big]-\lambda_{s}^{\alpha_{s}} d_{s, 2}\\ &=&\frac{e^{-\lambda_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 2} e^{\lambda_{s} x} {\int}^{\infty}_{0} e^{-\lambda_{s} y}y^{1-\alpha_{s}}dy\Big]-\lambda_{s}^{\alpha_{s}} d_{s, 2}\\ &=&\frac{e^{-\lambda_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 2} e^{\lambda_{s} x} {\int}^{\infty}_{0} e^{-\mu}\mu^{1-\alpha_{s}}\lambda^{\alpha_{s}-2}d\mu\Big]-\lambda_{s}^{\alpha_{s}} d_{s, 2}\\ &=&\frac{e^{-\lambda_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 2} e^{\lambda_{s} x} \lambda^{\alpha_{s}-2}{\Gamma}(2-\alpha_{s})\Big]-\lambda_{s}^{\alpha_{s}} d_{s, 2}\\ &=&d_{s, 2} \lambda_{s}^{\alpha_{s}}-\lambda_{s}^{\alpha_{s}} d_{s, 2}=0, \end{array} $$

$$ \begin{array}{@{}rcl@{}} D^{\xi_{s},\alpha_{s}}_{+}[d_{s, 1} e^{x}]&=&\frac{e^{\xi_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}{\int}^{\infty}_{x}\frac{d_{s, 1} e^{-\xi_{s}\zeta}e^{\zeta}}{(\zeta-x)^{\alpha_{s}-1}}d\zeta-\xi_{s}^{\alpha_{s}}d_{s, 1}e^{x}\\ &=&\frac{e^{\xi_{s} x}}{\Gamma(2-\xi_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 1}{\int}^{\infty}_{0} e^{(1-\xi_{s})(x+y)}y^{1-\alpha_{s}}dy\Big]-\xi_{s}^{\alpha_{s}}d_{s, 1}e^{x}\\ &=&\frac{e^{\xi_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 1} e^{(1-\xi_{s})x}{\int}^{\infty}_{0} e^{-(\xi_{s}-1)y}y^{1-\alpha_{s}}dy\Big]-\xi_{s}^{\alpha_{s}}d_{s, 1}e^{x}\\ &=&\frac{e^{\xi_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 1} e^{(1-\xi_{s})x}{\int}^{\infty}_{0} e^{-\mu}\mu^{1-\alpha_{s}}(\xi_{s}-1)^{\alpha_{s}-2}d\mu\Big]\\&&-\xi_{s}^{\alpha_{s}}d_{s, 1}e^{x}\\ &=&\frac{e^{\xi_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 1} e^{(1-\xi_{s})x} (\xi_{s}-1)^{\alpha_{s}-2}{\Gamma}(2-\alpha_{s})\Big]-\xi_{s}^{\alpha_{s}}d_{s, 1}e^{x}\\ &=&d_{s, 1} (\xi_{s}-1)^{\alpha_{s}} e^{x}-\xi_{s}^{\alpha_{s}}d_{s, 1}e^{x}, \end{array} $$

$$ \begin{array}{@{}rcl@{}} D^{\xi_{s},\alpha_{s}}_{+}[d_{s, 2}]&=&\frac{e^{\xi_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}{\int}^{\infty}_{x}\frac{d_{s, 2} e^{-\xi_{s}\zeta}}{(\zeta-x)^{\alpha_{s}-1}}d\zeta-\xi_{s}^{\alpha_{s}}d_{s, 2}\\ &=&\frac{e^{\xi_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 2}{\int}^{\infty}_{0} e^{-\xi_{s}(x+y)}y^{1-\alpha_{s}}dy\Big]-\xi_{s}^{\alpha_{s}}d_{s, 2}\\ &=&\frac{e^{\xi_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 2} e^{-\xi_{s} x}{\int}^{\infty}_{0} e^{-\xi_{s} y}y^{1-\alpha_{s}}dy\Big]-\xi_{s}^{\alpha_{s}}d_{s, 2}\\ &=&\frac{e^{\xi_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 2} e^{-\xi_{s} x}{\int}^{\infty}_{0} e^{-\mu}\mu^{1-\alpha}{\xi_{s}}^{\alpha_{s}-2}d\mu\Big]-\xi_{s}^{\alpha_{s}}d_{s, 2}\\ &=&\frac{e^{\xi_{s} x}}{\Gamma(2-\alpha_{s})}\frac{\partial^{2}}{\partial x^{2}}\Big[d_{s, 2} e^{-\xi_{s} x} {\xi_{s}}^{\alpha_{s}-2}{\Gamma}(2-\alpha_{s})\Big]-\xi_{s}^{\alpha_{s}}d_{s, 2}\\ &=&d_{s, 2} {\xi_{s}}^{\alpha_{s}}-\xi_{s}^{\alpha_{s}}d_{s, 2}=0. \end{array} $$

For European call options, by assuming that V_s(x_l, t) = 0 and $V_{s}(x_{r},t)=e^{x_{r}}-Ke^{-r(T-t)}$, we have

$$ \begin{array}{@{}rcl@{}} {\mathcal L}_{s} F(x,t)&=&\frac{\partial F(x,t)}{\partial t}+c_{s, 1}\frac{\partial F(x,t)}{\partial x}+c_{s, 2}D^{\xi_{s},\alpha_{s}}_{+}F(x,t)\\ &&+c_{s, 3}D^{\lambda_{s},\alpha_{s}}_{-}F(x,t)-rF(x,t)+\sum\limits_{j=1}^{\bar{S}}q_{s,j}F(x,t)\\ &=&\frac{\partial d_{s, 1}}{\partial t}e^{x}+\frac{\partial d_{s, 2}}{\partial t}+c_{s, 1}d_{s, 1}e^{x} +c_{s, 2}[d_{s, 1}(\xi_{s}-1)^{\alpha_{s}}e^{x}-d_{s, 1}\xi_{s}^{\alpha_{s}}e^{x}]\\ &&+c_{s, 3}[d_{s, 1}(\lambda_{s}+1)^{\alpha_{s}}e^{x}-d_{s, 1}\lambda_{s}^{\alpha_{s}}e^{x}]-r(d_{s, 1}e^{x}+d_{s, 2})\\ &=&\frac{rKe^{-r(T-t)}}{e^{x_{r}}-e^{x_{l}}}(e^{x_{l}} - e^{x})+d_{s, 1}[c_{s, 1}+c_{s, 2}(\xi_{s} - 1)^{\alpha_{s}} +c_{s, 3}(\lambda_{s}+1)^{\alpha_{s}}]e^{x}\\ &&-(c_{s, 2}\xi_{s}^{\alpha_{s}}+c_{s, 3}\lambda_{s}^{\alpha_{s}}+r)d_{s, 1}e^{x}-rd_{s, 2}. \end{array} $$

Hence, f_s(x, t) can be obtained analytically. For European put options, f_s(x, t) can be obtained similarly.