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A primal-dual active set approach to the valuation of American options in regime-switching models: numerical solutions and convergence analysis

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Abstract

In this study, we explore the valuation challenge posed by American options subject to regime switching, utilizing a model defined by a complex system of parabolic variational inequalities within an infinite domain. The initial pricing model is transformed into a linear complementarity problem (LCP) in a bounded rectangular domain, achieved through the application of a priori estimations and the introduction of an appropriate artificial boundary condition. To discretize the LCP, we employ a finite difference method (FDM), and address the resulting discretized system using a primal-dual active set (PDAS) strategy. The PDAS approach is particularly advantageous for its ability to concurrently determine the option’s price and the optimal exercise boundary. This paper conducts an extensive convergence analysis, evaluating both the truncation error associated with the FDM and the iteration error of the PDAS. Comprehensive numerical simulations are performed to validate the method’s accuracy and efficiency, underscoring its significant potential for application in the field of financial mathematics.

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Data are available on reasonable request.

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Acknowledgements

All authors would like to thank the reviewers for their valuable comments and suggestions, which significantly contributed to the improvement of this manuscript. The work of H. Song was supported by the National Key Research and Development Program of China under Grant No.2020YFA0713602, the NSF of Jilin Province under Grant No.20200201269JC, and the Fundamental Research Funds for the Central Universities. The work of Y. Li was supported by the Shenzhen Science and Technology Program under Grant No. 20220816165920001, and the Chinese University of Hong Kong, Shenzhen under Grant No. PF01000861.

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, Jilin, China

    Xin Wen, Haiming Song & Zihan Gao

  2. School of Science and Engineering, Chinese University of Hong Kong, Shenzhen, 518172, Guangdong, China

    Yutian Li

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  1. Xin Wen
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Correspondence to Yutian Li.

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Appendices

Proof of  Lemma 5

Proof

It is evident that \(\lambda _{i,q} \ge 1\). The Taylor expansion gives

$$\begin{aligned} \begin{aligned} \min \{\lambda _{1,q},\lambda _{2,q}\}&=1+\min \{\sigma _1^2,\sigma _2^2\}\alpha \left( 1-\cos \frac{\pi }{N}\right) \\&=1+\min \{\sigma _1^2,\sigma _2^2\}\Delta \tau \left( \frac{\pi ^2}{8L^2}+O\left( (\Delta x)^2\right) \right) . \end{aligned} \end{aligned}$$

Hence, there exists a constant \(\varrho >0\) such that the equation \(\min \{\lambda _{1,q},\lambda _{2,q}\} \le 1+\varrho \) holds when \(\Delta \tau \) and \(\Delta x\) are sufficiently small. On one hand, if \(\frac{1+\beta \Delta \tau }{\min \{\lambda _{1,q},\lambda _{2,q}\}}\ge 1\), we have

$$\begin{aligned} \frac{T}{1+\varrho } \le \frac{\Delta \tau }{\min {\{\lambda _{1,q},\lambda _{2,q}\}}}\sum _{p=1}^{M} \left( \frac{1+\beta \Delta \tau }{\min {\{\lambda _{1,q},\lambda _{2,q}\}}}\right) ^{M-p} \le T(1+\beta \Delta \tau )^{M} \le T e^{\beta T}. \end{aligned}$$

On the other hand, if \(\frac{1+\beta \tau }{\min \{\lambda _{1,q},\lambda _{2,q}\}}< 1\), we can get

$$\begin{aligned} \frac{T}{(1+\varrho )^{M}} \le \frac{\Delta \tau }{\min {\{\lambda _{1,q},\lambda _{2,q}\}}} \sum \limits _{p=1}^{M}\left( \frac{1+\beta \Delta \tau }{\min {\{\lambda _{1,q},\lambda _{2,q}\}}}\right) ^{M-p}\le T. \end{aligned}$$

Therefore, we conclude that

$$\begin{aligned} \frac{\Delta \tau }{\min {\{\lambda _{1,q},\lambda _{2,q}\}}} \sum _{p=1}^{M}\left( \frac{1+\beta \Delta \tau }{\min {\{\lambda _{1,q},\lambda _{2,q}\}}}\right) ^{M-p} \le T e^{\beta T}. \end{aligned}$$

\(\square \)

Proof of  Lemma 6

Proof

Replacing the variable W in (12) with w, we get

$$\begin{aligned} \varvec{L}_{i,j}^{m+1}w =\frac{1}{\Delta \tau }\left( w^{m+1}_{i,j}-w^{m}_{i,j}-\frac{\sigma _i^2}{2}\alpha \delta ^2w^{m+1}_{i,j} -\Delta \tau a_{il}w_{l,j}^{m}e^{(-1)^{l}(\xi \tau _m+\eta x_j)}\right) . \end{aligned}$$

Moreover, making use of the Taylor expansions of \(w^{m}_{i,j}\), \(w^{m+1}_{i,j+1}\) and \(w_{i,j-1}^{m+1}\) at the point \((x_{j},\tau _{m+1})\), we obtain

$$\begin{aligned} w^m_{i,j}&=w^{m+1}_{i,j}-\Delta \tau \Bigg (\frac{\partial w}{\partial \tau }\Bigg )^{m+1}_{i,j}+O((\Delta \tau )^2),\\ \delta ^2w^{m+1}_{i,j}&:=w^{m+1}_{i,j+1}-2w^{m+1}_{i,j}+w^{m+1}_{i,j-1}\\&=w^{m+1}_{i,j}+\Delta x\Bigg (\frac{\partial w}{\partial x}\Bigg )^{m+1}_{i,j} +\frac{1}{2}(\Delta x)^2\Bigg (\frac{\partial ^2 w}{\partial x^2}\Bigg )^{m+1}_{i,j} \\ {}&\quad +\frac{1}{6}(\Delta x)^3\Bigg (\frac{\partial ^3 w}{\partial x^3}\Bigg )^{m+1}_{i,j}+O((\Delta x)^4)\\&\quad -2w^{m+1}_{i,j}+w^{m+1}_{i,j}-\Delta x\Bigg (\frac{\partial w}{\partial x}\Bigg )^{m+1}_{i,j} +\frac{1}{2}(\Delta x)^2\Bigg (\frac{\partial ^2 w}{\partial x^2}\Bigg )^{m+1}_{i,j}\\&\quad -\frac{1}{6}(\Delta x)^3\Bigg (\frac{\partial ^3 w}{\partial x^3}\Bigg )^{m+1}_{i,j} +O((\Delta x)^4)\\&=(\Delta x)^2\Bigg (\frac{\partial ^2 w}{\partial x^2}\Bigg )^{m+1}_{i,j} +O((\Delta x)^4). \end{aligned}$$

Hence, for the regime 1, we have

$$\begin{aligned} \gamma ^{m+1}_{1,j}&=\varvec{L}_{1,j}^{m+1}w-\mathcal {L}_{1}w(x_j,\tau _{m+1})\nonumber \\&=\frac{1}{\Delta \tau }\Bigg (w^{m+1}_{1,j}-w^{m+1}_{1,j} +\Delta \tau \left( \frac{\partial w}{\partial \tau }\right) ^{m+1}_{1,j}+O((\Delta \tau )^2) \nonumber \\&\quad -\frac{\sigma _1^2}{2}\alpha \Bigg ((\Delta x)^2\Big (\frac{\partial ^2 w}{\partial x^2}\Big )^{m+1}_{1,j} +O((\Delta x)^4)\Bigg ) \nonumber \\&\quad -\Delta \tau a_{12}w_{2,j}^{m}e^{(-1)^{2}(\xi \tau _{m}+\eta x_j)}\Bigg ) -\frac{\partial w_1}{\partial \tau }\Bigg |_{(x_j,\tau _{m+1})} \nonumber \\&\quad +\frac{\sigma _{1}^{2}}{2}\frac{\partial ^2 w_1}{\partial x^2}\Bigg |_{(x_j,\tau _{m+1})}+ a_{12}w_{2,j}^{m+1}e^{(-1)^{2}(\xi \tau _{m+1}+\eta x_j)} \nonumber \\&=O(\Delta \tau +(\Delta x)^2)+a_{12}e^{(\xi \tau _{m}+\eta x_j)}\left( w_{2,j}^{m+1}e^{\xi \Delta \tau }-w_{2,j}^m\right) . \end{aligned}$$
(28)

Since \(a_{12}e^{(\xi \tau _{m}+\eta x_j)}\) is bounded, applying the Taylor expansion to \(w_2^{m+1}e^{\xi \Delta \tau }-w_2\) yields

$$\begin{aligned}&w_{2,j}^{m+1}e^{\xi \Delta \tau }-w_{2,j}^{m}\\&\quad =w_{2,j}^{m+1}(e^{\xi \Delta \tau }-1)+(w_{2,j}^{m+1}-w_{2,j}^m)\\&\quad =w_{2,j}^{m+1}(\Delta \tau \xi +O((\Delta \tau )^2))+\left( w_{2,j}^{m+1}-w_{2,j}^{m+1}+\Delta \tau \frac{\partial w_2}{\partial \tau }\Big |_{(x_j,\tau _{m+1})}+O((\Delta \tau )^2)\right) \\&\quad =\left( w_{2,j}^{m+1}\xi +\frac{\partial w_2}{\partial \tau }\Big |_{(x_j,\tau _{m+1})}\right) \Delta \tau +(w_{2,j}^{m+1}+1)O((\Delta \tau )^2). \end{aligned}$$

Applying the boundedness of \(w_2\) and \(\dfrac{\partial w_2}{\partial \tau }\), then for (28), we have

$$\begin{aligned} \gamma ^{m+1}_{1,j}\le C_{\gamma }(\Delta \tau +(\Delta x)^2). \end{aligned}$$

By a similar manner for regime 2, we can also conclude that

$$\begin{aligned} \gamma _{2,j}^{m+1}\le C_{\gamma }(\Delta \tau +(\Delta x)^2). \end{aligned}$$

This completes the proof. \(\square \)

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Wen, X., Song, H., Li, Y. et al. A primal-dual active set approach to the valuation of American options in regime-switching models: numerical solutions and convergence analysis. Comp. Appl. Math. 43, 345 (2024). https://doi.org/10.1007/s40314-024-02862-9

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