Primal-dual active set method for evaluating American put options on zero-coupon bonds

Abstract

An efficient numerical method is propoesd for a parabolic linear complementarity problem (LCP) arising in the valuation of American options on zero-coupon bonds under the Cox–Ingersoll–Ross (CIR) model. With variable substitutions, we first transform the original pricing problem into a degenerated linear complementarity problem on a bounded domain, and present a corresponding variational inequality (VI). We then give the full discretization scheme of VI constructed by finite element and finite difference methods in spatial and temporal directions, respectively. Within the framework of VI, the stability and the rate of convergence are obtained. Moreover, for the resulted discretised variational inequality, we present a primal-dual active set (PDAS) method to solve it. Numerical results are carried out to test the usefulness of the proposed method compared with existing methods.

The proof of Theorem 4

In this part, we will give the proof of Theorem 4 in detail. At first, let $\eta ^{m}=u^{m}-I_h u^{m}$, we can easily obtain that

$$\begin{aligned} \begin{aligned}&\big (\omega _1\partial e^{m+1},e^{m+1}\big )+a\big (e^{m+1},e^{m+1}\big )\\&=\big (\omega _1\partial e^{m+1},\eta ^{m+1}\big )+a\big (e^{m+1},\eta ^{m+1}\big )\\&\quad +\big (\omega _1\partial u^{m+1},I_h u^{m+1}-u_{h,\tau }^{m+1}\big )+a\big (u^{m+1},I_h u^{m+1}-u_{h,\tau }^{m+1}\big )\\&\quad -\big (\omega _1\partial u_{h,\tau }^{m+1},I_h u^{m+1}-u_{h,\tau }^{m+1}\big )-a\big (u_{h,\tau }^{m+1},I_h u^{m+1}-u_{h,\tau }^{m+1}\big ). \end{aligned} \end{aligned}$$

(A.1)

Taking $v=u_{h,\tau }^{m+1}$ in (8) and $v_h=I_h u^{m+1}$ in (11) for $\tau =\tau _{m+1}$

$$\begin{aligned} \Bigg (\omega _1 \frac{\partial u^{m+1}}{\partial \tau },u_{h,\tau }^{m+1}-u^{m+1}\Bigg )+a\Bigg (u^{m+1},u_{h,\tau }^{m+1}-u^{m+1}\Bigg )\ge & {} 0,\\ \Bigg (\omega _1\partial u_{h,\tau }^{m+1},I_h u^{m+1}-u_{h,\tau }^{m+1}\Bigg )+a\Bigg (u_{h,\tau }^{m+1},I_h u^{m+1}-u_{h,\tau }^{m+1}\Bigg )\ge & {} 0. \end{aligned}$$

Adding the so-obtained inequalities to (A.1), we find by easy calculation that

$$\begin{aligned} \begin{aligned}&\Bigg (\omega _1\partial e^{m+1},e^{m+1}\Bigg )+a\Bigg (e^{m+1},e^{m+1}\Bigg )\\ \le&\Bigg (\omega _1\partial e^{m+1},\eta ^{m+1}\Bigg )+a\Bigg (e^{m+1},\eta ^{m+1}\Bigg )-\Bigg (\omega _1\partial u^{m+1},\eta ^{m+1}\Bigg )\\&-a\Bigg (u^{m+1},\eta ^{m+1}\Bigg )-\Bigg (\omega _1\Bigg (\frac{\partial u^{m+1}}{\partial \tau }-\partial u^{m+1}\Bigg ),e^{m+1}\Bigg )\\ \le&\sum _{j=1}^4p_m^j, \end{aligned} \end{aligned}$$

(A.2)

where

$$\begin{aligned}{} & {} p_m^1=\big (\omega _1\partial e^{m+1},\eta ^{m+1}\big ),~~p_m^2=a\big (e^{m+1},\eta ^{m+1}\big ),\\{} & {} p_m^3=-\big (\omega _1\partial u^{m+1},\eta ^{m+1}\big )-a\big (u^{m+1},\eta ^{m+1}\big ),~~p_m^4=-\Bigg (\omega _1\Bigg (\frac{\partial u^{m+1}}{\partial \tau }-\partial u^{m+1}\Bigg ),e^{m+1}\Bigg ). \end{aligned}$$

Multiplication by $\Delta \tau$ and summation then gives:

$$\begin{aligned} \sum _{m=0}^{M-1}\Bigg (\omega _1 (e^{m+1}-e^m),e^{m+1}\Bigg )+\sum _{m=0}^{M-1}a(e^{m+1},e^{m+1})\Delta \tau \le \sum _{j=1}^4 E_j, \end{aligned}$$

(A.3)

where

$$\begin{aligned} E_j=\sum _{m=0}^{M-1}|p_m^j|\Delta \tau . \end{aligned}$$

The first term of (A.3) can be simplified that:

$$\begin{aligned} \begin{aligned}&\sum _{m=0}^{M-1}\Bigg (\omega _1 \big (e^{m+1}-e^m\big ),e^{m+1}\Bigg )=\sum _{m=0}^{M-1}\big (\omega _1 e^{m+1},e^{m+1}\big )-\sum _{m=0}^{M-1}\big (\omega _1 e^{m},e^{m+1}\big )\\ =&\sum _{m=0}^{M-1}\Vert e^{m+1}\Vert ^2_{0,\omega _1}-\sum _{m=0}^{M-1}\big (\omega _1e^m,e^{m+1}\big )\\ \ge&\sum _{m=0}^{M-1}\Vert e^{m+1}\Vert _{0,\omega _1}^2-\sum _{m=0}^{M-1}\frac{\Vert e^{m+1}\Vert _{0,\omega _1}^2+\Vert e^m\Vert _{0,\omega _1}^2}{2}=\frac{\Vert e^M\Vert _{0,\omega _1}^2-\Vert e^0\Vert _{0,\omega _1}^2}{2}, \end{aligned} \end{aligned}$$

which implies that

$$\begin{aligned} 2\sum _{m=0}^{M-1}a\big (e^{m+1},e^{m+1}\big )\Delta \tau \le 2\sum _{j=1}^4 E_j-\Vert e^M\Vert _{0,\omega _1}^2+\Vert e^0\Vert _{0,\omega _1}^2. \end{aligned}$$

(A.4)

Now, we will give the estimation of $E_j,~~j=1,\cdots ,4$, respectively. (1) The estimation of $E_1$ Summation by parts gives

$$\begin{aligned} \sum _{m=0}^{M-1}\big (\omega _1\partial e^{m+1},\eta ^{m+1}\big )\Delta \tau =-\sum _{m=0}^{M-1}\big (\omega _1 e^{m+1},\partial \eta ^{m+1}\big )\Delta \tau +\big (\omega _1 e^M,\eta ^M\big )-\big (\omega _1 e^0,\eta ^0\big ). \end{aligned}$$

Now, by the following conclusions (Johnson 1976; Vuik 1990):

$$\begin{aligned}{} & {} \Bigg \Vert \frac{\partial \eta }{\partial \tau }\Bigg \Vert _{L^2(J;H^2(\Omega ))}\le Ch\Bigg \Vert \frac{\partial u}{\partial \tau }\Bigg \Vert _{L^2(J;H^1(\Omega ))},\\{} & {} \Vert \partial \eta ^m\Vert _0\le C(\Delta \tau )^{-\frac{1}{2}}h\Bigg \Vert \frac{\partial u}{\partial \tau }\Bigg \Vert _{L^2(J;H^1(\Omega ))},\\{} & {} \max _m\Vert \eta ^m\Vert _0\le Ch\Vert u\Vert _{L^\infty (J;H^1(\Omega ))}, \end{aligned}$$

we can easily obtain that

$$\begin{aligned}{} & {} \Vert \partial \eta ^m\Vert _{0,\omega _1}^2\le \Vert \omega _1\Vert _{L^\infty } \Vert \partial \eta ^m\Vert _0^2\le C(\Delta \tau )^{-1}h^2\Bigg \Vert \frac{\partial u}{\partial \tau }\Bigg \Vert ^2_{L^2(J;H^1(\Omega ))},\\{} & {} \max _m\Vert \eta ^m\Vert _{0,\omega _1}\le Ch\Vert u\Vert _{L^\infty (J;H^1(\Omega ))}. \end{aligned}$$

Moreover, by the equation (10), we obtain that

$$\begin{aligned} a(v,v)\ge C\Vert v\Vert _{1,\omega _2}^2\ge \tilde{C}\Vert v\Vert _{0,\omega _1}^2,~~\forall v\in H^1_0(\Omega ). \end{aligned}$$

(A.5)

where

$$\begin{aligned} \tilde{C}=\min \Bigg (\frac{\sigma ^2}{2},1\Bigg )\cdot {\frac{\nu }{1+\delta }} \end{aligned}$$

Therefore, using Cauchy’s inequality and the above inequations, we find that

$$\begin{aligned} E_1\le & {} \sum _{m=0}^{M-1}\big |(\omega _1e^{m+1},\partial \eta ^{m+1})\big |\Delta \tau +\big |(\omega _1 e^M,\eta ^M)\big |+\big |(\omega _1e^0,\eta ^0)\big |\\\le & {} \frac{1}{8}\sum _{m=0}^{M-1}\Vert e^m\Vert _{0,\omega _1}^2\Delta \tau +2\sum _{m=0}^{M-1}\Vert \partial \eta ^{m+1}\Vert _{0,\omega _1}^2\Delta \tau +\frac{1}{8}\Vert e^M\Vert _{0,\omega _1}^2\\{} & {} +\Vert e^0\Vert _{0,\omega _1}^2+2\Vert \eta ^M\Vert _{0,\omega _1}^2+\frac{1}{4}\Vert \eta ^0\Vert _{0,\omega _1}^2\\\le & {} \frac{1}{8}\sum _{m=0}^{M-1}\Vert e^m\Vert _{0,\omega _1}^2\Delta \tau +Ch^2\Bigg \Vert \frac{\partial u}{\partial \tau }\Bigg \Vert _{{L^2(J;H^1(\Omega ))}}^2+\frac{1}{8}\Vert e^M\Vert _{0,\omega _1}^2\\{} & {} +Ch^2\Vert u\Vert _{{L^\infty (J;H^1(\Omega ))}}^2+\Vert e^0\Vert _{0,\omega _1}^2\\\le & {} \frac{\tilde{C_1}}{8}\sum _{m=0}^{M-1}a({e^{m+1},e^{m+1}})\Delta \tau +\frac{1}{8}\Vert e^M\Vert _{0,\omega _1}^2+\Vert e^0\Vert _{0,\omega _1}^2\\{} & {} +Ch^2\Bigg \Vert \frac{\partial u}{\partial \tau }\Bigg \Vert _{L^2(J;H^1(\Omega ))}^2+Ch^2\Vert u\Vert _{L^\infty (J;H^1(\Omega ))}^2. \end{aligned}$$

(2) The estimation of $E_2$

$$\begin{aligned} E_2= & {} \sum _{m=0}^{M-1}\big |a(e^{m+1},\eta ^{m+1})\big |\Delta \tau \\\le & {} \left\{ \frac{1}{8}\sum _{m=0}^{M-1}\left[ \frac{\sigma ^2}{2}\big (\omega _2e_r^{m+1},e_r^{m+1}\big )]+2\sum _{m=0}^{M-1}[\frac{\sigma ^2}{2}\big (\omega _2\eta _r^{m+1},\eta _r^{m+1}\big )\right] \right. \\{} & {} \left. +\frac{1}{8}\sum _{m=0}^{M-1}\big (\omega _2e^{m+1},e^{m+1}\big )+2\sum _{m=0}^{M-1}\big (\omega _2\eta ^{m+1},\eta ^{m+1}\big )\right\} \Delta \tau \\\le & {} \frac{1}{8}\sum _{m=0}^{M-1}a\big (e^{m+1},e^{m+1}\big )\Delta \tau +C\Vert \eta ^{m+1}\Vert _{1,\omega _2}^2\\\le & {} \frac{1}{8}\sum _{m=0}^{M-1}a\big (e^{m+1},e^{m+1}\big )\Delta \tau +Ch^2\Vert u\Vert ^2_{L^\infty (J;H^2(\Omega )}. \end{aligned}$$

(3) The estimation of $E_3$

$$\begin{aligned} E_3= & {} \sum _{m=0}^{M-1}\big |-(\omega _1\partial u^{m+1},\eta ^{m+1})-a(u^{m+1},\eta ^{m+1})\big |\Delta \tau . \end{aligned}$$

Integration by parts for the second one:

$$\begin{aligned}{} & {} \sum _{m=0}^{M-1}\left\{ |(\omega _1\partial u^{m+1},\eta ^{m+1})|\Delta \tau +|a(u^{m+1},\eta ^{m+1})|\Delta \tau \right\} \\ \le{} & {} \sum _{m=0}^{M-1}\Big (\Vert \partial u^{m+1}\Vert _{0,\omega _1}\cdot \Vert \eta ^{m+1}\Vert _{0,\omega _1}+C_{1}\Vert u^{m+1}\Vert _{1,\omega _2}\cdot \Vert \eta ^{m+1}\Vert _{1,\omega _2}\Big )\Delta \tau \\ \le{} & {} \sum _{m=0}^{M-1}C\Big (\Vert \partial u^{m+1}\Vert _{0,\omega _1}+\Vert u^{m+1}\Vert _{1,\omega _2}\Big )\cdot \Vert \eta ^{m+1}\Vert _{1,\omega _2}\Delta \tau . \end{aligned}$$

Therefore, the estimation of $E_3$ is as follows:

$$\begin{aligned} E_3\le{} & {} \sum _{m=0}^{M-1}C(\Vert \partial u^{m+1}\Vert _{0,\omega _1}+\Vert u^{m+1}\Vert _{1,\omega _2})\cdot \Vert \eta ^{m+1}\Vert _{1,\omega _2}\Delta \tau \\ \le{} & {} \sum _{m=0}^{M-1}Ch^2\left( \Vert \partial u^{m+1}\Vert _{L^\infty (J;L^\infty (\Omega ))}+\Vert u\Vert _{L^\infty (J;H^2(\Omega ))}\right) \cdot \Vert u\Vert _{L^\infty (J;H^2(\Omega ))}. \end{aligned}$$

(4) The estimation of $E_4$

$$\begin{aligned} E_4\le \sum _{m=0}^{M-1}\left| \left( \omega _1\left[ \partial u^{m+1}-\frac{\partial u^{m+1}}{\partial \tau }\right] ,e^{m+1}\right) \right| \Delta \tau . \end{aligned}$$

By means of equation (A.5) and Cauchy’s inequality, we obtain that:

$$\begin{aligned} \left( \omega _1\left[ \partial u^{m+1}-\frac{\partial u^{m+1}}{\partial \tau }\right] ,e^{m+1}\right)\le & {} \frac{1}{4}\big (\omega _1e^{m+1},e^{m+1}\big )\\ {}{} & {} \quad +\left( \omega _1\left[ \partial u^{m+1}-\frac{\partial u^{m+1}}{\partial \tau }\right] ,\partial u^{m+1}-\frac{\partial u^{m+1}}{\partial \tau }\right) \\\le & {} \frac{\tilde{C_1}}{4}a\big (e^{m+1},e^{m+1}\big )+\left\| \partial u^{m+1}-\frac{\partial u^{m+1}}{\partial \tau }\right\| _{0,\omega _1}^2. \end{aligned}$$

For the item

$$\begin{aligned} \left\| \partial u^{m+1}-\frac{\partial u^{m+1}}{\partial \tau }\right\| _{0,\omega _1}{} & {} =\left\| \frac{u^{m+1}-u^m}{\Delta \tau }-\frac{\partial u}{\partial \tau }(\tau _{m+1})\right\| _{0,\omega _1}\\{} & {} =\left\| \frac{1}{\Delta \tau }\int _{\tau _m}^{\tau _{m+1}}\left[ \frac{\partial u}{\partial \tau }(\tau )-\frac{\partial u}{\partial \tau }(\tau _{m+1})\right] d\tau \right\| _{0,\omega _1}\\{} & {} =\left\| \frac{1}{\Delta \tau }\int _{\tau _m}^{\tau _{m+1}}\left[ \int _{\tau _{m+1}}^\tau \frac{\partial ^2 u}{\partial \tau ^2}ds\right] d\tau \right\| _{0,\omega _1}\\{} & {} \le \frac{1}{\Delta \tau }\int _{\tau _m}^{\tau _{m+1}}\int _{\tau _m}^{\tau _{m+1}}\left\| \frac{\partial ^2 u}{\partial \tau ^2}\right\| _{0,\omega _1}dsd\tau , \end{aligned}$$

according to the literature (Vuik 1990), we similarly obtain that

$$\begin{aligned} \int _{\tau _m}^{\tau _{m+1}}\left\| \frac{\partial ^2 u}{\partial \tau ^2}\right\| _{0,\omega _1}ds\le {\sqrt{\Delta \tau }}\left\| \frac{\partial ^2 u}{\partial \tau ^2}\right\| _{0,\omega _1}. \end{aligned}$$

Therefore, the estimation of $E_4$ is as follows:

$$\begin{aligned} E_4\le \frac{\tilde{C_1}}{4}\sum _{m=0}^{M-1}a(e^{m+1},e^{m+1})\Delta \tau +C(\Delta \tau )^2\left\| \frac{\partial ^2 u}{\partial \tau ^2}\right\| _{L^2(J;H^2(\Omega ))}^2. \end{aligned}$$

So far, we have obtained the estimates of $E_j,~j=1,2,3,4$. Now, by calculating the right end of (A.4), we have

$$\begin{aligned}{} & {} \left( \frac{7-3\tilde{C_1}}{4}\right) \sum _{m=0}^{M-1}a\big (e^{m+1},e^{m+1}\big )\Delta \tau \le 3\Vert \text {e}^0\Vert _{0,\omega _1}^2-\frac{3}{4}\Vert e^M\Vert _{0,\omega _1}^2+C(h^2+(\Delta \tau )^2)\\ \end{aligned}$$

It is easily obtained from the assumption that

$$\begin{aligned} \sum _{m=0}^{M-1}a\big (e^{m+1},e^{m+1}\big )\Delta \tau \le{} & {} C\left\{ \Vert \text {e}^0\Vert _{0,\omega _1}^2+(h^2+(\Delta \tau )^2)\right\} \\ \end{aligned}$$

According to Poincare inequality, we have

$$\begin{aligned} \sum _{m=0}^{M-1}a\big (e^{m+1},e^{m+1}\big )\Delta \tau \ge \gamma \sum _{m=0}^{M-1}\Vert e^{m+1}\Vert _1^2\Delta \tau . \end{aligned}$$

Combining the above equations, we obtain

$$\begin{aligned} \sum _{m=0}^{M-1}\Vert e^{m+1}\Vert _1^2\Delta \tau \le{} & {} C\left\{ \Vert \text {e}^0\Vert _{0,\omega _1}^2+(h^2+(\Delta \tau )^2)\right\} . \end{aligned}$$

Finally, using the complex trapezoidal formula, so that

$$\begin{aligned} \Vert u-u_{h,\tau }\Vert _{L^2(J;H^1(\Omega ))}^2= & {} \sum _{m=0}^{M-1}\int _{\tau _m}^{\tau _{m+1}}\Vert u-u_{h,\tau }\Vert _{1}^2d\tau \\= & {} \sum _{m=0}^{M-1}\frac{\Delta \tau (\Vert \text {e}^{m+1}\Vert _1^2+\Vert \text {e}^{m}\Vert _1^2)}{2}+C(\Delta \tau )^2\\= & {} \sum _{m=0}^{M-1}\Delta \tau \Vert \text {e}^{m+1}\Vert _1^2+\frac{1}{2}\Delta \tau \Vert \text {e}^0\Vert _1^2-\frac{1}{2}\Delta \tau \Vert \text {e}^M\Vert _1^2+C(\Delta \tau )^2\\\le & {} C(h^2+(\Delta \tau )^2)+C\Vert \text {e}^0\Vert _0^2+\frac{1}{2}\Delta \tau \Vert \text {e}^0\Vert _1^2\\\le & {} C(h^2+(\Delta \tau )^2)+Ch^4|u^0|_2^2+\frac{1}{2}Ch^2\Delta \tau |u^0|_2^2\\\le & {} C(h+\Delta \tau )^2.\\ \end{aligned}$$