An integral equation approach for pricing American put options under regime-switching model

Figures & data

Figure 1. The option pricing domain with two optimal exercise boundaries.

Figure 2. The optimal exercise boundaries when the regime-switching model degenerate to Black–Scholes model.

Table 1. Optimal exercise price of perpetual American puts.

Analytic results [59]		Lu and Purti's results [41]		Our results
$S_{f1}$	$S_{f2}$	$S_{f1}$	$S_{f2}$	$S_{f1}$	$S_{f2}$
0.44055	0.61156	0.44059	0.61163	0.44053	0.61129

Table 2. Comparison of the CPU time.

	CPU time
Our equation	0.57 s
Kim's equation	6.74 s

Table 3. Comparison of the CPU time.

	CPU time
Our equation	6.80 s
Monte Carlo simulation	15.4 s

Table 4. Comparison of the American put option price in regime 1.

	$V_1$
S	CN	Tree	MC	Our result
90	14.6919	14.5189	14.5025	14.5021
100	9.9245	9.9059	9.9127	9.8985
110	6.7017	6.7010	6.7026	6.6958
120	4.5257	4.5228	4.5286	4.5255

Table 5. Comparison of the American put option price in regime 2.

	$V_2$
S	CN	Tree	MC	Our result
90	11.6126	11.5425	11.5369	11.4804
100	6.7423	6.7423	6.7401	6.7459
110	3.9244	3.9239	3.9824	3.8905
120	2.3083	2.3088	2.3057	2.2992

Figure 3. The optimal exercise boundaries.

Figure 4. The optimal exercise boundaries with different volatilities($A\in (0,\mathrm{\infty })$).

Table 6. Parameters for regime-switching volatilities.

		${\lambda }_1=0.5$	${\lambda }_2=0.5$	${\lambda }_1=0.3$	${\lambda }_2=0.5$	${\lambda }_1=0.5$	${\lambda }_2=0.3$
${\sigma }_1$	${\sigma }_2$	$S_{f1}$	$S_{f2}$	$S_{f1}$	$S_{f2}$	$S_{f1}$	$S_{f2}$
0.3	0.3	76.6394	76.2191	75.7424	75.8513	77.5621	76.5810
0.3	0.5	72.5664	68.0419	71.1661	67.1328	73.5478	69.3678
0.5	0.3	71.1619	82.4703	69.2605	82.0769	73.3150	82.6605
0.5	0.5	70.6750	71.0693	68.5912	70.2763	72.6235	71.8233

Figure 5. Two optimal exercise boundaries with different transition rates.

Table 7. Optimal exercise decision in a known state.

	State 1			State 2
	S=40	S=20	S=10	S=40	S=20	S=10
$S_{f1}=35$	Continue	Exercise	Exercise
	System 5(5) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_1^2{S}^2\frac{{\mathrm{\partial }}^2{V}_1}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}-r{V}_1={\lambda }_1(V_1-V_2),}\\ V_1(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_1(S,t)=0,\\ V_1(S_{f_1}(t),t)=K-S_{f_1}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}(S_{f_1}(t),t)=-1,}\end{array}\end{array}$(5)	K-S
$S_{f2}=15$				Continue	Continue	Exercise
				System 6(6) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}\end{array}$(6)	System 7(7) $\{\begin{array}{l}{V}_1(S,t)=K-S,\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=K-S,\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}$(7)	K-S

Table 8. Option value when the state is hidden.

	S=40	S=20	S=10
$S_{f1}=35$	Continue	Exercise	Exercise
$V_1$	System 5(5) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_1^2{S}^2\frac{{\mathrm{\partial }}^2{V}_1}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}-r{V}_1={\lambda }_1(V_1-V_2),}\\ V_1(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_1(S,t)=0,\\ V_1(S_{f_1}(t),t)=K-S_{f_1}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}(S_{f_1}(t),t)=-1,}\end{array}\end{array}$(5)	K-S
$S_{f2}=15$	Continue	Continue	Exercise
$V_2$	System 6(6) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}\end{array}$(6)	System 7(7) $\{\begin{array}{l}{V}_1(S,t)=K-S,\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=K-S,\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}$(7)	K-S
V	${\pi }_1{V}_1+{\pi }_2{V}_2$

Q. Zhang and X. Guo, Closed-Form solutions for perpetual American put options with regime switching, SIAM J. Appl. Math. 64 (JAN 2004), pp. 2034–2049.

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X. Lu and E. Putri, A semi-analytic valuation of American options under a two-state regime-switching economy, Phys. A Stat. Mech. Appl. 538 (OCT 2019), pp. 122968.

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