Pricing and hedging defaultable participating contracts with regime switching and jump risk

Abstract

This paper develops a transform-based approach for the pricing of participating life insurance contracts with a constant or floating guaranteed rate. Our analysis incorporates credit, market (jump), and economic (regime switching) risks, where the evolution of the reference portfolio is described by a regime switching double exponential jump-diffusion model. We provide semi-analytical formulas for the contract value by using a Laplace or Laplace–Fourier transform that involves matrix Wiener–Hopf factors. Then, the price is obtained by implementing the matrix Wiener–Hopf factorization and by performing a numerical Laplace and Fourier inversion. By comparing the results with Monte Carlo simulations, we show that our pricing method is easy to implement and accurate. We also show that the contract with a floating guaranteed rate is riskier but more profitable than the contract with a constant guaranteed rate. Two hedging strategies are introduced to hedge jump and regime switching risks in the participating contracts.

Appendix

1.1 Computation of subcontract terms for the constant guarantee contract

Using Proposition 1 and Lemma 1, we show that, for any $u>0$,

$$\begin{aligned} \widehat{\mathrm{GP}}(u)= & {} L_0 \int \limits _0^\infty \mathrm{e}^{-uT} E_{\mathbb {Q}}\left( \mathrm{e}^{-\int \limits _0^T (r_s-\tilde{r}_g) \mathrm {d}s} mathbb{1}_{\tau \ge T}\right) \mathrm {d}T \\= & {} L_0 E_{\mathbb {Q}}\left( \int \limits _0^{\tau } \mathrm{e}^{-\int \limits _0^T (r_s+u-\tilde{r}_g) \mathrm {d}s} \mathrm {d}T \right) \\= & {} L_0 E_{\mathbb {Q}} \left( \int \limits _0^{\infty } \mathrm{e}^{-\int \limits _0^T (r_s+u-\tilde{r}_g) \mathrm {d}s} \mathrm {d}T- \mathrm{e}^{-\int \limits _0^{\tau } (r_s+u-\tilde{r}_g) \mathrm {d}s} \int \limits _0^{\infty }E_{\mathbb {Q}}\right. \\&\left. \left. \left( \mathrm{e}^{-\int \limits _{\tau }^{\tau +T} (r_s+u-\tilde{r}_g) \mathrm {d}s}\right| \mathscr {F}_{\tau } \right) \mathrm {d}T\right) \\= & {} L_0 \left( Y_0 W^{(\hat{r}+u-\tilde{r}_g,-)}_{Z} \mathrm{e}^{Q^{(\hat{r}+u-\tilde{r}_g,-)}_{Z} \left( x-\ln \dfrac{\lambda L_0}{A_0}\right) }H(0)-J_0\right) \\&(Q-\text {diag}\{\hat{r}+u-\tilde{r}_g\})^{-1}mathbb{1}_n \end{aligned}$$

and

$$\begin{aligned} \widehat{\mathrm{RP}}(u)= & {} \int \limits _0^\infty \mathrm{e}^{-uT} E_{\mathbb {Q}}\left( \mathrm{e}^{-\int \limits _0^{\tau } r_s \mathrm {d}s}A_{\tau }mathbb{1}_{\tau < T} \right) \mathrm {d}T \\= & {} A_0 E_{\mathbb {Q}}\left( \int \limits _{\tau }^{\infty } \mathrm{e}^{-uT} \mathrm{e}^{-\int \limits _0^{\tau } (r_s-\tilde{r}_g) \mathrm {d}s} \mathrm{e}^{Z_{\tau }} \mathrm {d}T \right) \\= & {} \dfrac{A_0}{u}E_{\mathbb {Q}}\left( \mathrm{e}^{-\int \limits _0^{\tau } (r_s+u-\tilde{r}_g) \mathrm {d}s+Z_{\tau }}\right) \\= & {} \dfrac{A_0}{u} Y_0 W^{(\hat{r}+u-\tilde{r}_g,-)}_{Z} \mathrm{e}^{Q^{(\hat{r}+u-\tilde{r}_g,-)}_{Z} \left( x-\ln \dfrac{\lambda L_0}{A_0}\right) + \ln \dfrac{\lambda L_0}{A_0} I_{2n}} H(1)mathbb{1}_{n}. \end{aligned}$$

For any $1<\alpha _1<\tilde{\theta }+1$ and $u>\max (\mathfrak {R}(\phi ^Z_1(i v +1 -\alpha _1))+\tilde{r}_g-\hat{r}_1, \ldots , \mathfrak {R}(\phi ^Z_n(i v +1 -\alpha _1))+\tilde{r}_g-\hat{r}_n,0)$, we obtain:

$$\begin{aligned} \widetilde{\widehat{\mathrm{DO}}}(u,v)= & {} -A_0 \int \limits _0^\infty \mathrm{e}^{-u T} \int \limits _{-\infty }^\infty \mathrm{e}^{i v z_1}\mathrm{e}^{-\alpha _1 z_1}E_{\mathbb {Q}}\left( \mathrm{e}^{-\int \limits _0^T (r_s-\tilde{r}_g) \mathrm {d}s}\left( \mathrm{e}^{z_1}-\mathrm{e}^{Z_T}\right) _+ mathbb{1}_{\tau \ge T} \right) \mathrm {d}z_1\mathrm {d}T\\= & {} -A_0 E_{\mathbb {Q}}\left( \int \limits _0^{\infty } \mathrm{e}^{-\int \limits _0^T (r_s+u-\tilde{r}_g) \mathrm {d}s}mathbb{1}_{\tau \ge T} \int \limits _{Z_T}^{\infty }\mathrm{e}^{-(\alpha _1-iv) z_1}(\mathrm{e}^{z_1}-\mathrm{e}^{Z_T})\mathrm {d}{z_1}\mathrm {d}T \right) \\= & {} -\dfrac{A_0}{(\alpha _1-i v)(\alpha _1-i v -1)} E_{\mathbb {Q}}\left( \int \limits _0^{\infty } \mathrm{e}^{-\int \limits _0^T (r_s+u-\tilde{r}_g) \mathrm {d}s} mathbb{1}_{\tau \ge T}\mathrm{e}^{-(\alpha _1-iv-1)Z_T}\mathrm {d}T \right) \\= & {} -\dfrac{A_0}{(\alpha _1-i v)(\alpha _1-i v -1)} \left( E_{\mathbb {Q}}\left( \int \limits _0^{\infty } \mathrm{e}^{-\int \limits _0^T (r_s+u-\tilde{r}_g) \mathrm {d}s} \mathrm{e}^{-(\alpha _1-i v-1)Z_T}\mathrm {d}T\right) \right. \\&\quad -\sum \limits _{k=1}^n E_{\mathbb {Q}}\left( \mathrm{e}^{-\int \limits _0^{\tau } (r_s+u-\tilde{r}_g) \mathrm {d}s} \mathrm{e}^{-(\alpha _1-i v-1)Z_{\tau }} mathbb{1}_{J_{\tau }=e_k}\right. \\&\quad \left. \left. E_{\mathbb {Q}}\left( \left. \int \limits _0^{\infty } \mathrm{e}^{-\int \limits _{\tau }^{\tau +T} (r_s+u-\tilde{r}_g) \mathrm {d}s} \mathrm{e}^{-(\alpha _1-i v-1)(Z_{T+\tau }-Z_{\tau })}\mathrm {d}T\right| J_{\tau }=e_k\right) \right) \right) \\= & {} -\dfrac{A_0}{(\alpha _1-i v)(\alpha _1-i v -1)} \left( Y_0 W^{(\hat{r}+u-\tilde{r}_g,-)}_{Z} \mathrm{e}^{Q^{(\hat{r}+u-\tilde{r}_g,-)}_{Z} \left( x-\ln \dfrac{\lambda L_0}{A_0}\right) -(\alpha _1-i v-1)\ln \dfrac{\lambda L_0}{A_0} I_{2n}} \right. \\&\quad \left. H(i v+1-\alpha _1)-J_0\right) (Q-\text {diag}\{\hat{r}+u-\tilde{r}_g-\phi ^Z_k(i v +1 -\alpha _1)\})^{-1}mathbb{1}_n, \end{aligned}$$

where we use the strong Markov property of Z in the fourth equality and we obtain the fifth equality by using Proposition 1 and Lemma 1.

For any $\alpha _2>0$ and $u>\max (\mathfrak {R}(\phi ^Z_1(\alpha _2-i v+1))+\tilde{r}_g-\hat{r}_1,\ldots ,\mathfrak {R}(\phi ^Z_n(\alpha _2-i v+1))+\tilde{r}_g-\hat{r}_n,0)$, the same derivation gives $\widetilde{\widehat{\mathrm{GB}}}$ as follows:

$$\begin{aligned} \widetilde{\widehat{ \mathrm{GB}}}(u,v)= & {} \delta \alpha A_0 \int \limits _0^\infty \mathrm{e}^{-u T} \int \limits _{-\infty }^\infty \mathrm{e}^{i v z_2}\mathrm{e}^{-\alpha _2 z_2} E_{\mathbb {Q}}\left( \mathrm{e}^{-\int \limits _0^T (r_s-\tilde{r}_g) \mathrm {d}s} \left( \mathrm{e}^{Z_T}-\mathrm{e}^{-z_2}\right) _+ mathbb{1}_{\tau \ge T} \right) \mathrm {d}z_2\mathrm {d}T\\= & {} \delta \alpha A_0 E_{\mathbb {Q}}\left( \int \limits _0^{\infty } \mathrm{e}^{-\int \limits _0^T (r_s+u-\tilde{r}_g) \mathrm {d}s} mathbb{1}_{\tau \ge T} \int \limits _{-Z_T}^{\infty }\mathrm{e}^{-(\alpha _2-iv) z_2}(\mathrm{e}^{Z_T}-\mathrm{e}^{-z_2})\mathrm {d}{z_2}\mathrm {d}T \right) \\= & {} \dfrac{\delta \alpha A_0}{(\alpha _2-i v)(\alpha _2-i v +1)} E_{\mathbb {Q}}\left( \int \limits _0^{\infty } \mathrm{e}^{-\int \limits _0^T (r_s+u-\tilde{r}_g) \mathrm {d}s} mathbb{1}_{\tau \ge T}\mathrm{e}^{(\alpha _2-iv+1)Z_T}\mathrm {d}T \right) \\= & {} \dfrac{\delta \alpha A_0}{(\alpha _2-i v)(\alpha _2-i v +1)} \left( E_{\mathbb {Q}}\left( \int \limits _0^{\infty } \mathrm{e}^{-\int \limits _0^T (r_s+u-\tilde{r}_g) \mathrm {d}s} \mathrm{e}^{(\alpha _2-i v+1)Z_T}\mathrm {d}T\right) \right. \\&\quad -\sum \limits _{k=1}^n E_{\mathbb {Q}}\left( \mathrm{e}^{-\int \limits _0^{\tau } (r_s+u-\tilde{r}_g) \mathrm {d}s} \mathrm{e}^{(\alpha _2-i v+1)Z_{\tau }} mathbb{1}_{J_{\tau }=e_k} \right. \\&\quad \left. \left. E_{\mathbb {Q}}\left( \left. \int \limits _0^{\infty } \mathrm{e}^{-\int \limits _{\tau }^{\tau +T} (r_s+u-\tilde{r}_g) \mathrm {d}s} \mathrm{e}^{(\alpha _2-i v+1)(Z_{T+\tau }-Z_{\tau })}\mathrm {d}T\right| J_{\tau }=e_k\right) \right) \right) \\= & {} \dfrac{\delta \alpha A_0}{(\alpha _2-i v)(\alpha _2-i v +1)} \left( Y_0 W^{(\hat{r}+u-\tilde{r}_g,-)}_{Z} \mathrm{e}^{Q^{(\hat{r}+u-\tilde{r}_g,-)}_{Z} \left( x-\ln \dfrac{\lambda L_0}{A_0}\right) +(\alpha _2-i v+1)\ln \dfrac{\lambda L_0}{A_0} I_{2n}} \right. \\&\quad \left. H(\alpha _2-i v+1)-J_0\right) (Q-\text {diag}\{\hat{r}+u-\tilde{r}_g-\phi ^Z_k(\alpha _2-i v+1)\})^{-1}mathbb{1}_n. \end{aligned}$$

1.2 Computation of subcontract terms for the floating guarantee contract

For any $u>0$, we have:

$$\begin{aligned} \widehat{\mathrm{GP}}(u)= & {} \int \limits _0^\infty \mathrm{e}^{-uT} L_0 \mathrm{e}^{r^f T} \mathbb {Q}(\tau \ge T) \mathrm {d}T \\= & {} L_0 E_{\mathbb {Q}}\left( \int \limits _0^{\tau } \mathrm{e}^{-(u-r^f)T} \mathrm {d}T \right) \\= & {} \dfrac{L_0}{u-r^f}\left( 1-E_{\mathbb {Q}}\left( \mathrm{e}^{-(u-r^f)\tau }\right) \right) \\= & {} \dfrac{L_0}{u-r^f}\left( 1-Y_0 W^{(\tilde{u}-r^f,-)}_Z \mathrm{e}^{Q^{(\tilde{u}-r^f,-)}_Z\left( x-\ln \dfrac{\lambda L_0}{A_0}\right) }H(0)mathbb{1}_{n}\right) , \end{aligned}$$

and

$$\begin{aligned} \widehat{\mathrm{RP}}(u)= & {} \int \limits _0^\infty \mathrm{e}^{-uT} E_{\mathbb {Q}}\left( A_0 \mathrm{e}^{Z_{\tau }+r^f \tau }mathbb{1}_{\tau < T} \right) \mathrm {d}T \\= & {} A_0 E_{\mathbb {Q}}\left( \int \limits _{\tau }^{\infty } \mathrm{e}^{-uT} \mathrm{e}^{Z_{\tau }+r^f \tau } \mathrm {d}T \right) \\= & {} \dfrac{A_0}{u}E_{\mathbb {Q}}\left( \mathrm{e}^{-(u-r^f)\tau +Z_{\tau }}\right) \\= & {} \dfrac{A_0}{u} Y_0 W^{(\tilde{u}-r^f,-)}_Z \mathrm{e}^{Q^{(\tilde{u}-r^f,-)}_Z\left( x-\ln \dfrac{\lambda L_0}{A_0}\right) + \ln \dfrac{\lambda L_0}{A_0} I_{2n}} H(1)mathbb{1}_{n}. \end{aligned}$$

For any $1<\alpha _1<\tilde{\theta }+1$ and $u>\max (\mathfrak {R}(\phi ^Z_1(i v +1 -\alpha _1))+r^f,\ldots ,\mathfrak {R}(\phi ^Z_n(i v +1 -\alpha _1))+r^f,0)$, we obtain:

$$\begin{aligned} \widetilde{\widehat{ \mathrm{DO}}}(u,v)= & {} -A_0 \int \limits _0^\infty \mathrm{e}^{-u T} \int \limits _{-\infty }^\infty \mathrm{e}^{i v z_1}\mathrm{e}^{-\alpha _1 z_1} \mathrm{e}^{r^f T} E_{\mathbb {Q}}\left( \left( \mathrm{e}^{z_1}-\mathrm{e}^{Z_T}\right) _+ mathbb{1}_{\tau \ge T} \right) \mathrm {d}z_1\mathrm {d}T\\= & {} -A_0 E_{\mathbb {Q}}\left( \int \limits _0^{\infty } \mathrm{e}^{-(u-r^f) T}mathbb{1}_{\tau \ge T} \int \limits _{Z_T}^{\infty }\mathrm{e}^{-(\alpha _1-iv) z_1}(\mathrm{e}^{z_1}-\mathrm{e}^{Z_T})\mathrm {d}{z_1}\mathrm {d}T \right) \\= & {} -\dfrac{A_0}{(\alpha _1-i v)(\alpha _1-i v -1)} E_{\mathbb {Q}}\left( \int \limits _0^{\infty } \mathrm{e}^{-(u-r^f)T} mathbb{1}_{\tau \ge T}\mathrm{e}^{-(\alpha _1-iv-1)Z_T}\mathrm {d}T \right) \\= & {} -\dfrac{A_0}{(\alpha _1-i v)(\alpha _1-i v -1)} \left( E_{\mathbb {Q}}\left( \int \limits _0^{\infty } \mathrm{e}^{-(u-r^f)T}\mathrm{e}^{-(\alpha _1-i v-1)Z_T}\mathrm {d}T\right) \right. \\&\quad \left. -\sum \limits _{k=1}^n E_{\mathbb {Q}}\left( \mathrm{e}^{-(u-r^f) \tau } \mathrm{e}^{-(\alpha _1-i v-1)Z_{\tau }} mathbb{1}_{J_{\tau }=e_k}E_{\mathbb {Q}} \right. \right. \\&\left. \left. \left( \left. \int \limits _0^{\infty } \mathrm{e}^{-(u-r^f) T}\mathrm{e}^{-(\alpha _1-i v-1)(Z_{T+\tau }-Z_{\tau })}\mathrm {d}T\right| J_{\tau }=e_k\right) \right) \right) \\= & {} -\dfrac{A_0}{(\alpha _1-i v)(\alpha _1-i v -1)} \left( Y_0 W^{(\tilde{u}-r^f,-)}_Z \mathrm{e}^{Q^{(\tilde{u}-r^f,-)}_Z \left( x-\ln \dfrac{\lambda L_0}{A_0}\right) -(\alpha _1-i v-1)\ln \dfrac{\lambda L_0}{A_0} I_{2n}} \right. \\&\quad \left. H(i v+1-\alpha _1)-J_0\right) (Q-\text {diag}\{u-r^f-\phi ^Z_k(i v +1 -\alpha _1)\})^{-1}mathbb{1}_n. \end{aligned}$$

For any $\alpha _2>0$ and $u>\max (\mathfrak {R}(\phi ^Z_1(\alpha _2-i v+1))+r^f,\ldots ,\mathfrak {R}(\phi ^Z_n(\alpha _2-i v+1))+r^f,0)$, we also obtain:

$$\begin{aligned} \widetilde{\widehat{\mathrm{GB}}}(u,v)= & {} \delta \alpha A_0 \int \limits _0^\infty \mathrm{e}^{-u T} \int \limits _{-\infty }^\infty \mathrm{e}^{i v z_2}\mathrm{e}^{-\alpha _2 z_2} \mathrm{e}^{r^f T}E_{\mathbb {Q}}\left( \left( \mathrm{e}^{Z_T}-\mathrm{e}^{-z_2}\right) _+ mathbb{1}_{\tau \ge T} \right) \mathrm {d}z_2\mathrm {d}T\\= & {} \delta \alpha A_0 E_{\mathbb {Q}}\left( \int \limits _0^{\infty } \mathrm{e}^{-(u-r^f) T}mathbb{1}_{\tau \ge T} \int \limits _{-Z_T}^{\infty }\mathrm{e}^{-(\alpha _2-iv) z_2}(\mathrm{e}^{Z_T}-\mathrm{e}^{-z_2})\mathrm {d}{z_2}\mathrm {d}T \right) \\= & {} \dfrac{\delta \alpha A_0}{(\alpha _2-i v)(\alpha _2-i v +1)} E_{\mathbb {Q}}\left( \int \limits _0^{\infty } \mathrm{e}^{-(u-r^f)T} mathbb{1}_{\tau \ge T}\mathrm{e}^{(\alpha _2-iv+1)Z_T}\mathrm {d}T \right) \\= & {} \dfrac{\delta \alpha A_0}{(\alpha _2-i v)(\alpha _2-i v +1)} \left( E_{\mathbb {Q}}\left( \int \limits _0^{\infty } \mathrm{e}^{-(u-r^f)T}\mathrm{e}^{(\alpha _2-i v+1)Z_T}\mathrm {d}T\right) \right. \\&\quad \left. -\sum \limits _{k=1}^n E_{\mathbb {Q}}\left( \mathrm{e}^{-(u-r^f) \tau } \mathrm{e}^{(\alpha _2-i v+1)Z_{\tau }} mathbb{1}_{J_{\tau }=e_k}E_{\mathbb {Q}} \right. \right. \\&\left. \left. \left( \left. \int \limits _0^{\infty } \mathrm{e}^{-(u-r^f) T}\mathrm{e}^{(\alpha _2-i v+1)(Z_{T+\tau }-Z_{\tau })}\mathrm {d}T\right| J_{\tau }=e_k\right) \right) \right) \\= & {} \dfrac{\delta \alpha A_0}{(\alpha _2-i v)(\alpha _2-i v +1)} \left( Y_0 W^{(\tilde{u}-r^f,-)}_Z \mathrm{e}^{Q^{(\tilde{u}-r^f,-)}_Z \left( x-\ln \dfrac{\lambda L_0}{A_0}\right) +(\alpha _2-i v+1)\ln \dfrac{\lambda L_0}{A_0} I_{2n}} \right. \\&\quad \left. H(\alpha _2-i v+1)-J_0\right) (Q-\text {diag}\{u-r^f-\phi ^Z_k(\alpha _2-i v+1)\})^{-1}mathbb{1}_n. \end{aligned}$$

1.3 Model calibration

With the time-series data of reference funds value, many calibration methods can be used, such as the maximum likelihood estimation, the GMM estimation, and the MCMC method. For instance, if we use the maximum likelihood estimation, we can easily obtain the log-likelihood function for the observable Markov chain case and make use of the Hamilton filter (see Hamilton 1989) obtaining the log-likelihood function for the hidden Markov chain case. Denote by $x_i, l_i$ the log-returns of the observed reference funds value and the economic state at time $t_i, i=0,1,\ldots ,N$, respectively, where $t_i=i \Delta$. Let $\Theta =(\hat{\mu },\hat{\sigma },\hat{\lambda },\hat{p},\hat{\eta },\hat{\theta },Q)$ be the parameter set of the regime switching Kou model. In the observable Markov chain case, the log-likelihood function is

$$\begin{aligned} \log f(x_1,\ldots ,x_N,l_1,\ldots ,l_N|\Theta )=\sum \limits _{i=1}^N \log f(x_i|\Theta ,l_i)+\sum \limits _{i=1}^N \log P(l_i|\Theta ,l_{i-1}), \end{aligned}$$

where $f(x_i|\Theta ,l_i)$ is the probability density function of the regime switching Kou process at state $l_i$, which can be computed by inverting the characteristic function of $X^i$, and $P(l_i|\Theta ,l_{i-1})$ is the transition probability from state $l_{i-1}$ at time $t_{i-1}$ to state $l_{i}$ at time $t_i$, which can be computed from $\langle l_{i-1}\mathrm{e}^{Q\Delta },l_i \rangle$. In the hidden Markov chain case, the log-likelihood function is in the form of

$$\begin{aligned} \log f(x_1,\ldots ,x_N|\Theta )&=\log f(x_1|\Theta )+\log f(x_2|\Theta ,x_1)+\log f(x_3|\Theta ,x_1,x_2)\nonumber \\&\quad +\cdots +\log f(x_N|\Theta ,x_1,\ldots ,x_{N-1}). \end{aligned}$$

(9)

The conditional density function $f(x_k|\Theta ,x_1,\ldots ,x_{k-1})$ is computed by

$$\begin{aligned} f(x_k|\Theta ,x_1,\ldots ,x_{k-1})= & {} \sum \limits _{i=1}^{n}\sum \limits _{j=1}^{n} P(l_{k-1}=e_i|\Theta ,x_1,\ldots ,x_{k-1}) \\&\times P(l_k=e_j|\Theta ,l_{k-1}=e_i) \times f(x_k|\Theta ,l_k=e_j,l_{k-1}=e_i), \end{aligned}$$

where the transition probability $P(l_k=e_j|\Theta ,l_{k-1}=e_i)$ is given by the $(i,j)_{\text {th}}$ element of $\mathrm{e}^{Q\Delta }$, and $f(x_k|\Theta ,l_k=e_j,l_{k-1}=e_i)$ is computed by inverting the characteristic function of $X^j$. For $P(l_{k-1}=e_i|\Theta ,x_1,\ldots ,x_{k-1})$, it can be obtained from $f(x_{k-1}|\Theta ,x_1,\ldots ,x_{k-2})$ as follows:

$$\begin{aligned}&P(l_{k-1}=e_i|\Theta ,x_1,\ldots ,x_{k-1})\\&=\dfrac{\sum \nolimits _{j=1}^{n} P(l_{k-2}=e_j|\Theta ,x_1,\ldots ,x_{k-2})P(l_{k-1}=e_i|\Theta ,l_{k-2}=e_j)f(x_{k-1}|\Theta ,l_{k-1}=e_i,l_{k-2}=e_j)}{f(x_{k-1}|\Theta ,x_1,\ldots ,x_{k-2})}. \end{aligned}$$

Then, with the initial value

$$\begin{aligned}&f(x_1|\Theta )=\sum \limits _{i=1}^{n}\sum \limits _{j=1}^{n} P(l_{0}=e_i|\Theta )\times P(l_1=e_j|\Theta ,l_0=e_i) \\&\qquad \qquad \quad \quad \times f(x_1|\Theta ,l_1=e_j,l_{0}=e_i), \end{aligned}$$

in which we make $P(l_{0}=e_i|\Theta )$ as the stationary probability of the Markov chain J, the iterations give the log-likelihood function (9). Then, we obtain the parameters by maximizing the log-likelihood function. One more robust calibration method is the “peaks over threshold” (POT) approach, see, for instance, Embrechts et al. (2011), Hainaut (2016) and Hainaut and Moraux (2018). This method fits the jump parameters and non-jump parameters, separately. We can make some adjustments to fit our regime switching case. We first fit the data with the regime switching Brownian motion model: $x_k \sim \hat{\mu }_i^g \Delta +\hat{\sigma }_i^g W_{\Delta }$ if $l_k=e_i$. Let $g_i(\beta _j)=\hat{\sigma }^g_i \sqrt{\Delta } \Phi ^{-1}(\beta _j), j=1,2$ be the thresholds to filter jumps at state $e_i$, in which the jumps are believed to happen at time $t_k$ with state $e_i$ when $x_k-\hat{\mu }^g_i \Delta >g_i(\beta _1)$ or $<g_i(\beta _2)$. $\Phi$ is the standard normal distribution function, and the levels $\beta _1, \beta _2$ are computed by making the sample skewness and the sample kurtosis of $W_{\Delta }$ be close to the normal distribution. Once the jumps are detected, we can estimate the parameters $(\hat{\mu },\hat{\sigma },\hat{\lambda },\hat{p},\hat{\eta },\hat{\theta })$ at each state:

$$\begin{aligned} \left\{ \begin{aligned}&(\hat{\mu }_i,\hat{\sigma }_i) =\arg \max \sum \limits _{k=1}^N \log f(x_k|\hat{\mu }_i,\hat{\sigma }_i)\mathbf {1}_{\text {no jump at}\ t_k \ \text {with state}\ e_i} \\&(\hat{p}_i,\hat{\eta }_i,\hat{\theta }_i) =\arg \max \sum \limits _{k=1}^N \log v(x_k|\hat{p}_i,\hat{\eta }_i,\hat{\theta }_i)\mathbf {1}_{\text {jump at}\ t_k \ \text {with state}\ e_i} \\&\hat{\lambda }_i =\arg \max \sum \limits _{k=1}^N (\log (\hat{\lambda }_i \Delta ) \mathbf {1}_{\text {jump at}\ t_k \ \text {with state}\ e_i}+\log (1-\hat{\lambda }_i \Delta )\mathbf {1}_{\text {no jump at}\ t_k \ \text {with state}\ e_i})\\ \end{aligned} \right. , \end{aligned}$$

where $f(x_k|\hat{\mu }_i,\hat{\sigma }_i)$ is the probability density function for the Brownian motion part of $X^i$ and $v(x_i|\hat{p}_i,\hat{\eta }_i,\hat{\theta }_i)$ is the probability density function of the double exponential distribution in $X^i$.

For the calibration with the historical data of the participating contract price, given the observed price of participating contracts $P_i$ for different minimum guaranteed rate $\tilde{r}_g^i$, participation coefficient $\delta _i$, wealth distribution coefficient $\alpha _i$, and maturity $T_i$, $i=1,\ldots ,N$, we can make use of the least squares method to obtain the parameter estimation, by minimizing the in-sample quadratic pricing error,

$$\begin{aligned} \Theta =\arg \min \limits _{\mathbb {Q}_{\Theta }\in \mathscr {Q}} \sum \limits _{i=1}^N (C^{\Theta }(\tilde{r}_g^i,\delta _i,\alpha _i,T_i)-P_i)^2, \end{aligned}$$

where $C^{\Theta }(\tilde{r}_g^i,\delta _i,\alpha _i,T_i)$ denotes the participating contract price computed for the regime switching Kou model under the risk-neutral measure $\mathbb {Q}_{\Theta }$ with the parameter set $\Theta$, and $\mathscr {Q}$ is the set of martingale measures. However, this calibration method is not robust enough. One solution is to introduce a regularization method. One penalty term is added to the target function and the optimization problem changes into

$$\begin{aligned} \Theta =\arg \min \limits _{\mathbb {Q}_{\Theta }\in \mathscr {Q}} \sum \limits _{i=1}^N \left( C^{\Theta }(\tilde{r}_g^i,\delta _i,\alpha _i,T_i)-P_i\right) ^2+\widetilde{\lambda } F(\mathbb {Q}_{\Theta },\mathbb {P}), \end{aligned}$$

where $\widetilde{\lambda }$ is the penalty parameter, $\mathbb {P}$ is the historical measure, the functional term F can be the relative entropy or Kullback–Leibler distance of the pricing measure $\mathbb {Q}_{\Theta }$ with respect to the historical measure $\mathbb {P}$. See Tankov (2003) for more details.