Surrender and path-dependent guarantees in variable annuities: integral equation solutions and benchmark methods

Abstract

We investigate the evaluation problem of variable annuities by considering guaranteed minimum maturity benefits, with constant or path-dependent guarantees of up-and-out barrier and lookback type, and guaranteed minimum accumulation benefit riders, with different forms of the surrender amount. We propose to solve the non-standard Volterra integral equations associated with the policy valuations through a randomized trapezoidal quadrature rule combined with an interpolation technique. Such a rule improves the converge rate with respect to the classical trapezoidal quadrature, while the interpolation technique allows us to obtain an efficient algorithm that produces a very accurate approximation of the early exercise boundary. The method accuracy is assessed by constructing two benchmarks: The first one, developed in a lattice framework, is characterized by a novel algorithm for the lookback path-dependent guarantee obtained thanks to the lattice convergence properties, while the application is straightforward in the other cases; the second one is based on the least-squares Monte Carlo simulations.

Appendices

Appendix A

Applying Proposition C.6 reported in Jeon and Kwak (2018) to problem (20), we obtain

$$\begin{aligned}{} & {} {\mathcal {L}}\overline{V}_3(t,x)=-\left( \varphi r-(\alpha -\kappa )e^{-\kappa (T-t)}x\right) \mathbbm {1}_{\left\{ x<e^{\kappa (T-t)}b_3(t)\right\} }\\{} & {} \quad -\left( -\frac{\varphi r}{\lambda }x^{1-\lambda }+\left( 1+\frac{1}{\lambda }\right) \varphi r (b_3(t))^{\lambda }x-(\alpha -\kappa )e^{-\kappa (T-t)}(b_3(t))^{1+\lambda }x\right) \mathbbm {1}_{\left\{ x>\frac{e^{-\kappa (T-t)}}{b_3(t)}\right\} } \end{aligned}$$

and

$$\begin{aligned} \overline{V}_3(T,x)=(\varphi -x)\mathbbm {1}_{\{x<\varphi \}}+\left( -\frac{\varphi x^{1-\lambda }}{\lambda }+\frac{\varphi ^{1+\lambda }x}{\lambda }\right) \mathbbm {1}_{\left\{ x>\frac{1}{\varphi }\right\} }, \end{aligned}$$

on the domain $\mathcal {\overline{D}}_3$. Applying then Proposition C.4 in Jeon and Kwak (2018), we obtain

$$\begin{aligned} \overline{V}_3(t,x)= & {} \int _{0}^{\infty }(\varphi -u)\mathbbm {1}_{\{u<\varphi \}}\mathcal {G}\Big (T-t,\frac{x}{u} \Big )\frac{1}{u}\,du\\{} & {} \quad +\int _{0}^{\infty }\left( -\frac{\varphi u^{1-\lambda }}{\lambda }+\frac{\varphi ^{1+\lambda }u}{\lambda }\right) \mathbbm {1}_{\left\{ u>\frac{1}{\varphi }\right\} }\mathcal {G}\Big (T-t,\frac{x}{u} \Big )\frac{1}{u}\,du\\{} & {} \quad +\int _{t}^{T}\int _{0}^{\infty }\left( \varphi r-(\alpha -\kappa )e^{-\kappa (T-\eta )}u\right) \mathbbm {1}_{\left\{ u<e^{\kappa (T-\eta )}b_3(\eta )\right\} }\mathcal {G}\Big (\eta -t,\frac{x}{u}\Big )\frac{1}{u}\,du\,d\eta \\{} & {} \quad +\int _{t}^{T}\int _{0}^{\infty }\Bigg (-\frac{\varphi r}{\lambda }u^{1-\lambda }+\left( 1+\frac{1}{\lambda }\right) \varphi r (b_3(\eta ))^{\lambda }u\\{} & {} \quad -(\alpha -\kappa )e^{-\kappa (T-\eta )}(b_3(\eta ))^{1+\lambda }u\Bigg )\mathbbm {1}_{\left\{ u>\frac{e^{-\kappa (T-\eta )}}{b_3(\eta )}\right\} }\mathcal {G}\Big (\eta -t,\frac{x}{u}\Big )\frac{1}{u}dud\eta . \end{aligned}$$

By means of Lemma C.1 in Jeon and Kwak (2018) and making the substitutions $\tau =T-t$, $\xi =T-\eta$, and $\widetilde{b}_3(\tau )= b_3(T-\tau )$, we obtain

$$\begin{aligned} \overline{V}_3(t,x)= & {} \varphi e^{-r\tau }\Phi \left( -d^-\left( \tau ,\frac{x}{\varphi }\right) \right) -xe^{-\alpha \tau }\Phi \left( -d^+\left( \tau ,\frac{x}{\varphi }\right) \right) \\{} & {} \quad +\frac{1}{\lambda }\Bigg [\varphi ^{1+\lambda }e^{-\alpha \tau }x\Phi (d^+(\tau ,\varphi x))\\{} & {} \quad -\varphi e^{-r\tau }x^{1-\lambda }\Phi \left( \frac{\log (\varphi x)+(r-\alpha -\sigma ^2(-0.5+\lambda ))\tau }{\sigma \sqrt{\tau }}\right) \Bigg ]\\{} & {} \quad +\varphi r \int _{0}^{\tau } e^{-r(\tau -\xi )}\Phi \left( -d^-\left( \tau -\xi ,\frac{x}{e^{\kappa \xi }\tilde{b}_3(\xi )}\right) \right) \,d\xi \\{} & {} \quad -(\alpha -\kappa )x \int _{0}^{\tau } e^{-\kappa \xi }e^{-\alpha (\tau -\xi )}\Phi \left( -d^+\left( \tau -\xi ,\frac{x}{e^{\kappa \xi }\tilde{b}_3(\xi )}\right) \right) d\xi \\{} & {} \quad -\frac{\varphi r}{\lambda }x^{1-\lambda } \int _{0}^{\tau } e^{-r(\tau -\xi )}\\{} & {} \quad \times \Phi \left( \frac{\log \left( xe^{\kappa \xi }\tilde{b}_3(\xi ) \right) +(r-\alpha -\sigma ^2(-0.5+\lambda ))(\tau -\xi )}{\sigma \sqrt{\tau -\xi }}\right) \,d\xi \\{} & {} \quad +\left( \frac{1+\lambda }{\lambda }\right) \varphi rx \int _{0}^{\tau } \tilde{b}_3(\xi )^{\lambda }e^{-\alpha (\tau -\xi )}\Phi \Big (d^+\Big (\tau -\xi ,xe^{\kappa \xi }\tilde{b}_3(\xi )\Big )\Big )d\xi \\{} & {} \quad -(\alpha -\kappa )x \int _{0}^{\tau } \tilde{b}_3(\xi )^{1+\lambda }e^{-\kappa \xi }e^{-\alpha (\tau -\xi )}\Phi \Big (d^+\Big (\tau -\xi ,xe^{\kappa \xi }\tilde{b}_3(\xi )\Big )\Big )d\xi . \end{aligned}$$

$\square$

Appendix B

1.1 Model 2

Suppose to consider Model 2 so that we consider equation (17). For the sake of simplicity, we define the quantity

$$\begin{aligned}{} & {} V^E_2\left( \tau ,e^{\kappa \tau }\widetilde{B}_{2}(\tau )\right) = Ge^{-r\tau }\Phi \left( -d^-\left( \tau ,\frac{e^{\kappa \tau }\widetilde{B}_{2}(\tau )}{G}\right) \right) \\{} & {} \quad -\widetilde{B}_{2}(\tau )e^{-(\alpha -\kappa )\tau }\Phi \left( -d^+\left( \tau ,\frac{e^{\kappa \tau }\widetilde{B}_{2}(\tau )}{G}\right) \right) , \end{aligned}$$

and, for $w=0,1,\dots ,i-1$, we introduce the quantities $z_w=t_w+\omega _w\Delta$ and $\tilde{z}_w=t_w+\tilde{\omega }_w\Delta$, so that $z_w, \tilde{z}_w\in [t_w, t_{w+1}]$. We observe that $\widetilde{B}_{2}(t_0)=\min \Big (1,\frac{r}{\max (\alpha -\kappa ,0)}\Big )G$, and for each $t_i,i=1,\ldots ,N$, we can write

$$\begin{aligned} \tilde{B}_{2}(t_i)= & {} G-V^E_2(t_i,e^{\kappa t_i}\tilde{B}_{2}(t_i))\nonumber \\{} & {} \quad -rG\frac{\Delta }{2}\sum _{w=0}^{i-1}\Bigg [e^{-r(t_i-z_w)}\Phi \Bigg (-d^-\Bigg (t_i-z_w,\frac{e^{\kappa t_i}\widetilde{B}_{2}(t_i)}{e^{\kappa z_w}\widetilde{B}_{2}(z_w)}\Bigg )\Bigg )\nonumber \\{} & {} \quad +e^{-r(t_i-\tilde{z}_w)}\Phi \Bigg (-d^-\Bigg (t_i-\tilde{z}_w,\frac{e^{\kappa t_i}\widetilde{B}_{2}(t_i)}{e^{\kappa \tilde{z}_w}\widetilde{B}_{2}(\tilde{z}_w)}\Bigg )\Bigg ) \Bigg ]\nonumber \\{} & {} \quad +(\alpha -\kappa )e^{\kappa t_i} \widetilde{B}_{2}(t_i)\frac{\Delta }{2}\sum _{w=0}^{i-1}\Bigg [e^{-\kappa z_w}e^{-\alpha (t_i-z_w)}\Phi \Bigg (-d^+\Bigg (t_i-z_w,\frac{e^{\kappa t_i}\widetilde{B}_{2}(t_i)}{e^{\kappa z_w}\widetilde{B}_{2}(z_w)}\Bigg )\Bigg )\nonumber \\{} & {} \quad +e^{-\kappa \tilde{z}_w}e^{-\alpha (t_i-\tilde{z}_w)}\Phi \Bigg (-d^+\Bigg (t_i-\tilde{z}_w,\frac{e^{\kappa t_i}\widetilde{B}_{2}(t_i)}{e^{\kappa \tilde{z}_w}\widetilde{B}_{2}(\tilde{z}_w)}\Bigg )\Bigg )\Bigg ]\nonumber \\{} & {} \quad +\Bigg (\frac{e^{\kappa t_i}\widetilde{B}_{2}(t_i)}{H}\Bigg )^{1-\lambda }\Bigg [ rG\frac{\Delta }{2}\sum _{w=0}^{i-1}\Bigg [e^{-r(t_i-z_w)}\Phi \Bigg (-d^-\Bigg (t_i-z_w,\frac{e^{-\kappa t_i}\widetilde{B}_{2}(t_i)^{-1}H^2}{e^{\kappa z_w}\widetilde{B}_{2}(z_w)}\Bigg )\Bigg )\nonumber \\{} & {} \quad +e^{-r(t_i-\tilde{z}_w)}\Phi \Bigg (-d^-\Bigg (t_i-\tilde{z}_w,\frac{e^{-\kappa t_i}\widetilde{B}_{2}(t_i)^{-1}H^2}{e^{\kappa \tilde{z}_w}\widetilde{B}_{2}(\tilde{z}_w)}\Bigg )\Bigg ) \Bigg ]\nonumber \\{} & {} \quad -(\alpha -\kappa )e^{-\kappa t_i} \widetilde{B}_{2}(t_i)^{-1}H^2\frac{\Delta }{2}\nonumber \\{} & {} \quad \times \sum _{w=0}^{i-1}\Bigg [e^{-\kappa z_w}e^{-\alpha (t_i-z_w)}\Phi \Bigg (-d^+\Bigg (t_i-z_w,\frac{e^{-\kappa t_i}\widetilde{B}_{2}(t_i)^{-1}H^2}{e^{\kappa z_w}\widetilde{B}_{2}(z_w)}\Bigg )\Bigg )\nonumber \\{} & {} \quad +e^{-\kappa \tilde{z}_w}e^{-\alpha (t_i-\tilde{z}_w)}\Phi \Bigg (-d^+\Bigg (t_i-\tilde{z}_w,\frac{e^{-\kappa t_i}\widetilde{B}_{2}(t_i)^{-1}H^2}{e^{\kappa \tilde{z}_w}\widetilde{B}_{2}(\tilde{z}_w)}\Bigg )\Bigg )\Bigg ]\Bigg ]. \end{aligned}$$

(A.1)

Considering simultaneously, for all i, the nonlinear Eq. (A.1), we obtain a nonlinear system of N equations in 3N unknowns. In order to avoid this problem, we approximate each $\widetilde{B}_{2}(z_{w})$ and $\widetilde{B}_{2}(\tilde{z}_{w})$ with the interpolated values,

$$\begin{aligned} \widetilde{B}_{2}(z_{w})\approx F_A\Big (t_{w},t_{w+1},\widetilde{B}_{2}(t_{w}),\widetilde{B}_{2}(t_{w+1}),z_{w}\Big ), \end{aligned}$$

and

$$\begin{aligned} \widetilde{B}_{2}(\tilde{z}_{w})\approx F_B\Big (t_{w},t_{w+1},\widetilde{B}_{2}(t_{w}),\widetilde{B}_{2}(t_{w+1}),\tilde{z}_w\Big ), \end{aligned}$$

where $F_A$ and $F_B$ are two functions indicating a linear interpolation, thus reducing the 3N unknowns to the N unknowns represented by the quantities $\widetilde{B}_{2}(t_{i})$, with $i=1,\ldots ,N$. Solving the arising system of N equations in N unknowns and considering the exact value of $\widetilde{B}_{2}(t_0)$, we obtain the approximate solution

$$\begin{aligned} \left\{ \tilde{B}_{2}(t_0),\tilde{B}_{2}(t_1),\tilde{B}_{2}(t_2),\dots ,\tilde{B}_{2}(t_N)\right\} . \end{aligned}$$

As for Model 1, the provision $\alpha$ making the contract fair may be evaluated applying the proposed algorithm and considering the additional equation

$$\begin{aligned} W=VA_{2}(0,W). \end{aligned}$$

A system of $N+1$ equations in $N+1$ unknowns is solved in order to find both the early exercise boundary and the provision.

1.2 Model 3

Suppose to consider Model 3, so that we consider Eq. (22). For the sake of simplicity, we define the quantity

$$\begin{aligned} V^E_3(\tau ,x)= & {} \varphi e^{-r\tau }\Phi \left( -d^-\left( \tau ,\frac{x}{\varphi }\right) \right) -xe^{-\alpha \tau }\Phi \left( -d^+\left( \tau ,\frac{x}{\varphi }\right) \right) \\{} & {} \quad +\frac{1}{\lambda }\Bigg [\varphi ^{1+\lambda }e^{-\alpha \tau }x\Phi \left( d^+(\tau ,\varphi x)\right) \\{} & {} \quad -\varphi e^{-r\tau }x^{1-\lambda }\Phi \left( \frac{\log (\varphi x)+(r-\alpha -\sigma ^2(-0.5+\lambda ))\tau }{\sigma \sqrt{\tau }}\right) \Bigg ]. \end{aligned}$$

We observe that $\widetilde{b}_{3}(t_0)=\min \Big (1,\frac{r}{\max (\alpha -\kappa ,0)}\Big )\varphi$, and for each $t_i,i=1,\ldots ,N$, using, as above, the same definition of $z_w$ and $\tilde{z}_w$, we can write

$$\begin{aligned} \tilde{b}_3(t_i)= & {} \varphi -V^E_3\left( t_i,e^{\kappa t_i}\widetilde{b}_{3}(t_i)\right) \nonumber \\{} & {} \quad -\varphi r\frac{\Delta }{2}\sum _{w=0}^{i-1}\Bigg [e^{-r(t_i-z_w)}\Phi \Bigg (-d^-\Bigg (t_i-z_w,\frac{e^{\kappa t_i}\widetilde{b}_{3}(t_i)}{e^{\kappa z_w}\widetilde{b}_{3}(z_w)}\Bigg )\Bigg )\nonumber \\{} & {} \quad +e^{-r(t_i-\tilde{z}_w)}\Phi \Bigg (-d^-\Bigg (t_i-\tilde{z}_w,\frac{e^{\kappa t_i}\widetilde{b}_{3}(t_i)}{e^{\kappa \tilde{z}_w}\widetilde{b}_{3}(\tilde{z}_w)}\Bigg )\Bigg )\nonumber \\{} & {} \quad +(\alpha -\kappa )e^{\kappa t_i}\widetilde{b}_{3}(t_i)\frac{\Delta }{2}\sum _{w=0}^{i-1}\Bigg [e^{-\kappa z_w}e^{-\alpha (t_i-z_w)}\Phi \Bigg (-d^+\Bigg (t_i-z_w,\frac{e^{\kappa t_i}\widetilde{b}_{3}(t_i)}{e^{\kappa z_w}\widetilde{b}_{3}(z_w)}\Bigg )\Bigg )\nonumber \\{} & {} \quad +e^{-\tilde{z}_w}e^{-\alpha (t_i-\tilde{z}_w)}\Phi \Bigg (-d^+\Bigg (t_i-\tilde{z}_w,\frac{e^{\kappa t_i}\widetilde{b}_{3}(t_i)}{e^{\kappa \tilde{z}_w}\widetilde{b}_{3}(\tilde{z}_w)}\Bigg )\Bigg )\Bigg ] +\frac{\varphi r}{\lambda }\Big (e^{\kappa t_i} \widetilde{b}_{3}(t_i)\Big )^{1-\lambda } \frac{\Delta }{2} \nonumber \\{} & {} \quad \times \sum _{w=0}^{i-1}\left[ e^{-r(t_i-z_w)}\Phi \left( \frac{\log \left( e^{\kappa t_i}\widetilde{b}_{3}(t_i)e^{\kappa z_w}\tilde{b}_3(z_w) \right) +(r-\alpha -\sigma ^2(-0.5+\lambda ))(t_i-z_w)}{\sigma \sqrt{t_i-z_w}}\right) \right. \nonumber \\{} & {} \quad \left. + e^{-r(t_i-\tilde{z}_w)}\Phi \left( \frac{\log \left( e^{\kappa t_i}\widetilde{b}_{3}(t_i)e^{\kappa \tilde{z}_w}\tilde{b}_3(\tilde{z}_w) \right) +(r-\alpha -\sigma ^2(-0.5+\lambda ))(t_i-\tilde{z}_w)}{\sigma \sqrt{t_i-\tilde{z}_w}}\right) \right] \nonumber \\{} & {} \quad -\left( \frac{1+\lambda }{\lambda }\right) \varphi re^{\kappa t_i}\widetilde{b}_{3}(t_i)\frac{\Delta }{2}\sum _{w=0}^{i-1}\Bigg [\widetilde{b}_{3}(z_w)^\lambda e^{-\alpha (t_i-z_w)}\Phi \Bigg (d^+\Bigg (t_i-z_w,e^{\kappa t_i}\widetilde{b}_{3}(t_i)e^{\kappa z_w}\widetilde{b}_{3}(z_w)\Bigg )\Bigg )\nonumber \\{} & {} \quad +\widetilde{b}_{3}(\tilde{z}_w)^\lambda e^{-\alpha (t_i-\tilde{z}_w)}\Phi \Bigg (d^+\Bigg (t_i-\tilde{z}_w,e^{\kappa t_i}\widetilde{b}_{3}(t_i)e^{\kappa \tilde{z}_w}\widetilde{b}_{3}(\tilde{z}_w)\Bigg )\Bigg )\Bigg ]\nonumber \\{} & {} \quad +(\alpha -\kappa )e^{\kappa t_i}\widetilde{b}_{3}(t_i)\frac{\Delta }{2}\sum _{w=0}^{i-1}\Bigg [\widetilde{b}_{3}(z_w)^{1+\lambda }e^{-\kappa z_w} e^{-\alpha (t_i-z_w)}\Phi \Bigg (d^+\Bigg (t_i-\tilde{z}_w,e^{\kappa t_i}\widetilde{b}_{3}(t_i)e^{\kappa z_w}\widetilde{b}_{3}(z_w)\Bigg )\Bigg )\nonumber \\{} & {} \quad +\widetilde{b}_{3}(\tilde{z}_w)^{1+\lambda }e^{-\kappa \tilde{z}_w} e^{-\alpha (t_i-\tilde{z}_w)}\Phi \Bigg (d^+\Bigg (t_i-\tilde{z}_w,e^{\kappa t_i}\widetilde{b}_{3}(t_i)e^{\kappa \tilde{z}_w}\widetilde{b}_{3}(\tilde{z}_w)\Bigg )\Bigg )\Bigg ]. \end{aligned}$$

(A.2)

Considering simultaneously, for all i, the nonlinear equation (A.2), we obtain a nonlinear system of N equations in 3N unknowns. In order to avoid this problem, we approximate each $\widetilde{B}_{3}(z_{w})$ and $\widetilde{B}_{3}(\tilde{z}_{w})$ with the interpolated values,

$$\begin{aligned} \widetilde{B}_{3}(z_{w})\approx F_A\Big (t_{w},t_{w+1},\widetilde{B}_{3}(t_{w}),\widetilde{B}_{3}(t_{w+1}),z_{w}\Big ), \end{aligned}$$

and

$$\begin{aligned} \widetilde{B}_{3}(\tilde{z}_{w})\approx F_B\Big (t_{w},t_{w+1},\widetilde{B}_{3}(t_{w}),\widetilde{B}_{3}(t_{w+1}),\tilde{z}_w\Big ), \end{aligned}$$

where $F_A$ and $F_B$ are two functions indicating a linear interpolation, thus reducing the 3N unknowns to the N unknowns represented by the quantities $\widetilde{B}_{3}(t_{i})$, with $i=1,\ldots ,N$. Solving the arising system of N equations in N unknowns and considering the exact value of $\widetilde{B}_{3}(t_0)$, we obtain the approximate solution

$$\begin{aligned} \left\{ \tilde{B}_{3}(t_0),\tilde{B}_{3}(t_1),\tilde{B}_{3}(t_2),\dots ,\tilde{B}_{3}(t_N)\right\} . \end{aligned}$$

The provision $\alpha$ making the contract fair may be evaluated applying the proposed algorithm and considering the additional equation

$$\begin{aligned} W=VA_{3}(0,W). \end{aligned}$$

A system of $N+1$ equations in $N+1$ unknowns is solved in order to find both the early exercise boundary and the provision. $\square$