2.1. The financial market

We consider a financial market with one risk-free asset, assumed to have a zero interest rate, and $d$ risky assets, whose prices per share at time $t$ are denoted by $S_{1}(t)$ , …, $S_{d}(t)$ . The zero interest rate assumption is made without loss of generality, as explained in Remark 2.1 below. The vector process $\left\{ \boldsymbol{S}(t)\right\}_{0\leq t\leq T}=\left\{ \left( S_{1}(t),\ldots ,S_{d}(t)\right) ^{\intercal}\right\} _{0\leq t\leq T}$ , with $\boldsymbol{S}(0)\gt\boldsymbol{0}$ understood componentwise, satisfies the system of stochastic differential equations (SDEs)

(2.1) \begin{equation}\frac{dS_{i}(t)}{S_{i}(t{\kern.5pt}{-}{\kern-.8pt})}=\mu _{i}(t)dt+\sum_{j=1}^{m}\sigma_{i,j}(t)dW_{j}(t)+\int_{\mathbb{R}^{n}}\theta _{i}(t,\boldsymbol{x})\tilde{N}(dt,d\boldsymbol{x}),\qquad t\in \left[ 0,T\right] ,i=1,\ldots ,d. \end{equation}

The vectors $\boldsymbol{\mu }(t)=\left( \mu _{1}(t),\ldots ,\mu_{d}(t)\right) ^{\intercal }\in \mathbb{R}^{d}$ and $\boldsymbol{\theta }(t,\boldsymbol{x})=\left( \theta _{1}(t,\boldsymbol{x}),\ldots ,\theta _{d}(t,\boldsymbol{x})\right) ^{\intercal }\gt-\boldsymbol{1}$ and the matrix $\Sigma (t)=\left( \sigma _{i,j}(t)\right) \in \mathbb{R}^{d\times m}$ are all assumed to be uniformly bounded and $\mathbb{F}$ predictable, so that (2.1) admits a unique strong solution, which is further assumed to be positive; see, for example, Theorem 37 in Chapter II of Protter (Reference Protter2005). We assume that, a.e. with respect to $dP\mathcal{\otimes }dt$ , $\Sigma (t)$ has full rank, and there exist constants $\zeta \gt\varepsilon \gt0$ such that $\zeta I_{d}-\Sigma (t)\Sigma ^{\intercal }(t)$ and $\ \Sigma (t)\Sigma^{\intercal }(t)-\varepsilon I_{d}\ $ are positive semidefinite, where $I_{d} $ represents the identity matrix of dimension $d$ . Other than these assumptions, the model is general and covers most popular jump-diffusion models.

2.2. The mortality rate process

We consider a general MLS. By the very definition, an appropriate mortality model is central to establishing the pricing framework. The literature on this topic is extensive, largely inspired by reduced-form models in credit risk theory. We refer the reader to Dahl (Reference Dahl2004), Biffis (Reference Biffis2005), Bauer and Kramer (Reference Bauer and Kramer2016), and Bravo and Nunes (Reference Bravo and Nunes2021), to mention just several. These works model mortality rates as affine processes for analytical and computational tractability.

Following this strand of literature, we consider a general model for the mortality rate process $\{\lambda (t)\}_{0\leq t\leq T}$ , evolving according to

(2.2) \begin{equation}d\lambda (t)=\mu _{\lambda }(t)dt+\sum_{j=1}^{m}\sigma _{\lambda,j}(t)dW_{j}(t)+\int_{\mathbb{R}^{n}}\theta _{\lambda }(t,\boldsymbol{x})\tilde{N}(dt,d\boldsymbol{x}),\qquad t\in \left[ 0,T\right] ,\end{equation}

where $\mu _{\lambda }(t)\in \mathbb{R}$ , $\theta _{\lambda }(t,\boldsymbol{x})\in \mathbb{R}$ , and $\boldsymbol{\sigma }_{\lambda }(t)=\left( \sigma_{\lambda ,1}(t),\ldots ,\sigma _{\lambda ,m}(t)\right) ^{\intercal }\in\mathbb{R}^{m}$ are assumed to be $\mathbb{F}$ predictable processes and such that the integrals involved exist and (2.2) admits a unique strong solution; see, for example, Theorem 1.19 of Øksendal and Sulem (Reference Øksendal and Sulem2019). This covers, in particular, the class of affine mortality models. Note that $\{\lambda (t)\}_{0\leq t\leq T}$ in equation (2.2) should be interpreted as the mortality rate of a representative population rather than that of an individual. The model can also be applied to extreme mortality risk, as in Li et al. (Reference Li, Liu, Tang and Yuan2023), with $\{\lambda (t)\}_{0\leq t\leq T}$ interpreted accordingly as excess mortality.

It is straightforward to further specify the coefficients to ensure the positivity of the solution $\{\lambda (t)\}_{0\leq t\leq T}$ . However, we choose not to impose this restriction but instead allow $\{\lambda(t)\}_{0\leq t\leq T}$ to take negative values, in which case the coefficients can be adjusted to render the corresponding probability arbitrarily small. As recently discussed by Liu et al. (Reference Liu, Liu, Tang and Zhu2026), allowing for negative mortality rates, much like negative interest rates, has practical relevance. For example, in estimation, it can help address accounting issues in which over-counting at a previous time point needs to be corrected by subtracting the excess. Often, these accounting adjustments lead to sudden upward or downward changes in estimated mortality rates, which in turn justify both positive and negative jumps in the mortality rate process. This feature can be captured, for example, by using an asymmetric double exponential distribution, as assumed by Bravo and Nunes (Reference Bravo and Nunes2021) in their proposed affine jump–diffusion model for the mortality rate process.

2.3. Utility indifference pricing

Since the market for MLSs is typically incomplete, we employ the utility indifference pricing approach for addressing this incompleteness. Consider an agent—always specified as the issuer or provider of the MLS in this work—who faces a liability represented by a bounded $\mathcal{F}_{T}$ measurable random variable $C$ , payable at the terminal time $T$ . This means that the liability is a general payout contingent on the performance of both the financial market and the mortality rate. Our objective is to determine the utility indifference price $p$ at time 0 of the payoff $C$ . This price $p$ is defined as the value at which the agent is indifferent, meaning that the expected utility under optimal trading is unchanged between (i) receiving $p$ at time 0 and settling the payoff $C$ at time $T$ and (ii) not receiving $p$ at time 0 and therefore not settling the payoff $C$ at time $T$ .

Suppose the agent, with initial wealth of $w$ at time 0, invests at each time $t\in \left[ 0,T\right] $ an amount $\pi _{i}(t)$ in the $i$ th risky asset, $i\in \left\{ 1,\ldots ,d\right\} $ . In particular, we assume that the process $\left\{ \boldsymbol{\pi }(t)\right\} _{0\leq t\leq T}=\{\left(\pi _{1}(t),\ldots ,\pi _{d}(t)\right) ^{\intercal }\}_{0\leq t\leq T}$ belongs to an admissible set $\mathcal{A}$ , which consists of all $\mathbb{F}$ predictable processes with values in a bounded and closed subset of $\mathbb{R}^{d}$ . The wealth $X^{w,\boldsymbol{\pi }}(t)$ , accumulated from time 0 to $t$ , is then given by

(2.3) \begin{equation}X^{w,\boldsymbol{\pi }}(t)=w+\int_{0}^{t}\boldsymbol{\pi ^{\intercal }}(s)\left( \boldsymbol{\mu }(s)ds+\Sigma (s)d\boldsymbol{W}(s)+\int_{\mathbb{R}^{n}}\boldsymbol{\theta }(s,\boldsymbol{x})\tilde{N}(ds,d\boldsymbol{x})\right) , \end{equation}

which is well defined due to the boundedness of the strategies and the coefficients in (2.1).

Assume that the agent has an exponential utility, with a risk aversion parameter $\alpha \gt0$ :

(2.4) \begin{equation}U(y)=-e^{-\alpha y},\qquad y\in \mathbb{R}. \end{equation}

Exponential utility is widely employed in finance, insurance, and economics, largely due to its analytical tractability. Although it may be unrealistic in certain scenarios, it offers several practical advantages. In particular, it accommodates real-valued wealth, which is essential when modeling both the purchase and sale of securities, and it yields optimal decisions that are independent of initial wealth, thereby supporting pricing consistency across market participants with heterogeneous endowments.Footnote ⁴

Define the value function

(2.5) \begin{equation}V(w,q)=\sup_{\boldsymbol{\pi }\in \mathcal{A}}E\left[ U\left( X^{w,\boldsymbol{\pi }}(T)-qC\right) \right] ,\qquad q\geq 0. \end{equation}

Thus, the utility indifference price $p$ is the solution to

(2.6) \begin{equation}V(w,0)=V(w+p,1). \end{equation}

Given the agent’s exponential utility function, we notice that $V(w,q)=e^{-\alpha w}V(0,q)$ . Hence, relation (2.6) reduces to $e^{-\alpha w}V(0,0)=e^{-\alpha (w+p)}V(0,1)$ , which yields a representation of the indifference price as

(2.7) \begin{equation}p=\frac{1}{\alpha }\ln \frac{V(0,1)}{V(0,0)}. \end{equation}

Along the same lines, we can derive the indifference price $p$ ( $q$ ) for the payoff $qC$ with $q\gt0$ , and it is easy to further show that $p$ ( $q$ ) is convex in the quantity $q$ . See Section 2.3 of Carmona (Reference Carmona2009) for related discussions.

4.1. Various payoff structures

The financial market considered consists of a risk-free asset paying no interest and a risky asset whose price follows a stochastic volatility process. Furthermore, a mortality rate process $\{\lambda (t)\}_{0\leq t\leq T}$ is modeled as an affine jump-diffusion process to account for potential mortality shocks. Then, $e^{-\Lambda (T)}$ , with $\Lambda(t)=\int_{0}^{t}\lambda (s)ds$ for $t\geq 0$ , would represent the realized survival probability over [0, $T$ ]. Since $e^{-\Lambda (T)}$ could theoretically exceed 1 due to the possibility of large negative jumps in the mortality rate process, we redefine the realized survival probability at maturity $T$ as $e^{-\Lambda _{0}(T)}$ , where $\Lambda _{0}(T)=\max \left\{\Lambda (T),0\right\} $ . Nevertheless, in our specific model, the probability of $e^{-\Lambda (T)}$ exceeding 1 is extremely low, so in practice, $\Lambda _{0}(T)=\Lambda (T)$ , making this adjustment just a theoretical safeguard.

For compactness, given $0\leq x_{\min }\leq x_{\max }\leq \infty $ and $x\in\mathbb{R}$ , we define the two-sided truncation of $x$ as $(x)_{x_{\min},x_{\max }}=\min \left\{ \max \left\{ x,x_{\min }\right\} ,x_{\max}\right\} $ . In particular, $\Lambda _{0}(T)=(\Lambda (T))_{0,\infty }$ .

We now introduce various payoff structures to be considered:

• • S-forward: The payoff at maturity is given by

  \begin{equation*}C=a(e^{-\Lambda _{0}(T)}-b),\qquad a\gt0,b\in (0,1).\end{equation*}
  This can be interpreted as $a$ units of a zero coupon swap, where a fixed survival probability $b$ is exchanged for the realized survival probability $e^{-\Lambda _{0}(T)}$ at maturity $T$ . This derivative provides a hedge against the risk of a specified population deviating from a reference survival probability. For example, a pension fund or an insurance company facing longevity risk might agree to pay a fixed amount based on an initially expected survival rate. In return, the counterparty would pay a variable amount determined by the actual observed survival rates over the term of the contract. S-forward contracts have been extensively studied in the actuarial literature; see, for example, Bravo and Nunes (Reference Bravo and Nunes2021), Tang and Li (Reference Tang and Li2021), and Zeddouk and Devolder (Reference Zeddouk and Devolder2024).

• • Longevity caplet: The payoff at maturity is given by

  \begin{equation*}C=a(e^{-\Lambda _{0}(T)}-b)_{0,\infty },\qquad a\gt0,b\in (0,1).\end{equation*}
  This is an option-type longevity derivative that pays $a(e^{-\Lambda_{0}(T)}-b)$ if the realized survival probability $e^{-\Lambda _{0}(T)}$ at maturity $T$ exceeds a prespecified cap rate $b$ , and zero otherwise. Unlike an S-forward, this derivative triggers a payoff only if the realized survival rate at maturity exceeds the threshold $b$ . This payoff structure may be more appealing to insurers, as it transfers only extreme longevity risk and thus reduces the capital requirement. See Bauer et al. (Reference Bauer, Börger and Ruß2010) and Bravo and Nunes (Reference Bravo and Nunes2021) for related discussions.

• • Unit-linked security: The payoff at maturity is given by

  \begin{equation*}C=1_{(e^{-\Lambda _{0}(T)}\gt b)}(S(T))_{G_{\min },G_{\max }},\qquad b\in (0,1),\end{equation*}
  where $0\leq G_{\min }\lt G_{\max }\leq \infty $ represent the minimum and maximum guaranteed amounts provided by the contract issuer, respectively, and $S$ ( $T$ ) is the terminal value of a stock price. This investment-linked payout ensures that if the reference survival probability exceeds an agreed threshold $b$ , the contract holder is paid based on the stock price at terminal $T$ , subject to minimum and maximum guarantees. The unit-linked security provides protection against both extreme longevity risk and financial market downside risk—an idea akin to the hybrid catastrophe bonds concept proposed by Barrieu and Loubergé (Reference Barrieu and Loubergé2009). Given potential interactions between mortality outcomes and market performance, this dual protection feature could increase demand for the security. By the way, it is easy to see that
  (4.1) \begin{equation}C=\underbrace{1_{(e^{-\Lambda _{0}(T)}\gt b)}G_{\min }}_{\text{guaranteed portion}}+\underbrace{1_{(e^{-\Lambda _{0}(T)}\gt b)}(S(T)-G_{\min})_{0,G_{\max }-G_{\min }}}_{\text{liability portion}}. \end{equation}

4.2. Dynamics of the asset price and mortality rate

Introduce a three-dimensional standard Brownian motion $\{\boldsymbol{W}(t)\}_{0\leq t\leq T}$ , to be used to model the dynamics of the risky asset price and mortality rate. As mentioned earlier, we consider a financial market consisting of a risk-free asset paying no interest and a risky asset. Suppose that the price per share of the risky asset, denoted by $\{S(t)\}_{0\leq t\leq T}$ , follows the stochastic volatility model, for $t\in \lbrack 0,T]$ ,

(4.2) \begin{align}\frac{dS(t)}{S(t)} & = \mu dt+\Sigma (t)d\boldsymbol{W}(t), \nonumber\\\Sigma (t) & = (v(t))_{v_{\min },v_{\max }}^{0.5}\left( \rho ,\sqrt{1-\rho^{2}}\psi ,\sqrt{1-\rho ^{2}}\sqrt{1-\psi ^{2}}\right) , \end{align}

(4.3) \begin{align} dv(t) &=(\beta _{0}+\beta _{1}v(t))dt+{\sigma _{v}}(v(t))_{v_{\min},v_{\max }}^{\gamma }dW_{2}(t), \end{align}

where $\mu ,\beta _{0},\beta _{1}\in \mathbb{R}$ , ${\sigma _{v},}\gamma \gt0$ , $\rho ,\psi \in ({-}1,1)$ , $0 \lt v_{\min }\lt v_{\max }\lt \infty $ , and $S(0),v(0)\gt0$ are all constants. The existence of a unique solution to (4.3) is evident, as the volatility function ${\sigma _{v}(}v)_{{v_{\min }},v_{\max}}^{\gamma }$ , due to the two-sided truncation, is Lipschitz continuous. This stochastic volatility model is a slightly modified version of model (3) in Jones (Reference Jones2003), who, in studying the dynamic behavior of the volatility of equity index returns, proposes a generalization of the Heston model to incorporate a time-varying leverage effect on the instantaneous variance. It is common practice in financial modeling to employ truncation on the volatility process; for related discussions, see, for example, Appendix B of Davydov and Linetsky (Reference Davydov and Linetsky2001).

We assume that the reference mortality rate process $\{\lambda (t)\}_{0\leq t\leq T}$ evolves according to

(4.4) \begin{equation}d\lambda (t)=(a_{0}+a_{1}\lambda (t))dt+\sigma _{\lambda}dW_{1}(t)+d\sum_{i=1}^{N(t)}J_{i},\qquad t\in \left[ 0,T\right] ,\end{equation}

where $a_{0},a_{1}\in \mathbb{R}$ and $\sigma _{\lambda }$ $\gt0$ are constants, $\{N(t)\}_{0\leq t\leq T}$ is a Poisson process with constant intensity ${\delta }\gt0$ , and $\{J_{i}\}_{i=1}^{\infty }$ is a sequence of independent and identically distributed jumps. Further assume that these jumps follow a common asymmetric double exponential distribution, with probability density function

(4.5) \begin{equation}\frac{\phi }{\varepsilon _{1}}e^{-u/\varepsilon _{1}}1_{(u\geq 0)}+\frac{1-\phi }{\varepsilon _{2}}e^{u/\varepsilon _{2}}1_{(u\lt0)},\qquad u\in\mathbb{R}, \end{equation}

for $\varepsilon _{1},\varepsilon _{2}\gt0$ and $\phi \in \lbrack 0,1]$ . As usual, assume that $\{\boldsymbol{W}(t)\}_{0\leq t\leq T}$ , $\{N(t)\}_{0\leq t\leq T}$ , and $\{J_{i}\}_{i=1}^{\infty }$ are mutually independent.

Notably, the reference mortality process $\{\lambda (t)\}_{0\leq t\leq T}$ and the asset price process $\{S(t)\}_{0\leq t\leq T}$ are allowed to be dependent, with the instantaneous correlation

\begin{align*}\mathrm{corr}\left( \frac{dS(t)}{S(t)},d\lambda (t)\right) =\rho \frac{E\left[ \sqrt{(v(t))_{v_{\min },v_{\max }}(a_{0}+a_{1}\lambda (t))}\right] }{\sqrt{E[(v(t))_{v_{\min },v_{\max }}](E\left[ a_{0}+a_{1}\lambda (t)\right]+\delta E\left[ J^{2}\right] )}}.\end{align*}

Furthermore, we suppose that the set of admissible strategies consists of all real-valued and $\mathbb{F}$ predictable processes with values in the interval $\left[ \pi _{\min },{\pi _{\max }}\right] $ , where $-\infty \lt\pi_{\min }\lt\pi _{\max }\lt\infty $ .

4.3. Numerical approximation of the utility indifference price

To obtain the utility indifference price $p$ in (2.7) for an MLS, we need the values of $V$ (0,1) and $V$ (0,0). By Theorem 3.1, calculating these values reduces to evaluating $-e^{\alpha Y(0)}$ with $C$ set to the security’s payoff and $C=0$ , respectively. Alternatively, we can evaluate $-\tilde{Y}(0)$ , where $\{\tilde{Y}(t)\}_{0\leq t\leq T}$ corresponds to the auxiliary BSDE (A.1). We estimate the price $p$ using this alternative approach and numerically solve the auxiliary BSDE (A.1) to approximate $-\tilde{Y}(0)$ . The reason for following this approach is that discrete-time approximations for forward-BSDEs typically require the generator of the BSDE to be Lipschitz continuous; see Section 1 of Bouchard and Elie (Reference Bouchard and Elie2008).

The asset price process $\{S(t)\}_{0\leq t\leq T}$ described in (4.2) does not contain jumps, which implies that, under the general formula (2.1), $\theta (t,x)=0$ a.e. with respect to $dP\mathcal{\otimes }d\eta \mathcal{\otimes }dt$ . Thus, (A.1) simplifies to

(4.6) \begin{eqnarray}\tilde{Y}(t) &=&e^{\alpha C}+\int_{t}^{T}\left\{ -\alpha \pi _{\ast}(s)\left( \mu \tilde{Y}(s)+\Sigma (s){\tilde{\boldsymbol{Z}}}(s)\right) +\frac{1}{2}\alpha ^{2}\pi _{\ast }^{2}(s)\left\Vert \Sigma (s)\right\Vert ^{2}\tilde{Y}(s)\right\} ds \notag \\[7pt]&&-\int_{t}^{T}{\tilde{\boldsymbol{Z}}}^{\intercal }(s)d\boldsymbol{W}(s)-\int_{t}^{T}\int_{\mathbb{R}}\tilde{K}(s,x)\tilde{N}(ds,dx),\qquad t\in\lbrack 0,T], \end{eqnarray}

where

\begin{equation*}\pi _{\ast }(t)=\left( \frac{1}{\alpha }\left\Vert \Sigma (t)\right\Vert ^{-2}\left( \mu +\Sigma (t)\tilde{\boldsymbol{Z}}(t)(\tilde{Y}(t))^{-1}\right) \right) _{\pi _{\min },\pi _{\max }}.\end{equation*}

We now describe a discrete-time approximation for the forward-BSDEs (4.2)–(4.6). This approximation is then implemented using an LSMC algorithm.

For every $t\in \lbrack 0,T]$ , define $\boldsymbol{\chi }(t)=\left(S(t),v(t),\lambda (t),\Lambda (t)\right) ^{\intercal }$ , where $S$ ( $t$ ), $v$ ( $t$ ), and $\lambda (t)$ are given by equations (4.2)–(4.4), respectively, and $\Lambda(t)=\int_{0}^{t}\lambda (s)ds$ . Let $0=t_{0}\lt t_{1}\lt \ldots \lt t_{m_{p}}=T$ be a partition of [0, $T$ ] with an equal spacing of $h=\frac{T}{m_{p}}$ . A discrete-time approximation of $\{\boldsymbol{\chi }(t)\}_{0\leq t\leq T}$ , denoted by $\{\boldsymbol{\chi }_{h}(t)\}_{0\leq t\leq T}$ $=\{\left(S_{h}(t),v_{h}(t),\lambda _{h}(t),\Lambda _{h}(t)\right) ^{\intercal}\}_{0\leq t\leq T}$ , is given by the Euler scheme. Consider a payoff of the form $C=c(\boldsymbol{\chi }(T))$ , where $c(\!\cdot\!)$ is a real-valued and bounded measurable function. Given the approximation $\{\boldsymbol{\chi }_{h}(t)\}_{0\leq t\leq T}$ , an approximated solution $( \tilde{Y}_{h}(t),{\tilde{\boldsymbol{Z}}}_{h}(t)) _{0\leq t\leq T}$ to the BSDE (4.6) is given by the backward recursion

\begin{equation*}\left\{\begin{array}{l}\displaystyle\tilde{Y}_{h}(T)=e^{\alpha c(\boldsymbol{\chi }_{h}(T))}, \\[8pt] \displaystyle {\tilde{\boldsymbol{Z}}}_{h}(t_{i})=\frac{1}{h}E\left[ \left. \tilde{Y}_{h}(t_{i+1})\left( \boldsymbol{W}(t_{i+1})-\boldsymbol{W}(t_{i})\right)\right\vert \mathcal{F}_{t_{i}}\right] , \\[12pt] \displaystyle \tilde{Y}_{h}(t_{i})=E\big[ \tilde{Y}_{h}(t_{i+1})+g\big(\boldsymbol{\chi }_{h}(t_{i}),\tilde{Y}_{h}(t_{i+1}),{\tilde{\boldsymbol{Z}}}_{h}(t_{i})\big) h\big\vert \mathcal{F}_{t_{i}}\big] ,\end{array}\right.\end{equation*}

and $( \tilde{Y}_{h}(t),{\tilde{\boldsymbol{Z}}}_{h}(t)) =(\tilde{Y}_{h}(t_{i}),{\tilde{\boldsymbol{Z}}}_{h}(t_{i})) $ for $t_{i}\leq t\lt t_{i+1}$ , where $i=0,\ldots ,m_{p}-1$ and $g(\!\cdot\!)$ corresponds to the generator of (4.6). The conditional expectations are approximated using the LSMC method with four sets of $m_{b}$ dimensional basis functions, $\boldsymbol{b}_{l}(\!\cdot\!)=\left( b_{l,1}(\!\cdot\!),\ldots,b_{l,m_{b}}(\!\cdot\!)\right) ^{\intercal }$ for $l=0,1,2,3$ ; see Longstaff and Schwartz (Reference Longstaff and Schwartz2001).

We now describe how to simulate $m_{s}$ sample paths of $\{\tilde{Y}(t)\}_{0\leq t\leq T}$ using the LSMC algorithm. To facilitate the description, we define three independent random objects:

• • a three-dimensional normal random vector $\Delta \boldsymbol{W}_{h}$ with independent components, each having mean 0 and variance $h\gt0$ ;

• • a random variable $\Delta N_{h}$ following a Poisson distribution with mean ${\delta }h$ ;

• • a random variable $J$ following an asymmetric double exponential distribution (4.5) with parameters $(\varepsilon _{1},\varepsilon_{2},\phi )$ .

We apply the LSMC algorithm according to the following steps:

1. 1. Simulate $m_{p}\times m_{s}$ independent copies of $\Delta \boldsymbol{W}_{h}$ and $\Delta N_{h}$ and store them in $\Delta \boldsymbol{W}^{j}(i+1) $ and $\Delta N^{j}(i+1)$ , respectively, for $i=0,\ldots ,m_{p}-1$ and $j=1,\ldots ,m_{s}$ . Then, simulate $\Delta N^{j}(i+1)$ independent copies of $J$ for $i=0,\ldots ,m_{p}-1$ and $j=1,\ldots ,m_{s}$ .

2. 2. Generate $m_{s}$ independent copies of $\left\{ \boldsymbol{\chi }_{h}(t_{i})\right\} _{i=1}^{m_{p}}$ according to the Euler scheme and store them in $\{\boldsymbol{\chi }_{h}^{j}(t_{i})\}_{i=1}^{m_{p}}$ for $j=1,\ldots,m_{s}$ .

3. 3. Set $\hat{Y}\left( T,\boldsymbol{\chi }_{h}^{j}(T)\right) =e^{\alpha c(\boldsymbol{\chi }_{h}^{j}(T))}$ for $j=1,\ldots ,m_{s}$ .

4. 4. For $i=0,\ldots ,m_{p}-1$ ,

   1. (a) solve the least-squares regression

      \begin{equation*}\boldsymbol{\xi }_{l}=\mathop{\mathrm{arg\,min}}\limits_{\boldsymbol{\xi }\in \mathbb{R}^{m_{b}}}\left\{ \frac{1}{m_{s}}\sum_{j=1}^{m_{s}}\left\vert \frac{1}{h}\hat{Y}\left( t_{i+1},\boldsymbol{\chi }_{h}^{j}(t_{i+1})\right) \Delta W_{l}^{j}(i+1)-\boldsymbol{\xi }^{\intercal }\boldsymbol{b}_{l}\left(\boldsymbol{\chi }_{h}^{j}(t_{i})\right) \right\vert ^{2}\right\}\end{equation*}
      and set $\hat{Z}_{l}\left( t_{i},\boldsymbol{\chi }_{h}^{j}(t_{i})\right) =\boldsymbol{\xi }_{l}^{\top }\boldsymbol{b}_{l}\left( \boldsymbol{\chi }_{h}^{j}(t_{i})\right) $ for $l=1,2,3$ and $j=1,\ldots ,m_{s}$ ;

   2. (b) solve the least-squares regression

      \begin{eqnarray*}\boldsymbol{\xi }_{0} &=&\mathop{\mathrm{arg\,min}}\limits_{\boldsymbol{\xi }\in \mathbb{R}^{m_{b}}}\left\{ \frac{1}{m_{s}}\sum_{j=1}^{m_{s}}\left\vert \hat{Y}\left(t_{i+1},\boldsymbol{\chi }_{h}^{j}(t_{i+1})\right) \right. \right. \\&&\left. \left. +g\left( \boldsymbol{\chi }_{h}^{j}(t_{i}),\hat{Y}\left(t_{i+1},\boldsymbol{\chi }_{h}^{j}(t_{i+1})\right) ,{\hat{\boldsymbol{Z}}}\left( t_{i},\boldsymbol{\chi }_{h}^{j}(t_{i})\right) \right) h-\boldsymbol{\xi }^{\intercal }\boldsymbol{b}_{0}\left( \boldsymbol{\chi }_{h}^{j}(t_{i})\right) \vphantom{\tilde{Y}^{j}}\right\vert ^{2}\vphantom{\sum_{j=1}^{m}}\right\} ,\end{eqnarray*}
      and set $\hat{Y}\left( t_{i},\boldsymbol{\chi }_{h}^{j}(t_{i})\right) =\boldsymbol{\xi }_{0}^{\intercal }\boldsymbol{b}_{0}\left( \boldsymbol{\chi }_{h}^{j}(t_{i})\right) $ and $j=1,\ldots ,m_{s}$ .

The procedure above yields approximated sample paths for the solution $\{\tilde{Y}(t)\}_{0\leq t\leq T}$ of the auxiliary BSDE (A.1) with $C $ equal to the payoff of the security. We then set $V(0,1)=-\tilde{Y}(0)$ . The procedure is repeated using the same process with $C=0$ to generate sample paths of $\{\tilde{Y}(t)\}_{0\leq t\leq T}$ , which results in the estimate $V(0,0)=-\tilde{Y}(0)$ . The utility indifference price of the contract at $t=0$ is then obtained by plugging these values to (2.7).

4.4. Implementation and results

In our numerical studies, we compute the indifference prices of the three MLSs introduced earlier in Subsection 4.1: S-forward, longevity caplet, and unit-linked security. Specifically, we illustrate the impact of the mortality rate parameters, the correlation parameter, and short-selling constraints on these indifference prices. Our results demonstrate that the parameters of the mortality rate, particularly the drift coefficient $a_{1}$ and the jump intensity $\delta $ , have a more pronounced effect on pricing than those governing the asset price dynamics. Furthermore, the correlation between the risky asset price and the mortality rate significantly influences pricing, especially for unit-linked securities.

Baseline estimation

For the mortality rate in (4.4), we use the parameters calibrated by Bravo and Nunes (Reference Bravo and Nunes2021), based on annual mortality data from 1950 to 2017 for the total US population aged 65 years in 1990. For the price process of the risky asset described by (4.2)–(4.3), we estimate its parameters using daily closing prices of the S&P 500 index from 2005 to 2017, employing the SVDNF package in R (Arsenault-Mahjoubi et al., Reference Arsenault-Mahjoubi, Bégin and Boudreault2024). This ensures using time windows with the same ending year when estimating parameters for the asset price and mortality rate models. We refrain from a full joint re-estimation of both models due to the mismatch in data frequencies—mortality data are annual, while stock price data are daily.

For the three MLSs, we consider a time horizon $T=10$ . The absolute risk aversion parameter $\alpha $ in (2.4) is set to $0.02$ , which falls within the range of absolute risk aversion estimates surveyed by Babcock et al. (Reference Babcock, Choi and Feinerman1993). The values of $b$ and $G_{\min }$ are set to the median of the simulated values of $e^{-\Lambda _{0}(T)}$ and $S$ ( $T$ ), respectively, indicating that the survival probability threshold and the minimum guarantee reasonably reflect expectations of future values of reference survival probabilities and asset price values. The initial values $S(0)$ and $v$ (0) and the parameters $a$ , $G_{\max }$ , $v_{\min }$ , $v_{\max} $ , $\pi _{\min }$ , and $\pi _{\max }$ are set to arbitrary but reasonable values. In particular, the values of $\pi _{\min }$ and $\pi _{\max }$ are set far from 0 to ensure that they have virtually no impact on pricing. The number of partitions $m_{p}$ and the number of simulations $m_{s}$ are set to 150 and 25,000, respectively. In the LSMC implementation, we use Laguerre polynomials of degree up to $m_{b}=2$ as basis functions for each state variable. The choice of basis functions can be adjusted without significant impact on the results; see Longstaff and Schwartz (Reference Longstaff and Schwartz2001). Table 2 collects all the parameter values used in the baseline investigation.

Table 2.

Parameters used for the base numerical experiments.

The first set of results, shown in Table 3, illustrates the impact of portfolio constraints on the indifference prices of the three MLSs we consider. Regarding the unit-linked security, we report the prices of three payoffs: one for the full payoff $C$ on the left-hand side of (4.1) and the other two for the guaranteed and liability portions from the decomposition on the right-hand side of (4.1). From these results, we find that imposing a short-selling restriction ( $\pi _{\min }=0$ ) or an upper investment constraint ( $\pi_{\max }=200$ ) has little to no impact on the indifference prices.

Table 3.

Indifference prices for the mortality- and unit-linked securities with and without investment constraints.

Notably, the results in the last three rows indicate that the indifference prices for the three payoffs related to the unit-linked security satisfy super-additivity, though this has not been established in a general context. We also note that, although the mortality rate model includes jumps, the parameter estimates do not account for the mortality shocks caused by the COVID-19 pandemic. Nevertheless, we conduct a sensitivity analysis of the indifference prices, with particular attention to the jump intensity $\delta$ of the mortality rate process.

Sensitivity analysis

We now conduct a sensitivity test of indifference prices against various key parameters in the mortality rate and asset price models, including the correlation parameter $\rho $ , the jump intensity $\delta $ and the drift $a_{1}$ of the mortality rate, the expected rate of return $\mu $ , and the drift $\beta _{1}$ of the volatility process. Additionally, we consider the absolute risk aversion parameter $\alpha $ in (2.4). The ranges for some of these parameters are $\rho \in (-1,1)$ , $\delta \in \lbrack 0,2]$ , and $\alpha \in (0,0.03]$ , while the other parameters $a_{1}$ , $\mu $ , and $\beta _{1}$ vary within $\pm 10\%$ of their baseline values listed in Table 2.

The results of the sensitivity analysis are visualized in Figure 1. To supplement the visual analysis, we compute the arc elasticities of the indifference prices with respect to the model parameters using the following formula:

\begin{equation*}\text{arc-elasticity}=\left\vert \frac{y_{2}-y_{1}}{\frac{1}{2}(y_{1}+y_{2})}\cdot \frac{\frac{1}{2}(x_{1}+x_{2})}{x_{2}-x_{1}}\right\vert ,\end{equation*}

where $x_{1},x_{2}$ represent the values of a specific parameter under consideration, and $y_{1},y_{2}$ the resulting indifference prices.

The arc elasticities are computed for each adjacent pair of parameter values and indifference prices. In Table 4, we report the mean and median arc elasticity for each security with respect to each parameter under consideration. These are regarded as measures of central tendency due to the occurrence of outliers, particularly when the parameter value changes from a value close to zero. Since the elasticities are, by definition, nonnegative, the overall direction of the impact can be gleaned from Figure 1.

Table 4.

Arc elasticities of indifference prices with respect to various key mortality and financial parameters.

Figure 1.

Indifference prices for various values of key mortality and financial parameters.

For the S-forward and longevity caplet derivatives, whose payoffs depend solely on the mortality rate, the indifference prices are primarily influenced by mortality rate parameters, with financial model parameters having a “second-order” effect. This is evident from the relatively high (median) elasticities with respect to the mortality rate parameters $\delta $ and $a_{1}$ , compared to other parameters in the asset price model. Moreover, the indifference prices of these instruments exhibit mild elasticity with respect to $\rho $ .

In contrast, for the unit-linked security, the effect of mortality shocks is less pronounced, as indicated by the elasticities with respect to $\delta $ . Furthermore, the indifference prices corresponding to the unit-linked security, particularly the liability portion, are more sensitive to changes in $\rho $ .

The S-forward and longevity caplet are both highly elastic with respect to changes in the jump intensity $\delta $ of the mortality rate process. As $\delta $ increases, the likelihood of jumps in the mortality process rises, which in turn boosts the payoff. In contrast, the indifference price of the unit-linked security exhibits a less straightforward relationship with $\delta $ , exhibiting an upward trend that slows at higher levels of $\delta $ .

Apparently, the indifference prices of all three securities increase with the coefficient of absolute risk aversion $\alpha $ . The issuer, when becoming more risk-averse, demands a higher price to issue a contract to the buyer.

In Appendix A.3, we show that the performance of financial assets, even when assumed to be independent of the mortality rate, still influences the pricing of mortality derivatives.

In the baseline case, the mortality rate process $\{\lambda (t)\}_{0\leq t\leq T}$ is not mean-reverting since $a_{1}\gt0$ . This implies that, in the long run, its continuous part exhibits an upward trend, with a higher $a_{1}$ indicating a faster rate of increase. Our numerical experiments show that the indifference prices of the longevity caplet and the S-forward are highly sensitive to changes in $a_{1}$ . These prices decrease as $a_{1}$ increases because, as the mortality rate rises over time, the payoff declines, leading the issuer to demand a lower price. For the unit-linked security, a higher mortality rate reduces the likelihood of a payoff, and thus its price also tends to decrease with $a_{1}$ , though the effect is less significant than for the longevity caplet and the S-forward.