Valuing Equity-Linked Death Benefits on Multiple Life with Time until Death following a K n Distribution

. The purpose of this paper is to investigate the valuation of equity-linked death bene ﬁ t contracts and the multiple life insurance on two heads based on a joint survival model. Using the exponential Wiener process assumption for the stock price process and a K n distribution for the time until death, we provide explicit formulas for the expectation of the discounted payment of the guaranteed minimum death bene ﬁ t products, and we derive closed expressions for some options and numerical illustrations. We investigate multiple life insurance based on a joint survival using the bivariate Sarmanov distribution with K n (i.e., the Laplace transform of their density function is a ratio of two polynomials of degree at most) marginal distributions. We present analytical results of the joint-life status.


Introduction
Most classical insurance and bank products have experienced decrease in interest rates. This situation, due to the financial crisis, has led investors to give prominent attention in highreturn products in spite of the high risks involved. Consequently, banks and insurance companies have to innovate by offering attractive products that can yield high rates or allow investors to participate in some underlying asset's benefits. To avoid unwanted market declines, this alternative can be used by stock market investors. As a result, products linked or indexed to a specific value have emerged in the insurance and banking sectors (for instance, variable annuities, guaranteed minimum death benefit (GMDB), and guaranteed minimum living benefit (GMLB)). Although these products are more attractive and meet the expectations of most investors, their valuations are difficult and require an in-depth knowledge of actuarial and financial techniques. In response, [1] proposed a new valuation methodology based on decomposing a liability into two parts (the actuarial or model part and the financial or market part) and then valuing each part individually. Assuming that the underlying stock price follows an expo-nential Brownian motion, [2] analysed the valuation of GMDB using discounted payments to death. Additionally, they assumed that the time to death follows an exponential distribution. Analytical formulas for options such as lookback options and surrenders based on the assumption of independence between stock price and time of death were developed. Although their results are interesting, they are less attractive from a practical perspective, because the assumptions underlying their model (e.g., the exponential Brownian motion process and exponential distribution assumptions) are merely used to simplify the model rather than to ensure its accuracy. Gerber et al. [3] improved their model by adding a jump in the diffusion process and examining their results for equity-linked death benefits. Liang et al. [4] used the same argument as [2] to estimate guarantee equity-linked contracts. Another study looked at term insurance products with equity-linked or inflation-indexed exercise periods. In addition, an analysis of parameter sensitivities has been incorporated. Deelstra and Hieber [5] approximated the distribution of the remaining lifetime by either a series of Erlang's distributions or a Laguerre series expansion to study death-linked contingent claims paying a random financial return at a random time of death in the general case where financial returns follow a regime-switching model with two-sided phase-type jumps. The literature on GMDB valuation contains several other extensions of the pioneering work of [2,3] in other direction. For instance, the regime-switching jump volatility was considered in ( [6][7][8]) and the references therein.
Multiple researchers have proposed different distributions due to the difficulty of finding a corresponding distribution to the time until death. For example, [9] addressed this problem by proposing a Laguerre expansion, which was also applied to the valuation of equity-linked death benefits. Results obtained were more accurate when compared to the results of the existing literature. Phase-type distributions to model human lifetimes were used when phase-type jump is incorporated into the diffusion process by [10]. In terms of matrix representation, they derived a closed analytic expression for price. Because dependency modelling is a key concept in financial and actuarial modelling, we are interested in equity-linked death benefits for multiple life scenarios. In Kim et al.'s [11] study, phasetype distributions are applied to joint-life products and to group risk ordering and pricing within a pool of insureds by exploring the properties of phase-type distributions. Moutanabbir and Abdelrahman [12] utilised the bivariate Sarmanov distribution with phase-type marginal distributions to model dependence between lifetimes. The phase-type distributions are used in [13] to model human mortality. Recently, [14] considered mixed exponential distribution and studied the problem of GMDB valuation for married couple.
In thi paper, we study the problem of GMDB by considering the mixture of Erlang's distributions for time until death and model the underlying stock price process by exponential Wiener process, on the one hand, and the problem to valuing equity-linked death benefits on multiple life based on a joint survival using the bivariate Sarmanov distribution with K n marginal distributions, on the other hand.
The structure of this paper is as follows: the model is presented in Section 2. Section 3 describes the Erlang stopping of a Wiener process. Section 4 provides a valuation of basic options. In Section 5, multiple life insurance is discussed, followed by some numerical results in Section 6.

The Model
Consider the problem of GMDB rider that guarantees to the policyholder, max ðSðT x Þ, KÞ, where T x is the time until death random variable for a life aged x and K is the minimum guaranteed amount. Because max ðSðT x Þ, KÞ = SðT x Þ + max ½K − SðT x Þ + , where max ½K − SðT x Þ + = max ðK − SðT x Þ, 0Þ, the problem of valuing the guarantee becomes the problem of valuing a K-strike put option that is exercised at time T x . Since T x is a random variable, the put option is of neither the European style nor the American style. It is a life-contingent put option. Thus, we are interested in evaluating the expectation where δ denotes a constant force of interest and bðsÞ is an equity-indexed death benefit function. Let f T x denote the prob-ability density function of T x . Under the assumption that T x is independent of the stock price fSðtÞg, the above expectation is In this paper, T x is assumed to follow K n distributions. The class of K n , n ∈ ℕ, distributions is the family of probability distributions whose Laplace transform is given bỹ where β i s i is a polynomial of degree n − 2 or less. If τ is an arbitrary K n , random variable, then the mean and variance of the interclaim time random variables are given by respectively. The class of K n distributions is widely used in applied probability applications (see for instance [15,16]).
Under the assumption that T x is independent of the stock price process fSðtÞg, the problem of approximating the expectation (1) reduces to that of evaluating where τ is an arbitrary K n , random variable independent of fSðtÞg.
If λ 1 , λ 2 , ⋯, λ n are distinct, then using partial fractions, where This gives which is the density function of a mixture of exponential distributions, with weights a i /λ i , i = 1, ⋯, n.

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We can use the factorization Hence, the derivation formulas for are essential. Let Mðτ i Þ denote the running maximum of the Lévy process fXðtÞg up to time τ i . As shown in [2,3] and [17], the random variables Mðτ i Þ and Xðτ i Þ − Mðτ i Þ are independent (which is still true if δ = 0 (even though MðtÞ and ½MðtÞ − XðtÞ are not independent)).
The functions are referred to as discounted density functions; in the case of negative δ, the adjective inflated might be more appropriate.
Consider the process fXðtÞ = μt + σWðtÞ, t ≥ 0g, where WðtÞ is a standard Brownian motion and μ and σ > 0 are constants. The process XðtÞ is stopped at time τ i . Unless stated otherwise, in this paper, α i and β i are two real numbers, which are the solutions of the following quadratic equation: where σ is defined as the volatility per unit of time of the process fXðtÞ, t ≥ 0g. Let Δ 2 i = 1/ηðλ i + ðδ + μ 2 /4ηÞÞ. We have Proposition 1. As in [2], for each t > 0, The proof can be found in books such as [18,19].
The pdf of an inverse Gaussian (IG) random variable W with parameters b, ðb > 0Þ, and ν, ðν > 0Þ, i.e., ðW~IG ðb, νÞÞ, is and its nth moment is where K p is the modified Bessel function of the third kind.
3 Journal of Applied Mathematics If instead some of the λ 1 , λ 2 , ⋯, λ n are not distinct, then using partial fractions where Then using partial fractions, where This gives which is the density function of a mixture of the Erlang distributions, with weights a i,j /λ j i , i = 1, ⋯, k and j = 1, ⋯, n i : Hence, this paper will derive formulas for where we will be looking at an Erlang stopping time τ i .

Erlang Stopping of Exponential Wiener Process
Let SðtÞ denote the time price at time t of a share of stock or unit of a mutual fund. We assume that where XðtÞ = μt + σWðtÞ, where μ represents the drift per unit of time, σ is the volatility per unit of time, and WðtÞ is the Wiener process.

Theorem 2.
Assuming τ i is the Erlang distributed, i.e., τ iẼ rlangðn, λ i Þ, the distribution of the pair ðXðτ i Þ, Mðτ i ÞÞ is where α i and β i are given by (13). Proof.

Theorem 3.
Assuming τ i is the Erlang distributed, i.e., τ iẼ rlangðn, λ i Þ, f δ Xðτ i Þ and f δ Mðτ i Þ are given, respectively, by the following: where (2) For n = 1, Remark 4. For n = 1, the results of Theorem 3 are those obtained in [2]. The mixture of the Erlang distributions is a dense family of distributions, which makes our results more general.
Proof. Assume n ∈ ℕ − f0, 1g. According to the expression of f δ Xðτ i Þ,Mðτ i Þ given by Theorem 2, we have By changing the change of variables technique, we have With the incomplete Gamma function, we have

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Since we obtain φ i and cðn, kÞ are given by (32).
We also have with To finally have For n = 1,

Valuation of Options
As in Section 3, we denote by SðtÞ the time t's price of a share of stock or unit of a mutual fund. We assume where XðtÞ = μt + σWðtÞ. It is easy to show that EðSðtÞÞ = Sð0Þe vt , t ≥ 0, and v = μ + ðσ 2 /2Þ. In this section, we evaluate the expected discounted value of the payoff bðSðτ i ÞÞ, for various payoff or benefit functions bðsÞ. Under the assumption that the random variable τ i is independent of the process SðtÞ, the expectation (45) is Since we know that τ i~E rlangðn, λ i Þ, we have Journal of Applied Mathematics In the special case where bðsÞ = s, Equation (47) becomes Remark 5. If v = δ, it is straightforward to show that E½e −δτ i Sðτ i Þ = Sð0Þ which is the result in the risk-neutral pricing framework, where δ represents the risk-free interest rate in the complete market.
Here, m is a real number; m = 0 and m = 1 are two special cases of particular interest. The constant K is greater than Sð0Þ; the term "out-of-the-money" means that the option, if exercised now, is worth nothing. Let which is positive since K > Sð0Þ. Proof.
Here, K > Sð0Þ because the option is out-of-the-money. By applying (51) with m = 1 and m = 0, we have which is (55) with K = Sð0Þ. Thus, it follows from (54) that 4.5. Out-of-the-Money All-or-Nothing Put Option. The payoff function is Here, m is the real number, and K < Sð0Þ because the option is out-of-the-money. Since θ = ln ðK/Sð0ÞÞ < 0, it follows from the following.

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Proof.
4.6. At-the-Money Put Option. For K = Sð0Þ, we have By applying (60) with m = 0 and m = 1, we have By (63), we have 4.9. In-the-Money Put and Call Options Proof.

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We have Hence, To finally have

Multiple Life Insurance on Two Heads
In this section, we apply K n distributions in the context of joint-life modelling. The survival of the two lives is referred to as the status of interest or simply the status. There are two common types of status: the joint-life and the last survival status. Consider two random variables T x and T y which are assumed to be dependent. The random variables denote the future lifetimes of a life aged x and y, respectively. The dependence can be introduced using copulas or a common shock model. In this paper, we use the bivariate Sarmanov distribution which is given by where f x and f y are the marginal probability distribution functions of the future life random variables T x and T y , respectively. The kernel function fΨ, i = x, yg is assumed to be bounded and nonconstant such that E½Ψ i ðT i Þ = 0. The dependence parameter ω is a real number such that for all s, t ∈ ℝ \ f0g. If ω = 0, then we have achieved independence. The choice of a suitable kernel function is very important. In the literature, the most commonly used kernel functions are as follows (see [20] for details): Define ν i = Ð +∞ 0 sΨ i ðÞf i ðsÞds for i = x, y; then, the covariance and correlation coefficient are given by The maximum attainable correlation for a bivariate Sarmanov distribution is discussed in [21] for the different marginal distributions. In this paper, it is assumed that both T x and T y are following K n with In the rest of the paper, we will be using the Erlang-type kernel function. 9 Journal of Applied Mathematics Then, the joint distribution of T x and T y is given as or in a compact form with g 0 i ðsÞ = 1 for i = x, y and for all s with If both T x and T y follow a bivariate Sarmanov distribution, we have the following: Theorem 9. The CDF and survival functions follow Proof.
By the Fubini theorem, we have Hence, (ii) Computing of h 2 ðs, tÞ = Ð s Journal of Applied Mathematics Thus, (iii) Computing of Hðs, tÞ By the Fubini theorem, we have Hence,

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Thus, (vi) Computing of 5.1. Joint Status. The joint-life status is one that requires the survival of both lives. Accordingly, the status terminates upon the first death of one of the two lives. The jointlife status of two lives x and y will be denoted by ðxyÞ, and the moment of death random variable is given by T ðxyÞ = min ðT x , T y Þ.

Theorem 10. The survival function for T ðxyÞ is given by
Using the survival function, we get the following pdf: Proof.
Remark 11. Clearly, the above distribution is a combination of mixture of the Erlang distribution, since Equation (47) can be generalized as follows: 12 Journal of Applied Mathematics Þx r e −β 2,i x dx: ð104Þ where N = q x + q y , cðN, kÞ is given by (32); α 1,i and β 1,i are solutions of Equation (12), with λ i replaced by λ x i + λ y i ; α 2,i and β 2,i are also solutions of Equation (12), with λ i replaced by γ x + λ x i + γ y + λ y i .

The
Last-Survivor Status. The other common status is the last-survivor status. The last-survivor status is one that ends upon the death of both lives. That is, the status survives as long as at least one of the component members remains alive. The last-survivor status of two lives x and y will be denoted by ð xyÞ, and the moment of death random variable is given by T ð xyÞ = max ðT x , T y Þ.

Theorem 12. The CDF and survival functions follow
13 Journal of Applied Mathematics and the pdf is also given by Proof.
From Theorem 10 and Theorem 12, we can easily notice that the distributions of T ðxyÞ and T ð xyÞ have the same form just with different parameters, and one can deduce E½e

Some Numerical Results
This section presents some numerical results for call and put options.
6.1. Comments. The average age of death calculated with the values of parameters λ i in Table 1 is approximately 71 years. This age is around 67 in Tables 2-4. Clearly, the higher the In-the-money call option S 0 ð Þ = 120 K = 100 25.32719 In-the-money put option S 0 ð Þ = 100 K = 120 21.06781 In-the-money call option S 0 ð Þ = 120 K = 100 23.69608 In-the-money put option S 0 ð Þ = 100 K = 120 19.7145 In-the-money call option S 0 ð Þ = 120 K = 100 39.51472 In-the-money put option S 0 ð Þ = 100 K = 120 32.89137 14 Journal of Applied Mathematics average age of death, the lower the premium to be paid. This remains true with the modification of other parameters such as the expectation μ and the volatility σ. Tables 2 and 3 show that the premium increases with a slight increase in the volatility. This is similar to that of the expectation μ, but less sensitive than that of the volatility σ (see Tables 3 and 4). Therefore, parameter values play an important role in the applicability of the results.

Concluding Remarks
It has provided a contribution to the study of the valuation of equity-linked death benefits. Under the exponential Lévy process assumption for the stock price process and K n distribution for the time until death, explicit formulas are derived for the discounted payment of the guaranteed minimum death benefit products. A closed expression is established for both call and put options. Using a bivariate Sarmanov distribution with K n marginal distributions, we analyze multiple life insurance based on joint survival. Calls and puts are illustrated numerically. In future work, we plan to investigate the case of death following a matrix exponential distribution.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that they have no conflicts of interest. In-the-money call option S 0 ð Þ = 120 K = 100 31.94692 In-the-money put option S 0 ð Þ = 100 K = 120 26.57115 15 Journal of Applied Mathematics