Abstract
The purpose of this paper is to investigate the valuation of equity-linked death benefit 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 distribution for the time until death, we provide explicit formulas for the expectation of the discounted payment of the guaranteed minimum death benefit 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 (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.
1. 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 high-return 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 exponential 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 ([6–8]) 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, phase-type 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 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.
2. The Model
Consider the problem of GMDB rider that guarantees to the policyholder, , where is the time until death random variable for a life aged and is the minimum guaranteed amount. Because , where , the problem of valuing the guarantee becomes the problem of valuing a -strike put option that is exercised at time . Since 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 denotes a constant force of interest and is an equity-indexed death benefit function. Let denote the probability density function of . Under the assumption that is independent of the stock price , the above expectation is
In this paper, is assumed to follow distributions.
The class of , , distributions is the family of probability distributions whose Laplace transform is given by where , for , and is a polynomial of degree or less. If is an arbitrary , random variable, then the mean and variance of the interclaim time random variables are given by respectively. The class of distributions is widely used in applied probability applications (see for instance [15, 16]).
Under the assumption that is independent of the stock price process , the problem of approximating the expectation (1) reduces to that of evaluating where is an arbitrary , random variable independent of .
If are distinct, then using partial fractions, where
This gives which is the density function of a mixture of exponential distributions, with weights , .
We can use the factorization
Hence, the derivation formulas for are essential.
Let denote the running maximum of the Lévy process up to time . As shown in [2, 3] and [17], the random variables and are independent (which is still true if (even though and are not independent)).
The functions are referred to as discounted density functions; in the case of negative , the adjective inflated might be more appropriate.
Consider the process , where is a standard Brownian motion and and are constants. The process is stopped at time . Unless stated otherwise, in this paper, and are two real numbers, which are the solutions of the following quadratic equation: where is defined as the volatility per unit of time of the process .
Let . We have
Proposition 1. As in [2], for each , The proof can be found in books such as [18, 19].
The pdf of an inverse Gaussian (IG) random variable with parameters , and , i.e., , is and its th moment is where is the modified Bessel function of the third kind.
If instead some of the are not distinct, then using partial fractions where are distinct, .
Then using partial fractions, where
This gives which is the density function of a mixture of the Erlang distributions, with weights , and
We have
Hence, this paper will derive formulas for where we will be looking at an Erlang stopping time .
3. Erlang Stopping of Exponential Wiener Process
Let denote the time price at time of a share of stock or unit of a mutual fund. We assume that where , where represents the drift per unit of time, is the volatility per unit of time, and is the Wiener process.
Theorem 2. Assuming is the Erlang distributed, i.e., , the distribution of the pair is where and are given by (13).
Proof.
Let and Then,
We have where . Using Equation (16), for , we get
Substituting Equation (17) in Equation (30), we get the result for . For , and the result follows.
Theorem 3. Assuming is the Erlang distributed, i.e., , and are given, respectively, by the following: (1)For ,where (2)For ,
Remark 4. For , 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 . According to the expression of given by Theorem 2, we have
By changing the change of variables technique, we have
With the incomplete Gamma function, we have
To have
Since we obtain
and are given by (32).
We also have with
To finally have
For ,
4. Valuation of Options
As in Section 3, we denote by the time ’s price of a share of stock or unit of a mutual fund. We assume where . It is easy to show that , and .
In this section, we evaluate the expected discounted value of the payoff , for various payoff or benefit functions . Under the assumption that the random variable is independent of the process , the expectation (45) is
Since we know that , we have
In the special case where , Equation (47) becomes
Remark 5. If , it is straightforward to show that which is the result in the risk-neutral pricing framework, where represents the risk-free interest rate in the complete market.
4.1. Out-of-the-Money All-or-Nothing Call Option
The payoff function is
Here, is a real number; and are two special cases of particular interest. The constant is greater than ; the term “out-of-the-money” means that the option, if exercised now, is worth nothing. Let which is positive since .
Theorem 6. If , then
Proof. with
4.2. At-the-Money All-or-Nothing Call Option
For , we have
4.3. Out-of-the-Money Call Option
The payoff function is
Here, because the option is out-of-the-money. By applying (51) with and , we have
4.4. At-the-Money Call Option
The payoff function is which is (55) with . Thus, it follows from (54) that
4.5. Out-of-the-Money All-or-Nothing Put Option
The payoff function is
Here, is the real number, and because the option is out-of-the-money. Since , it follows from the following.
Theorem 7. If , then,
Proof. with
4.6. At-the-Money Put Option
For , we have
4.7. Out-of-the-Money Put Option
The payoff function is
By applying (60) with and , we have
4.8. At-the-Money Put Option
The payoff function is
By (63), we have
4.9. In-the-Money Put and Call Options
Theorem 8.
Proof.
We have
Hence,
To finally have
5. Multiple Life Insurance on Two Heads
In this section, we apply 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 and which are assumed to be dependent. The random variables denote the future lifetimes of a life aged and , 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 and are the marginal probability distribution functions of the future life random variables and , respectively. The kernel function is assumed to be bounded and nonconstant such that . The dependence parameter is a real number such that for all . If , 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): (i)Farlie-Gumbel-Morgenstern (FGM) copula case: , where is the cumulative distribution function associated to (ii)Exponential kernel case: (iii)The marginal kernel case:
Define for ; 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 and are following with
In the rest of the paper, we will be using the Erlang-type kernel function.
Then, the joint distribution of and is given as or in a compact form with for and for all with
If both and follow a bivariate Sarmanov distribution, we have the following:
Theorem 9. The CDF and survival functions follow
Proof. (i)Computing of By the Fubini theorem, we have Let ; we have Hence, (ii)Computing of Put ; we have Thus, (iii)Computing of which gives (82). For Equation (83), we have (iv)Computing of By the Fubini theorem, we have Let ; we have Hence, (v)Computing of Put ; we have 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 joint-life status of two lives and will be denoted by , and the moment of death random variable is given by .
Theorem 10. The survival function for 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:
For ,
For , where
is given by (32); and are solutions of Equation (12), with replaced by ; and are also solutions of Equation (12), with replaced by .
5.2. 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 and will be denoted by , and the moment of death random variable is given by .
Theorem 12. The CDF and survival functions follow and the pdf is also given by
Proof. From Theorem 10 and Theorem 12, we can easily notice that the distributions of and have the same form just with different parameters, and one can deduce similarly as in Remark 11.
6. 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 in Table 1 is approximately 71 years. This age is around 67 in Tables 2–4. Clearly, the higher the average age of death, the lower the premium to be paid. This remains true with the modification of other parameters such as the expectation and the volatility . Tables 2 and 3 show that the premium increases with a slight increase in the volatility. This is similar to that of the expectation , but less sensitive than that of the volatility (see Tables 3 and 4).
Therefore, parameter values play an important role in the applicability of the results.
7. 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 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 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.
Acknowledgments
Open access funding is enabled and organized by SANLiC Gold.