Valuation of Equity-Linked Death Benefits on Two Lives with Dependence

Abstract

The purpose of this paper is to investigate equity-linked death benefits for joint alive and last survivor individuals. Utilizing Farlie–Gumbel–Morgenstern (FGM) type dependency modeling framework, we first analyze the joint distribution of the couple (joint alive and last survival density) when marginal distributions follow mixed exponentials and weighted exponentials distributions. Then, we derive the price of the guaranteed minimum death benefit (GMDB) product. In addition, we provide closed analytical expressions of the price of some financial contingent claim contracts (classical and exotic options). Furthermore, we present some numerical results to support our theoretical results. We show in our numerical example that it is important to model the dependency between two lives (couple) since the price changes as the copula parameter changes.

1. Introduction

Consider the problem of a Guaranteed Minimum Death Benefit (GMDB) rider that guarantees the following payment to the customer’s estate when the customer dies, $max\left(S\left(T_x\right),K\right)$, where $T_x$ is the time-until-death random variable for a life aged x, and K is the guaranteed amount. Because

$$max\left(S\left(T_x\right),K\right)=S\left(T_x\right)+{\left(K-S\left(T_x\right)\right)}_{+},$$

the problem is equivalent to determining the price for an exotic put option when valuing such a product. Several studies have been conducted on classical European and American options with fixed maturities.

The transformation of the market and the risk appetite of investors and policyholders have made classical life contingency products less attractive to policyholders, requiring providers of insurance and financial contracts to develop more complex products than classical (traditional) products, such as variable annuities and minimum guaranteed benefit. These products combine actuarial and financial principles.

This makes the valuation of these products a complex problem that requires deep knowledge of both actuarial and financial pricing techniques (fair valuation). Examples of a new approach of fair valuation that combines both market-consistent and actuarial methods can be found in Dhaene et al. (2017).

In general, scholars have been working on developing accurate pricing models for these products over the past years. In Gerber et al. (2012), the authors addressed the Guaranteed Minimum Death Benefit pricing issue based on the assumption that the death time is exponential (mixed exponentials) and the expected discounted value of the payment was calculated. Extending further in Gerber et al. (2013), they derived the call, put, lookback, and barrier options prices based on the upward and downward stock’s movement.

The lookback option pricing problem was previously studied by Gerber and Shiu (2003). By using Laplace transform techniques, they derived the expected discounted value of dividend payments based on a standard Brownian motion model of the company’s aggregate net income. As a result, they were able to derive the price of some European lookback call and put options in both fixed and floating strike prices. The authors concluded their work by analysing the possibility of stochastic guaranteed levels. Similarly, Buchen and Konstandatos (2005) presented a new way to price lookback options using the Black and Scholes framework. They showed in their analysis that the lookback option is an integrated version of the barrier option. As a result of the work of these researchers, the portfolio present value can be used to price exotic options using an arbitrage-free framework.

Scholars find it challenging to choose a process that can adequately explain the time evolution of the underlying stock process and provide a closed tractable expression of any financial contingent contract. Stochastic process models used by researchers tend to be unrealistic (the Black–Scholes framework). According to Linetsky (2004), the price of a lookback option can be derived in terms of spectral expansion by linking the diffusion maximum and minimum to the hitting times and the spectral decomposition of diffusion hitting times. In addition, he showed that a closed analytical form can be obtained under the constant elasticity variance diffusion framework. Zhang et al. (2020) utilized the exponential Lévy process for modeling the stock price process to analyze the GMDB pricing problem. By using the Fast Fourier Transform, they derived the price of the GMDB and obtained the price for various payoffs. Their numerical results were compared to those computed using B-spline functions of different orders to demonstrate the efficiency and accuracy of their proposed algorithm.

With GMDB, Wang et al. (2021) analyzed the valuation problem of equity-linked annuities with regime-switching jump diffusion models. Their method of Fourier series expansion and Fourier transform has been used to derive closed expressions for some GMDB contracts. Their method’s effectiveness was demonstrated by numerical values that confirmed its efficiency. Accordingly, Ai and Zhang (2022) adopted an exponential regime switching Lévy process for the stock and, by using Fourier constant series expansion techniques, they derived explicit expressions for the price of life contingent lookback options embedded in variables GMDB. Unlike Wang et al. (2021), Kirkby and Nguyen (2021) focused their work on determining the payoff of equity-linked GMDBs and they were able to derive a closed form of the price of such products when the risky index process follows the exponential Lévy process.

The valuation of such an option requires a distribution that reflects the policyholder’s life expectancy or mortality well. Thus, Shevchenko and Luo (2016) assumed stochastic mortality behavior among policyholders. The authors examined the existence of a numerical pricing method via option stochastic control. They developed a method for valuing variable annuities based on Gauss–Hermite quadrature. In their analysis, both incomplete and complete markets were considered. The problem of finding the optimal fund to finance a pensioner of age $\left(x\right)$ was previously analyzed by Dufresne (2007). By assuming a mixed exponential distribution of the lifetime of an individual aged $\left(x\right)$, he derived analytic expressions for the stochastic life annuity distribution.

Most of the insurance and finance research has focused on the death or survival of only one family member. Buying insurance or financial contracts is generally done to protect savings, so it is important to look at the life statuses of both the wife and the husband. As a result, “joint life or last survival” insurance contracts are used in pensions. For a comprehensive overview of annuity for a married couple, we refer the readers to Brown and Poterba (1999) and Matvejevs and Matvejevs (2001).

In this paper, we investigate the valuation problem of GMDBs as in Gerber et al. (2012). For the time to maturity of the option, we take into account both the joint life and the last survival lifetime, unlike Gerber et al. (2012). Additionally, we assume that the lifetime random variables in the married couple are interdependent. Gerber et al. (2012) results are reviewed under a specific dependency structure and a hyper and weighted exponential distributions assumption. To the best of the authors’ knowledge, this is the first study to consider married couples when pricing GMDB contracts.

The paper is structured as follows. In Section 2, we set up the model and derive the distributions of both joint life and last survival status. Then, we derive the discounted density of Brownian motion in Section 3. We go on to discuss some classical options valuations problems in Section 4 and Section 5 and analyze the impact of the dependency as well as the type of life status under consideration on the price in Section 6. Finally, in Section 7, we conclude the paper with an explanation of the limitations and possible extensions of the work.

2. Model

2.1. Multiple-Life Insurance Model

In this section, we apply the mixed exponential distributions in the context of joint-life insurance modelling. This family of distributions allows us to derive some closed-form expression for many useful actuarial quantities. 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 status and the last survival status.

2.2. Joint Life 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 $\left(x\right)$ and $\left(y\right)$ will be denoted by $\left(x,y\right)$, and the moment of death random variable is given by $T_{\left(x,y\right)}=min\left(T_x,T_y\right)$. If the random variables $T_x$ and $T_y$ are dependent and model this dependency via the Farlie–Gumbel–Morgenstern (FGM) copula then, the joint distribution of $\left(T_x,T_y\right)$ is defined as follows:

$$f_{\left(T_x,T_y\right)}\left(u,v\right)=\left(f_{T_x}\left(u\right)-\theta h_{T_x}\left(u\right)\right)f_{T_y}\left(v\right)+2\theta h_{T_x}\left(u\right)f_{T_y}\left(v\right){\overline{F}}_{T_y}\left(v\right),$$

where $h_{T_x}\left(u\right)=f_{T_x}\left(u\right)\left(1-2F_{T_x}\left(u\right)\right)$ and $-1\le \theta \le 1$ is the FGM copula’s parameter.

Hereafter, we denote by

$$\begin{array}{ccc}\hfill {\eta }_{{\lambda }_i,{\lambda }_j{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }& =& {\alpha }_i{\alpha }_j{\beta }_k{\beta }_l\left({\lambda }_i+{\lambda }_j+{\gamma }_k+{\gamma }_l\right);{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k,{\gamma }_l}={\lambda }_i+{\lambda }_j+{\gamma }_k+{\gamma }_l,\hfill \\ \hfill {\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(i_1,k_1)& =& {\alpha }_i^{i_1}{\lambda }_i{\beta }_k^{k_1}{\gamma }_k\left(i_1{\lambda }_i+k_1{\gamma }_k\right);{\eta }_{{\lambda }_i,{\gamma }_k}(i_1,k_1)=i_1{\lambda }_i+k_1{\gamma }_k,\hfill \\ \hfill {\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(i_1,j_1,k_1)& =& {\alpha }_i^{i_1}{\lambda }_i{\alpha }_j^{j_1}{\lambda }_j{\beta }_k^{k_1}{\gamma }_k\left(i_1{\lambda }_i+j_1{\lambda }_j+k_1{\gamma }_k\right),\hfill \\ \hfill {\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(i_1,k_1,l_1)& =& {\alpha }_i^{i_1}{\lambda }_i{\beta }_k^{k_1}{\beta }_k{\beta }_l^{l_1}{\gamma }_k\left(i_1{\lambda }_i+k_1{\gamma }_k+l_1{\gamma }_l\right),\hfill \\ \hfill {\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}(i_1,j_1,k_1)& =& i_1{\lambda }_i+j_1{\lambda }_j+k_1{\gamma }_k;{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}(i_1,k_1,l_1)=i_1{\lambda }_i+k_1{\gamma }_k+l_1{\gamma }_l.\hfill \end{array}$$

Proposition 1.

If $T_x$ and $T_y$ follow hyper-exponential distributions with density functions,$f_{T_x}\left(t\right)={\sum }_{i=1}^n{\alpha }_i{\lambda }_i{e}^{-{\lambda }_{i}t}$, $f_{T_y}\left(t\right)={\sum }_{i=1}^m{\beta }_i{\gamma }_i{e}^{-{\gamma }_{i}t},$ for $t\ge 0$. Then,

$$\begin{array}{ccc}\hfill f_{T_{\left(x,y\right)}}\left(w\right)& =& \left(1+\theta \right)\sum _{k=1}^n\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1)e^{-{\eta }_{{\lambda }_k,{\gamma }_i}(1,1)w}-\theta \left(\sum _{k=1}^m\sum _{i=1}^n{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,1)e^{-{\eta }_{{\lambda }_i,{\gamma }_k}(2,1)w}\right.\hfill \\ & +& 2\left.\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\eta }_{{\lambda }_k,{\gamma }_i,{\gamma }_j}^{\alpha ,\beta }(1,1,1)e^{-{\eta }_{{\lambda }_k,{\gamma }_i,{\gamma }_j}(1,1,1)w}\right)\hfill \\ & +& 2\theta \left(\sum _{k=1}^m\sum _{i=1}^n{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(1,2)e^{-{\eta }_{{\lambda }_i,{\gamma }_k}(1,2)w}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,1)e^{-{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}(1,1,1)w}\right)\hfill \\ & -& 2\theta \left(\sum _{k=1}^m\sum _{i=1}^n{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,2)e^{-{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,2)w}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,2)e^{-{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}(1,1,2)w}\right.\hfill \\ & +& 2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\eta }_{{\lambda }_k,{\gamma }_i,{\gamma }_j}^{\alpha ,\beta }(2,1,1)e^{-{\eta }_{{\lambda }_k,{\gamma }_i,{\gamma }_j}^{\alpha ,\beta }(2,1,1)w}\hfill \\ & +& \left.4\sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=+1}^{n-1}\sum _{j=i+1}^n{\eta }_{{\lambda }_i,{\lambda }_j{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }{e}^{-{\eta }_{{\lambda }_i,{\lambda }_j{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }w}\right),\hfill \end{array}$$

where n and m represent the number of exponential distributions used to construct the hyper-exponential distribution, ${\alpha }_i$ and ${\beta }_i$ represent the probability weight of the distribution and satisfy the following constraint ${\sum }_{i=1}^n{\alpha }_i={\sum }_{j=1}^m{\beta }_j=1$.

Proof. 

See Appendix A. □

2.3. 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 $\left(x\right)$ and $\left(y\right)$ will be denoted by ($\overline{xy}$), and the moment of death random variable is given by $T_{\overline{\left(xy\right)}}=max\left(T_x,T_y\right)$.

Proposition 2.

If $T_x$ and $T_y$ follow hyper-exponential distributions with density functions,

$$f_{T_x}\left(t\right)=\sum _{i=1}^n{\alpha }_i{\lambda }_i{e}^{-{\lambda }_{i}t},f_{T_y}\left(t\right)=\sum _{i=1}^m{\beta }_i{\gamma }_i{e}^{-{\gamma }_{i}t},\mathit{for}t\ge 0.$$

Then, the density of $T_{\overline{\left(xy\right)}}$ is given by

$$\begin{array}{ccc}\hfill f_{\left(T_{\overline{\left(xy\right)}}\right)}\left(w\right)& =& \sum _{i=1}^n{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(1,0)e^{-{\eta }_{{\lambda }_i,{\gamma }_k}(1,0)w}+\left(1-\theta \right)\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(0,1)e^{-{\eta }_{{\lambda }_k,{\gamma }_i}(0,1)w}\hfill \\ & +& \theta \left(\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(0,2)e^{-{\eta }_{{\lambda }_k,{\gamma }_i}(0,2)w}+2\sum _{k=1}^{m-1}\sum _{l=k+1}^m{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(0,1,1)e^{-{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}(0,1,1)w}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^m\sum _{i=1}^n{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,1)e^{-{\eta }_{{\lambda }_i,{\gamma }_k}(2,1)w}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,1)e^{-{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}(1,1,1)w}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^n\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,2)e^{-{\eta }_{{\lambda }_k,{\gamma }_i}(1,2)w}+2\sum _{i=1}^n\sum _{k=1}^{m-1}\sum _{l=k+1}^m{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(1,1,1)e^{-{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}(1,1,1)w}\right)\hfill \\ & -& \theta \left(\sum _{k=1}^m\sum _{i=1}^n{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,2)e^{-{\eta }_{{\lambda }_i,{\gamma }_k}(2,2)w}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,2)e^{-{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}(1,1,2)w}\right.\hfill \\ & +& 2\sum _{i=1}^n\sum _{k=1}^{m-1}\sum _{l=k+1}^m{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(2,1,1)e^{-{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}(2,1,1)w}\hfill \\ & +& 4\left.\sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\eta }_{{\lambda }_i,{\lambda }_j{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }{e}^{-{\eta }_{{\lambda }_i,{\lambda }_j{\gamma }_k,{\gamma }_l}w}\right).\hfill \end{array}$$

Proof. 

See Appendix A. □

2.4. Special Case of the Weighted Exponential Distribution

The pdf of the weighted exponential (WE) distribution is unimodal (contrary to the pdf of the exponential distribution) and the corresponding hazard rate function (hrf) is increasing for all values of t. It also possesses various likelihood ratio properties. Additionally, all of its moments can be calculated explicitly—it follows that the related mean, variance, skewness, kurtosis, coefficient of variation, etc. can be computed easily. The technical details can be found in Gupta and Kundu (2001) and Das and Kundu (2016). On the practical side, the WE distribution is suitable for modelling lifetime data when wear-out or ageing is present, providing a real alternative to the exponential distribution for this aim. The success of this weighted version of the exponential distribution has inspired a generation of researchers and practitioners for more in this direction.

The following definitions can be found in Chesneau et al. (2022).

Definition 1

(Weighted Exponential distribution).

The random variable $T_x$ (respectively, $T_y$ ) is said to have $WE$ distribution, with the shape and scale parameters as $\alpha >0$ and $\lambda >0$, respectively, if the $PDF$ of $T_y$ is

$$f_{T_x}\left(x;\alpha ,\lambda \right)=\frac{\alpha +1}{\alpha }\lambda e^{-\lambda x}\left(1-e^{-\alpha \lambda x}\right);x>0\mathit{and}0\mathit{otherwise}.$$

Respectively,

$$f_{T_y}\left(y;\alpha ,\lambda \right)=\frac{\alpha +1}{\alpha }\lambda e^{-\lambda y}\left(1-e^{-\alpha \lambda y}\right);y>0\mathit{and}0\mathit{otherwise}.$$

We will denote it as $WE\left(\alpha ,\lambda \right)$.

Corollary 1.

If $T_x$ and $T_y$ follow hyper-exponential distributions with density functions,

$$\begin{array}{ccc}\hfill f_{T_x}\left(t\right)& =& {\alpha }_1{\lambda }_1{e}^{-{\lambda }_{1}t}+{\alpha }_2{\lambda }_2{e}^{-{\lambda }_{2}t}\left(1+\alpha \right)\lambda e^{-\left(1+\alpha \right)\lambda t},t\ge 0\hfill \\ \hfill f_{T_y}\left(t\right)& =& {\beta }_1{\gamma }_1{e}^{-{\gamma }_{1}t}+{\beta }_2{\gamma }_2{e}^{-{\gamma }_{2}t}\left(1+\beta \right)\gamma e^{-\left(1+\beta \right)\gamma t},t\ge 0\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {\alpha }_1={\displaystyle \frac{\alpha +1}{\alpha }},{\lambda }_1& =& \lambda ,{\alpha }_2=-{\displaystyle \frac{1}{\alpha }},{\lambda }_2=\left(1+\alpha \right)\lambda ,\hfill \\ \hfill {\beta }_1={\displaystyle \frac{\beta +1}{\beta }};{\gamma }_1& =& \gamma ,{\beta }_2=-{\displaystyle \frac{1}{\beta }}\mathit{and}{\gamma }_2=\left(1+\beta \right)\gamma ,\hfill \end{array}$$

then

$$\begin{array}{ccc}\hfill f_{T_{\left(x,y\right)}}\left(w\right)& =& \left(1+\theta \right)\sum _{k=1}^2\sum _{i=1}^2{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1)e^{-{\eta }_{{\lambda }_k,{\gamma }_i}(1,1)w}\hfill \\ & -& \theta \left(\sum _{k=1}^2\sum _{i=1}^2{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,1)e^{-{\eta }_{{\lambda }_i,{\gamma }_k}(2,1)w}+2\sum _{k=1}^2{\eta }_{{\gamma }_k,{\lambda }_1,{\lambda }_2}^{\alpha ,\beta }(1,1,1)e^{-{\eta }_{{\gamma }_k,{\lambda }_1,{\lambda }_2}(1,1,1)w}\right)\hfill \\ & +& 2\theta \left(\sum _{k=1}^2{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(1,2)e^{-{\eta }_{{\lambda }_i,{\gamma }_k}(1,2)w}+2\sum _{i=1}^2{\eta }_{{\lambda }_i,{\gamma }_1,{\gamma }_2}^{\alpha ,\beta }(1,1,1)e^{-{\eta }_{{\lambda }_i,{\gamma }_1,{\gamma }_2}(1,1,1)w}\right)\hfill \\ & -& 2\theta \left(\sum _{k=1}^2\sum _{i=1}^2{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,2)e^{-{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,2)w}+2\sum _{k=1}^2{\eta }_{{\lambda }_1,{\lambda }_1,{\gamma }_k}^{\alpha ,\beta }(1,1,2)e^{-{\eta }_{{\lambda }_1,{\lambda }_1,{\gamma }_k}(1,1,2)w}\right)\hfill \\ & -& 4\left(\sum _{k=1}^2{\eta }_{{\lambda }_k,{\gamma }_1,{\gamma }_2}^{\alpha ,\beta }(2,1,1)e^{-{\eta }_{{\lambda }_k,{\gamma }_1,{\gamma }_1}^{\alpha ,\beta }(2,1,1)w}+2{\eta }_{{\lambda }_1,{\lambda }_2{\gamma }_1,{\gamma }_2}^{\alpha ,\beta }{e}^{-{\eta }_{{\lambda }_1,{\lambda }_2{\gamma }_1,{\gamma }_2}^{\alpha ,\beta }w}\right).\hfill \end{array}$$

Proof. 

Replace the distribution of $T_x$ and $T_y$ in Proposition 1 with a weighted exponential to complete the proof. □

Corollary 2.

If $T_x$ and $T_y$ follow hyper-exponential distributions with density functions,

$$\begin{array}{ccc}\hfill f_{T_x}\left(t\right)& =& {\alpha }_1{\lambda }_1{e}^{-{\lambda }_{1}t}+{\alpha }_2{\lambda }_2{e}^{-{\lambda }_{2}t}\left(1+\alpha \right)\lambda e^{-\left(1+\alpha \right)\lambda t},t\ge 0\hfill \\ \hfill f_{T_y}\left(t\right)& =& {\beta }_1{\gamma }_1{e}^{-{\gamma }_{1}t}+{\beta }_2{\gamma }_2{e}^{-{\gamma }_{2}t}\left(1+\beta \right)\gamma e^{-\left(1+\beta \right)\gamma t},t\ge 0,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {\alpha }_1={\displaystyle \frac{\alpha +1}{\alpha }},{\lambda }_1& =& \lambda ,{\alpha }_2=-{\displaystyle \frac{1}{\alpha }},{\lambda }_2=\left(1+\alpha \right)\lambda ,\hfill \\ \hfill {\beta }_1={\displaystyle \frac{\beta +1}{\beta }};{\gamma }_1& =& \gamma ,{\beta }_2=-{\displaystyle \frac{1}{\beta }}\mathit{and}{\gamma }_2=\left(1+\beta \right)\gamma ,\hfill \end{array}$$

then

$$\begin{array}{ccc}\hfill f_{\left(T_{\overline{\left(xy\right)}}\right)}\left(w\right)& =& \sum _{i=1}^2{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(1,0)e^{-{\eta }_{{\lambda }_i,{\gamma }_k}(1,0)w}+\left(1-\theta \right)\sum _{i=1}^2{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(0,1)e^{-{\eta }_{{\lambda }_k,{\gamma }_i}(0,1)w}\hfill \\ & +& \theta \left(\sum _{i=1}^2{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(0,2)e^{-{\eta }_{{\lambda }_k,{\gamma }_i}(0,2)w}+2{\eta }_{{\lambda }_i,{\gamma }_1,{\gamma }_2}^{\alpha ,\beta }(0,1,1)e^{-{\eta }_{{\lambda }_i,{\gamma }_1,{\gamma }_2}(0,1,1)w}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^2\sum _{i=1}^2{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,1)e^{-{\eta }_{{\lambda }_i,{\gamma }_k}(2,1)w}+2\sum _{k=1}^2{\eta }_{{\lambda }_1,{\lambda }_2,{\gamma }_k}^{\alpha ,\beta }(1,1,1)e^{-{\eta }_{{\lambda }_1,{\lambda }_2,{\gamma }_k}(1,1,1)w}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^2\sum _{i=1}^2{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,2)e^{-{\eta }_{{\lambda }_k,{\gamma }_i}(1,2)w}+2\sum _{i=1}^2{\eta }_{{\lambda }_i,{\gamma }_1,{\gamma }_2}^{\alpha ,\beta }(1,1,1)e^{-{\eta }_{{\lambda }_i,{\gamma }_1,{\gamma }_2}(1,1,1)w}\right)\hfill \\ & -& \theta \left(\sum _{k=1}^2\sum _{i=1}^2{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,2)e^{-{\eta }_{{\lambda }_i,{\gamma }_k}(2,2)w}+2\sum _{k=1}^2{\eta }_{{\lambda }_1,{\lambda }_2,{\gamma }_k}^{\alpha ,\beta }(1,1,2)e^{-{\eta }_{{\lambda }_1,{\lambda }_2,{\gamma }_k}(1,1,2)w}\right)\hfill \\ & +& 2\left(\sum _{i=1}^2{\eta }_{{\lambda }_i,{\gamma }_1,{\gamma }_2}^{\alpha ,\beta }(2,1,1)e^{-{\eta }_{{\lambda }_i,{\gamma }_1,{\gamma }_2}(2,1,1)w}+2{\eta }_{{\lambda }_1,{\lambda }_2{\gamma }_1,{\gamma }_2}^{\alpha ,\beta }{e}^{-{\eta }_{{\lambda }_1,{\lambda }_2{\gamma }_1,{\gamma }_2}w}\right).\hfill \end{array}$$

Proof. 

Replace the distribution of $T_x$ and $T_y$ in Proposition 2 with a weighted exponential to complete the proof. □

3. Exponential Stopping of Brownian Motion

As in Gerber et al. (2012), let us define

$$X\left(t\right)=\mu t+\sigma W\left(t\right),t\ge 0,$$

(1)

where $W\left(t\right)$ is a standard Brownian motion (Wiener process), and $\mu$ and $\sigma >0$ are constants. Further, let $M\left(t\right)=max\left\{X\left(s\right):0\le s\le t\right\}$ denote the running maximum of the process. Let $f_{X\left(t\right),M\left(t\right)}\left(x,y\right)$, $y\ge max\left(x,0\right)$, denote the joint probability density function of $X\left(t\right)$ and $M\left(t\right)$. Then, the process $X\left(t\right)$ is stopped at time $T_{(x,y)}$ or $T_{\left(\overline{xy}\right)}$, an independent random variable with density defined in Propositions (1) and (2). Hereafter, for simplicity of notation we will denote $T_{(x,y)}$ by $\tau$ and $T_{\left(\overline{xy}\right)}$ by $\overline{\tau }$.

For ${\displaystyle -\delta <\underset{1\le i\le n,1\le j\le m}{min}\left\{{\lambda }_i,{\gamma }_j\right\}}$, we define the following functions:

$$\begin{array}{ccc}\hfill f_{X\left(\tau \right),M\left(\tau \right)}^{\delta }\left(x,y\right)& =& {\int }_0^{\infty }{e}^{-\delta t}{f}_{X\left(t\right),M\left(t\right)}\left(x,y\right)f_{\tau }\left(t\right)\mathrm{d}ty\ge max\left(x,0\right),\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill f_{X\left(\overline{\tau }\right),M\left(\overline{\tau }\right)}^{\delta }\left(x,y\right)& =& {\int }_0^{\infty }{e}^{-\delta t}{f}_{S\left(t\right),M\left(t\right)}\left(x,y\right)f_{\overline{\tau }}\left(t\right)\mathrm{d}ty\ge max\left(x,0\right).\hfill \end{array}$$

(3)

Such functions are referred to as discounted density functions, in the case of negative $\delta$, the adjective inflated might be more appropriate.

Under the conditions of Proposition 1, the joint discounted density of $X\left(\tau \right)$ and $M\left(\tau \right)$ is given by

$$\begin{array}{ccc}\hfill \frac{{\sigma }^2}{2}{f}_{X\left(\tau \right),M\left(\tau \right)}^{\delta }\left(x,y\right)& =& \left(1+\theta \right)\sum _{k=1}^n\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1)exp\left[-b_{i,k}^{\left(1\right)}x-\left(a_{i,k}^{\left(1\right)}-b_{i,k}^{\left(1\right)}\right)y\right]\hfill \\ & -& \theta \left(\sum _{k=1}^m\sum _{i=1}^n{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,1)exp\left[-b_{i,k}^{\left(2\right)}x-\left(a_{i,k}^{\left(2\right)}-b_{i,k}^{\left(2\right)}\right)y\right]\right.\hfill \\ & +& 2\left.\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\eta }_{{\lambda }_k,{\gamma }_i,{\gamma }_j}^{\alpha ,\beta }(1,1,1)exp\left[-b_{k,i,j}^{\left(3\right))}x-\left(a_{k,i,j}^{\left(3\right)}-b_{k,i,j}^{\left(3\right)}\right)y\right]\right)\hfill \\ & +& 2\theta \left(\sum _{i=1}^n\sum _{k=1}^m{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(1,2)exp\left[-b_{i,k}^{\left(4\right)}x-\left(a_{i,k}^{\left(4\right)}-b_{i,k}^{\left(4\right)}\right)y\right]\right.\hfill \\ & +& 2\left.\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,1)exp\left[-b_{k,i,j}^{\left(5\right))}x-\left(a_{k,i,j}^{\left(5\right)}-b_{k,i,j}^{\left(5\right)}\right)y\right]\right)\hfill \\ & -& 2\theta \left(\sum _{k=1}^m\sum _{i=1}^n{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,2)exp\left[-b_{k,i}^{\left(6\right)}x-\left(a_{k,i}^{\left(6\right)}-b_{k,i}^{\left(6\right)}\right)y\right]\right.\hfill \\ & +& 2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,2)exp\left[-b_{j,k,i}^{\left(7\right)}x-\left(a_{j,k,i}^{\left(7\right)}-b_{j,k,i}^{\left(7\right)}\right)y\right]\hfill \\ & +& 2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\eta }_{{\lambda }_k,{\gamma }_i,{\gamma }_j}^{\alpha ,\beta }(2,1,1)exp\left[-b_{k,i,j}^{\left(8\right)}x-\left(a_{k,i,j}^{\left(8\right)}-b_{k,i,j}^{\left(8\right)}\right)y\right]\hfill \\ & +& 4\left.\sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=+1}^{n-1}\sum _{j=i+1}^n{\eta }_{{\lambda }_i,{\lambda }_j{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }exp\left[-b_{i,j,k,l}^{\left(9\right)}x-\left(a_{i,j,k,l}^{\left(9\right)}-b_{i,j,k,l}^{\left(9\right)}\right)y\right]\right),\hfill \end{array}$$

where $a_{k,l,i,j}^{\left(n\right)}>0,b_{k,l,i,j}^{\left(n\right)}<0$ are solutions of quadratic equations in Formula (A2) and $y\ge max(x,0)$.

Let us recall the following formulas that can be found in the books of Jeanblanc et al. (2009) and Borodin and Salminen (2002):

$$\begin{array}{ccc}\hfill f_{X\left(t\right),M\left(t\right)}(x,y)& =& {\displaystyle \frac{2y-x}{2\sqrt{\pi D^3{t}^3}}}exp\left(\frac{\mu x-\frac{1}{2}{\mu }^{2}t-\frac{{\left(2y-x\right)}^2}{2t}}{2D}\right),\hfill \\ \hfill {\int }_0^{\infty }{e}^{-\eta t}{\displaystyle \frac{a{e}^{-\frac{a^2}{2t}}}{\sqrt{2\pi t^3}}}\mathrm{d}t& =& e^{-a\sqrt{2\eta }},\eta \ge 0.\hfill \end{array}$$

(4)

Let us prove the first term of the formula in the above result for the discounted joint density. Let $D=\frac{{\sigma }^2}{2}$; from Equations (2) and (4) we have

$$\begin{array}{ccc}& & \sum _{k=1}^n\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1){\int }_0^{\infty }{e}^{-\delta t}{\displaystyle \frac{2y-x}{2\sqrt{\pi D^3{t}^3}}}{e}^{\frac{\mu x-\frac{1}{2}{\mu }^{2}t-\frac{{\left(2y-x\right)}^2}{2t}}{2D}}{e}^{-{\eta }_{{\lambda }_k,{\gamma }_i}(1,1)t}\mathrm{d}t\hfill \\ & =& \frac{1}{D}\sum _{k=1}^n\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1)e^{\frac{\mu x}{2D}}{\int }_0^{\infty }{e}^{-ϵs}{\displaystyle \frac{a}{\sqrt{2\pi s^3}}}{e}^{-\frac{a^2}{2s}}\mathrm{d}t\hfill \\ & =& \frac{1}{D}\sum _{k=1}^n\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1)e^{\frac{\mu x}{2D}}{e}^{-a\sqrt{2ϵ}},\hfill \end{array}$$

where $ϵ=\frac{{\mu }^2}{4D^2}+\frac{{\eta }_{{\lambda }_k,{\gamma }_i}(1,1)+\delta }{D}$, $a=\frac{2y-x}{\sqrt{2}}$.

Now consider the quadratic equation,

$$D{\zeta }^2+\mu \zeta -\left({\eta }_{{\lambda }_k,{\gamma }_i}(1,1)+\delta \right)=0;$$

its solution is given by

$$\begin{array}{ccc}\hfill \Delta & =& {\mu }^2+4D\left({\eta }_{{\lambda }_k,{\gamma }_i}(1,1)+\delta \right)\hfill \\ \hfill a_{i,k}^{\left(1\right)}& =& \frac{-\mu +\sqrt{\Delta }}{2D},b_{i,k}^{\left(1\right)}=-\frac{\mu +\sqrt{\Delta }}{2D}\hfill \\ \hfill a_{i,k}^{\left(1\right)}+b_{i,k}^{\left(1\right)}& =& -\frac{\mu }{D},a_{i,k}^{\left(1\right)}-b_{i,k}^{\left(1\right)}=2\sqrt{ϵ}.\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill \frac{1}{D}\sum _{k=1}^n\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1)e^{\frac{\mu x}{2D}}{e}^{-a\sqrt{2ϵ}}& =& \frac{1}{D}\sum _{k=1}^n\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1)e^{\frac{-\left(a_{i,k}^{\left(1\right)}+b_{i,k}^{\left(1\right)}\right)x}{2}}{e}^{-\left(2y-x\right)\left(\frac{a_{i,k}^{\left(1\right)}-b_{i,k}^{\left(1\right)}}{2}\right)}\hfill \\ & =& \frac{1}{D}\sum _{k=1}^n\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1)exp\left[-b_{i,k}^{\left(1\right)}x-\left(a_{i,k}^{\left(1\right)}-b_{i,k}^{\left(1\right)}\right)y\right].\hfill \end{array}$$

This completes the proof.

It can similarly be proven that the discounted joint density $X\left(\overline{\tau }\right)$ is given by

$$\begin{array}{ccc}\hfill \frac{{\sigma }^2}{2}{f}_{X\left(\overline{\tau }\right),M\left(\overline{\tau }\right)}^{\delta }(x,y)& =& \sum _{i=1}^n{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(1,0)e^{-d_i^{\left(1\right)}x-\left(c_i^{\left(1\right)}-d_i^{\left(1\right)}\right)y}+\left(1-\theta \right)\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(0,1)e^{-d_i^{\left(2\right)}x-\left(c_i^{\left(2\right)}-d_i^{\left(2\right)}\right)y}\hfill \\ & +& \theta \left(\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(0,2)e^{-d_i^{\left(3\right)}x-\left(c_i^{\left(3\right)}-d_i^{\left(3\right)}\right)y}\right.\hfill \\ & +& \left.2\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(0,1,1)e^{-d_{i,j}^{\left(4\right)}x-\left(c_{i,j}^{\left(4\right)}-d_{i,j}^{\left(4\right)}\right)y}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^m\sum _{i=1}^n{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,1)exp\left[-d_{k,i}^{\left(5\right)}x-\left(c_{k,i}^{\left(5\right)}-d_{k,i}^{\left(5\right)}\right)y\right]\right.\hfill \\ & +& 2\left.\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,1)exp\left[-d_{k,i,j}^{\left(6\right)}x-\left(c_{k,i,j}^{\left(6\right)}-d_{k,i,j}^{\left(6\right)}\right)y\right]\right)\hfill \\ & +& \theta \left(\sum _{k=1}^n\sum _{i=1}^m{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,2)exp\left[-d_{k,i}^{\left(5\right)}x-\left(c_{k,i}^{\left(7\right)}-d_{k,i}^{\left(7\right)}\right)y\right]\right.\hfill \\ & +2& \left.\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(1,1,1)exp\left[-d_{k,i,j}^{\left(8\right)}x-\left(c_{k,i,j}^{\left(8\right)}-d_{k,i,j}^{\left(8\right)}\right)y\right]\right)\hfill \\ & -& \theta \left(\sum _{i=1}^m\sum _{j=1}^n{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,2)exp\left[-d_{i,j}^{\left(9\right)}x-\left(c_{i,j}^{\left(9\right)}-d_{i,j}^{\left(9\right)}\right)y\right]\right.\hfill \\ & +& 2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,2)exp\left[-d_{k,i,j}^{\left(10\right)}x-\left(c_{k,i,j}^{\left(10\right)}-d_{k,i,j}^{\left(10\right)}\right)y\right]\hfill \\ & +& 2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(2,1,1)exp\left[-d_{k,i,j}^{\left(11\right)}x-\left(c_{k,i,j}^{\left(11\right)}-d_{k,i,j}^{\left(11\right)}\right)y\right]\hfill \\ & +& 4\left.\sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\eta }_{{\lambda }_i,{\lambda }_j{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }{e}^{-d_{k,l,i,j}^{\left(12\right)}x-\left(c_{k,l,i,j}^{\left(12\right)}-d_{k,l,i,j}^{\left(12\right)}\right)y}\right),\hfill \end{array}$$

where $c_{k,l,i,j}^{\left(n\right)}>0,d_{k,l,i,j}^{\left(n\right)}<0$ are solutions of quadratic equations in Formula (A3) and $y\ge max(x,0)$.

Integrating over y (respectively over x) yields the discounted densities. Below, are the formulas

$$\begin{array}{ccc}\hfill f_{X\left(\tau \right)}^{\delta }\left(x\right)& =& \left(1+\theta \right)\sum _{k=1}^n\sum _{i=1}^m{\chi }_{i,k}^{\left(1\right)}{e}^{-a_{i,k}^{\left(1\right)}x{\mathrm{𝟙}}_{x\ge 0}-b_{i,k}^{\left(1\right)}x{\mathrm{𝟙}}_{x<0}}-\theta \left(\sum _{k=1}^m\sum _{i=1}^n{\chi }_{i,k}^{\left(2\right)}{e}^{-a_{i,k}^{\left(2\right)}x{\mathrm{𝟙}}_{x\ge 0}-b_{i,k}^{\left(2\right)}x{\mathrm{𝟙}}_{x<0}}\right.\hfill \\ & +& 2\left.\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\chi }_{k,i,j}^{\left(3\right)}{e}^{-a_{k,i,j}^{\left(3\right))}x{\mathrm{𝟙}}_{x\ge 0}-b_{k,i,j}^{\left(3\right)}x{\mathrm{𝟙}}_{x<0}}\right)+2\theta \left(\sum _{k=1}^n\sum _{i=1}^m{\chi }_{i,k}^{\left(4\right)}{e}^{-a_{i,k}^{\left(4\right)}x{\mathrm{𝟙}}_{x\ge 0}-b_{i,k}^{\left(4\right)}x{\mathrm{𝟙}}_{x<0}}\right.\hfill \\ & +& 2\left.\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\chi }_{k,i,j}^{\left(5\right)}{e}^{-a_{k,i,j}^{\left(5\right))}x{\mathrm{𝟙}}_{x\ge 0}-b_{k,i,j}^{\left(5\right)}x{\mathrm{𝟙}}_{x<0}}\right)-2\theta \sum _{k=1}^m\sum _{i=1}^n{\chi }_{k,i}^{\left(6\right)}{e}^{-a_{i,j}^{\left(6\right)}x{\mathrm{𝟙}}_{x\ge 0}-b_{k,i}^{\left(6\right)}x{\mathrm{𝟙}}_{x<0}}\hfill \\ & -& 4\theta \left(\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\chi }_{j,k,i}^{\left(7\right)}{e}^{-a_{j,k,i}^{\left(7\right)}x{\mathrm{𝟙}}_{x\ge 0}-b_{j,k,i}^{\left(7\right)}x{\mathrm{𝟙}}_{x<0}}+\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\chi }_{k,i,j}^{\left(8\right)}{e}^{-a_{k,i,j}^{\left(8\right)}x{\mathrm{𝟙}}_{x\ge 0}-b_{k,i,j}^{\left(8\right)}x{\mathrm{𝟙}}_{x<0}}\right)\hfill \\ \hfill & -& 8\theta \sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=+1}^{n-1}\sum _{j=i+1}^n{\chi }_{i,j,k,l}^{\left(9\right)}exp\left[-a_{i,j,k,l}^{\left(9\right)}x{\mathrm{𝟙}}_{x\ge 0}-b_{i,j,k,l}^{\left(9\right)}x{\mathrm{𝟙}}_{x<0}\right],\hfill \end{array}$$

(5)

and

$$\begin{array}{ccc}\hfill (-1)\times f_{M\left(\tau \right)}^{\delta }\left(y\right)& =& \left(1+\theta \right)\sum _{k=1}^n\sum _{i=1}^m\frac{{\chi }_{i,k}^{\left(1\right)}}{b_{i,k}^{\left(1\right)}}{e}^{-a_{i,k}^{\left(1\right)}y}{\mathrm{𝟙}}_{y\ge 0}-\theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{{\chi }_{i,k}^{\left(2\right)}}{b_{i,k}^{\left(2\right)}}{e}^{-a_{i,k}^{\left(2\right)}y}{\mathrm{𝟙}}_{y\ge 0}\right.\hfill \\ & +& 2\left.\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\chi }_{k,i,j}^{\left(3\right)}}{b_{k,i,j}^{\left(3\right)}}{e}^{-a_{k,i,j}^{\left(3\right))}y}{\mathrm{𝟙}}_{y\ge 0}\right)+2\theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{{\chi }_{i,k}^{\left(5\right)}}{b_{i,k}^{\left(4\right)}}{e}^{-a_{i,k}^{\left(4\right)}y}{\mathrm{𝟙}}_{y\ge 0}\right.\hfill \\ & +& 2\left.\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{k,i,j}^{\left(5\right)}}{b_{k,i,j}^{\left(5\right)}}{e}^{-a_{k,i,j}^{\left(5\right))}y}{\mathrm{𝟙}}_{y\ge 0}\right)-2\theta \sum _{k=1}^m\sum _{i=1}^n\frac{{\chi }_{k,i}^{\left(6\right)}}{b_{k,i}^{\left(6\right)}}{e}^{-a_{k,i}^{\left(6\right)}y}{\mathrm{𝟙}}_{y\ge 0}\hfill \\ & -& 4\theta \left(\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{j,k,i}^{\left(7\right)}}{b_{j,k,i}^{\left(7\right)}}{e}^{-a_{j,k,i}^{\left(7\right)}y}{\mathrm{𝟙}}_{y\ge 0}+\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\chi }_{k,i,j}^{\left(8\right)}}{b_{k,i,j}^{\left(8\right)}}{e}^{-a_{k,i,j}^{\left(8\right)}y}{\mathrm{𝟙}}_{y\ge 0}\right)\hfill \\ \hfill & -& 8\theta \sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=+1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{i,j,k,l}^{\left(9\right)}}{b_{i,j,k,l}^{\left(9\right)}}exp\left[-a_{i,j,k,l}^{\left(9\right)}y\right]{\mathrm{𝟙}}_{y\ge 0},\hfill \end{array}$$

(6)

where

$$\begin{array}{ccc}\hfill {\chi }_{i,k}^{\left(1\right)}& =& \frac{2{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1)}{{\sigma }^2\left(a_{i,k}^{\left(1\right)}-b_{i,k}^{\left(1\right)}\right)},{\chi }_{i,k}^{\left(2\right)}=\frac{2{\beta }_k{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,1)}{{\sigma }^2\left(a_{i,k}^{\left(2\right)}-b_{i,k}^{\left(2\right)}\right)},{\chi }_{k,i,j}^{\left(3\right)}=\frac{2{\eta }_{{\lambda }_k,{\gamma }_i,{\gamma }_j}^{\alpha ,\beta }(1,1,1)}{{\sigma }^2\left(a_{k,i,j}^{\left(3\right)}-b_{k,i,j}^{\left(3\right)}\right)},\hfill \\ \hfill {\chi }_{i,k}^{\left(4\right)}& =& \frac{2{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(1,2)}{{\sigma }^2\left(a_{i,k}^{\left(4\right)}-b_{i,k}^{\left(4\right)}\right)},{\chi }_{k,i,j}^{\left(5\right)}=\frac{2{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,1)}{{\sigma }^2\left(a_{k,i,j}^{\left(5\right)}-b_{k,i,j}^{\left(5\right)}\right)},{\chi }_{k,i}^{\left(6\right)}=\frac{2{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,2)}{{\sigma }^2\left(a_{k,i}^{\left(6\right)}-b_{k,i}^{\left(6\right)}\right)},\hfill \\ \hfill {\chi }_{j,k,i}^{\left(7\right)}& =& \frac{2{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,2)}{{\sigma }^2\left(a_{k,i,j}^{\left(7\right)}-b_{k,i,j}^{\left(7\right)}\right)},{\chi }_{k,i,j}^{\left(8\right)}=\frac{2{\eta }_{{\lambda }_k,{\gamma }_i,{\gamma }_j}^{\alpha ,\beta }(2,1,1)}{{\sigma }^2\left(a_{k,i,j}^{\left(8\right)}-b_{k,i,j}^{\left(8\right)}\right)},{\chi }_{i,j,k,l}^{\left(9\right)}=\frac{2{\eta }_{{\lambda }_i,{\lambda }_j{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }}{{\sigma }^2\left(a_{i,j,k,l}^{\left(9\right)}-b_{i,j,k,l}^{\left(9\right)}\right)}.\hfill \end{array}$$

(7)

Similarly, when we consider the last survival distribution, we have

$$\begin{array}{ccc}\hfill f_{X\left(\overline{\tau }\right)}^{\delta }\left(x\right)& =& \sum _{i=1}^n{\kappa }_i^{\left(1\right)}{e}^{-c_i^{\left(1\right)}x{\mathrm{𝟙}}_{x\ge 0}-d_i^{\left(1\right)}x{\mathrm{𝟙}}_{x<0}}+\left(1-\theta \right)\sum _{i=1}^m{\kappa }_i^{\left(2\right)}{e}^{-c_i^{\left(2\right)}x{\mathrm{𝟙}}_{x\ge 0}-d_i^{\left(2\right)}x{\mathrm{𝟙}}_{x<0}}\hfill \\ & +& \theta \left(\sum _{i=1}^m{\kappa }_i^{\left(3\right)}{e}^{-c_i^{\left(0\right)}x{\mathrm{𝟙}}_{x\ge 0}-d_i^{\left(3\right)}x{\mathrm{𝟙}}_{x<0}}+2\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\kappa }_{i,j}^{\left(4\right)}{e}^{-c_{i,j}^{\left(4\right)}x{\mathrm{𝟙}}_{x\ge 0}-d_{i,j}^{\left(4\right)}x{\mathrm{𝟙}}_{x<0}}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^m\sum _{i=1}^n{\kappa }_{k,i}^{\left(5\right)}{e}^{-c_{k,i}^{\left(5\right)}x{\mathrm{𝟙}}_{x\ge 0}-d_{k,i}^{\left(5\right)}x{\mathrm{𝟙}}_{x<0}}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\kappa }_{k,i,j}^{\left(6\right)}{e}^{-c_{k,i,j}^{\left(6\right)}x{\mathrm{𝟙}}_{x\ge 0}-d_{k,i,j}^{\left(6\right)}x{\mathrm{𝟙}}_{x<0}}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^n\sum _{i=1}^m{\kappa }_{k,i}^{\left(7\right)}{e}^{-c_{k,i}^{\left(7\right)}x{\mathrm{𝟙}}_{x\ge 0}-d_{k,i}^{\left(7\right)}x{\mathrm{𝟙}}_{x<0}}+2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\kappa }_{k,i,j}^{\left(8\right)}{e}^{-c_{k,i,j}^{\left(8\right)}x{\mathrm{𝟙}}_{x\ge 0}-d_{k,i,j}^{\left(8\right)}x{\mathrm{𝟙}}_{x<0}}\right)\hfill \\ & -& \theta \left(\sum _{i=1}^m\sum _{j=1}^n{\kappa }_{i,j}^{\left(9\right)}{e}^{-c_{i,j}^{\left(9\right)}x{\mathrm{𝟙}}_{x\ge 0}-d_{i,j}^{\left(9\right)}x{\mathrm{𝟙}}_{x\ge 0}}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\kappa }_{k,i,j}^{\left(10\right)}{e}^{-c_{k,i,j}^{\left(10\right)}x{\mathrm{𝟙}}_{x\ge 0}-d_{k,i,j}^{\left(10\right)}x{\mathrm{𝟙}}_{x<0}}\right.\hfill \\ & +& 2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\kappa }_{k,i,j}^{\left(11\right)}exp\left[-c_{k,i,j}^{\left(11\right)}x{\mathrm{𝟙}}_{x\ge 0}-d_{k,i,j}^{\left(11\right)}x{\mathrm{𝟙}}_{x<0}\right]\hfill \\ \hfill & +& 4\left.\sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\kappa }_{k,l,i,j}^{\left(12\right)}exp\left[-c_{k,l,i,j}^{\left(12\right)}x{\mathrm{𝟙}}_{x\ge 0}-d_{k,l,i,j}^{\left(12\right)}x{\mathrm{𝟙}}_{x<0}\right]\right)\hfill \end{array}$$

(8)

and

$$\begin{array}{ccc}\hfill (-1)\times f_{M\left(\overline{\tau }\right)}^{\delta }\left(y\right)& =& \sum _{i=1}^n\frac{{\kappa }_i^{\left(1\right)}}{d_i^{\left(1\right)}}{e}^{-c_i^{\left(1\right)}y}{\mathrm{𝟙}}_{y\ge 0}+\left(1-\theta \right)\sum _{i=1}^m\frac{{\kappa }_i^{\left(2\right)}}{d_i^{\left(2\right)}}{e}^{-c_i^{\left(2\right)}y}{\mathrm{𝟙}}_{x\ge 0}\hfill \\ & +& \theta \left(\sum _{i=1}^m\frac{{\kappa }_i^{\left(3\right)}}{d_i^{\left(3\right)}}{e}^{-c_i^{\left(0\right)}y}{\mathrm{𝟙}}_{y\ge 0}+2\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\kappa }_{i,j}^{\left(4\right)}}{d_{i,j}^{\left(4\right)}}{e}^{-c_{i,j}^{\left(4\right)}y}{\mathrm{𝟙}}_{y\ge 0}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{{\kappa }_{k,i}^{\left(5\right)}}{d_{k,i}^{\left(5\right)}}{e}^{-c_{k,i}^{\left(5\right)}y}{\mathrm{𝟙}}_{y\ge 0}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\kappa }_{k,i,j}^{\left(6\right)}}{d_{k,i,j}^{\left(6\right)}}{e}^{-c_{k,i,j}^{\left(6\right)}y}{\mathrm{𝟙}}_{y\ge 0}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{{\kappa }_{k,i}^{\left(7\right)}}{d_{k,i}^{\left(7\right)}}{e}^{-c_{k,i}^{\left(7\right)}y}{\mathrm{𝟙}}_{y\ge 0}+2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\kappa }_{k,i,j}^{\left(8\right)}}{d_{k,i,j}^{\left(8\right)}}{e}^{-c_{k,i,j}^{\left(8\right)}y}{\mathrm{𝟙}}_{y\ge 0}\right)\hfill \\ & -& \theta \left(\sum _{i=1}^m\sum _{j=1}^n\frac{{\kappa }_{i,j}^{\left(9\right)}}{d_{i,j}^{\left(9\right)}}{e}^{-c_{i,j}^{\left(9\right)}y}{\mathrm{𝟙}}_{y\ge 0}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\kappa }_{k,i,j}^{\left(10\right)}}{d_{k,i,j}^{\left(10\right)}}{e}^{-c_{k,i,j}^{\left(10\right)}y}{\mathrm{𝟙}}_{x\ge 0}\right.\hfill \\ & +& 2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\kappa }_{k,i,j}^{\left(11\right)}}{d_{k,i,j}^{\left(11\right)}}exp\left[-c_{k,i,j}^{\left(11\right)}y\right]{\mathrm{𝟙}}_{y\ge 0}\hfill \\ \hfill & +& 4\left.\sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\kappa }_{k,l,i,j}^{\left(12\right)}}{d_{k,l,i,j}^{\left(12\right)}}exp\left[-c_{k,l,i,j}^{\left(12\right)}y\right]{\mathrm{𝟙}}_{y\ge 0}\right),\hfill \end{array}$$

(9)

where

$$\begin{array}{ccc}\hfill {\kappa }_i^{\left(1\right)}& =& \frac{2{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(1,0)}{{\sigma }^2\left(c_i^{\left(1\right)}-d_i^{\left(1\right)}\right)},{\kappa }_i^{\left(2\right)}=\frac{2{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(0,1)}{{\sigma }^2\left(c_i^{\left(2\right)}-d_i^{\left(2\right)}\right)},{\kappa }_i^{\left(3\right)}=\frac{2{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(0,2)}{{\sigma }^2\left(c_i^{\left(3\right)}-d_i^{\left(3\right)}\right)},\hfill \\ \hfill {\kappa }_{i,j}^{\left(4\right)}& =& \frac{2{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(0,1,1)}{{\sigma }^2\left(c_{i,j}^{\left(4\right)}-d_{i,j}^{\left(4\right)}\right)},{\kappa }_{k,i}^{\left(5\right)}=\frac{2{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,1)}{{\sigma }^2\left(c_{k,i}^{\left(5\right)}-d_{k,i}^{\left(5\right)}\right)},{\kappa }_{k,i,j}^{\left(6\right)}=\frac{2{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,1)}{{\sigma }^2\left(c_{k,i,j}^{\left(6\right)}-d_{k,i,j}^{\left(6\right)}\right)},\hfill \\ \hfill {\kappa }_{k,i}^{\left(7\right)}& =& \frac{2{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,2)}{{\sigma }^2\left(c_{k,i}^{\left(7\right)}-d_{k,i}^{\left(7\right)}\right)},{\kappa }_{k,i,j}^{\left(8\right)}=\frac{2{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(1,1,1)}{{\sigma }^2\left(c_{k,i,j}^{\left(8\right)}-d_{k,i,j}^{\left(8\right)}\right)},{\kappa }_{i,j}^{\left(9\right)}=\frac{2{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,2)}{{\sigma }^2\left(c_{i,j}^{\left(9\right)}-d_{i,j}^{\left(9\right)}\right)},\hfill \\ \hfill {\kappa }_{k,i,j}^{\left(10\right)}& =& \frac{2{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,2)}{{\sigma }^2\left(c_{k,i,j}^{\left(10\right)}-d_{k,i,j}^{\left(10\right)}\right)},{\kappa }_{k,i,j}^{\left(11\right)}=\frac{2{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(2,1,1)}{{\sigma }^2\left(c_{k,i,j}^{\left(11\right)}-d_{k,i,j}^{\left(11\right)}\right)},{\kappa }_{k,l,i,j}^{\left(12\right)}=\frac{2{\eta }_{{\lambda }_i,{\lambda }_j{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }}{{\sigma }^2\left(c_{k,l,i,j}^{\left(12\right)}-d_{k,l,i,j}^{\left(12\right)}\right)}.\hfill \end{array}$$

(10)

(i)

By letting $n=m=2$ in Equations (5) and (6), we obtain the discounted densities for weighted exponential distribution in the case of joint alive status;

(ii)

By letting $n=m=2$ in Equations (8) and (9), we obtain the discounted densities for weighted exponential distribution in the case of last survival status.

As we have a closed expression of the discounted density, in both (joint life and last survival) scenarios, we are able to valuate some financial contingent claims and give a closed-analytical expressions. In the following section, we will analyse some basic options and derive some closed-expression of their prices.

4. Valuation of Basic Options

Hereafter, $S\left(t\right)$ represents the underlying stock price process defined by

$$S\left(t\right)=S\left(0\right)e^{X\left(t\right)},$$

(11)

where $X\left(t\right)$ is the linear Brownian motion defined in Equation (1). As shown in Gerber et al. (2012), we have

$$\mathbf{E}\left[S\left(t\right)\right]=S\left(0\right)e^{\nu t},$$

where

$$\nu =\mu +\frac{{\sigma }^2}{2}.$$

(12)

The functions $b\left(S\left(\tau \right)\right)$, $b\left(S\left(\overline{\tau }\right)\right)$ represent the payoff of the financial contingent claim, then, valuing GMDB entails determining

$$\mathbf{E}\left[e^{-\delta \tau }b\left(S\left(\tau \right)\right)\right]\mathrm{and}\mathbf{E}\left[e^{-\delta \overline{\tau }}b\left(S\left(\overline{\tau }\right)\right)\right].$$

(13)

For a different type of option payoff, we can find the corresponding function $b\left(\right)$ (see Gerber et al. 2012).

By assumption, $S\left(t\right)$ and $T_x$ or $T_y$ are independent. So $S\left(t\right)$ and $\tau$ or $\overline{\tau }$ are also independent. Thus, Equation (13) becomes

$$\begin{array}{ccc}\hfill \mathbf{E}\left[e^{-\delta \tau }b\left(S\left(\tau \right)\right)\right]& =\hfill & \hfill {\int }_{-\infty }^{\infty }b\left(s\left(0\right)e^x\right)f_{X\left(\tau \right)}^{\delta }\left(x\right)\mathrm{d}x,\\ \hfill \mathbf{E}\left[e^{-\delta \overline{\tau }}b\left(S\left(\overline{\tau }\right)\right)\right]& =\hfill & \hfill {\int }_{-\infty }^{\infty }b\left(s\left(0\right)e^x\right)f_{X\left(\overline{\tau }\right)}^{\delta }\left(x\right)\mathrm{d}x.\end{array}$$

(14)

For the special case where $b\left(x\right)=x$ we have

$$\begin{array}{ccc}\hfill \mathbf{E}\left[e^{-\delta \tau }b\left(S\left(\tau \right)\right)\right]& =& \left[\left(1+\theta \right)\sum _{k=1}^n\sum _{i=1}^m\right.\frac{{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1)}{\delta +{\eta }_{{\lambda }_k,{\gamma }_i}(1,1)-\nu }-\theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,1)}{\delta +{\eta }_{{\lambda }_i,{\gamma }_k}(2,1)-\nu }\right.\hfill \\ & +& 2\left.\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\eta }_{{\lambda }_k,{\gamma }_i,{\gamma }_j}^{\alpha ,\beta }(1,1,1)}{\delta +{\eta }_{{\lambda }_k,{\gamma }_i,{\gamma }_j}(1,1,1)-\nu }\right)+2\theta \left(\sum _{i=1}^n\sum _{k=1}^m\frac{{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(1,2)}{\delta +{\eta }_{{\lambda }_i,{\gamma }_k}(1,2)-\nu }\right.\hfill \\ & +& 2\left.\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,1)}{\delta +{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}(1,1,1)-\nu }\right)-2\theta \sum _{k=1}^m\sum _{i=1}^n\frac{{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,2)}{\delta +{\eta }_{{\lambda }_i,{\gamma }_k}(2,2)-\nu }\hfill \\ & -& 4\theta \left(\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,2)}{\delta +{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}(1,1,2)-\nu }\right.\hfill \\ & +& \left.\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\eta }_{{\lambda }_k,{\gamma }_i,{\gamma }_j}^{\alpha ,\beta }(2,1,1)}{\delta +{\eta }_{{\lambda }_k,{\gamma }_i,{\gamma }_j}(2,1,1)-\nu }\right)\hfill \\ \hfill & -& \left.8\theta \sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=+1}^{n-1}\sum _{j=i+1}^n\frac{{\eta }_{{\lambda }_i,{\lambda }_j{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }}{\delta +{\eta }_{{\lambda }_i,{\lambda }_j{\gamma }_k,{\gamma }_l}-\nu }\right]S\left(0\right).\hfill \end{array}$$

(15)

Proof. 

Without loss of generality, we will only prove the first part of Equation (15):

$$\begin{array}{ccc}\hfill \mathbf{E}\left[e^{-\delta \tau }b\left(S\left(\tau \right)\right)\right]& =& {\int }_{-\infty }^{\infty }b\left(s\left(0\right)e^x\right)f_{X\left(\tau \right)}^{\delta }\left(x\right)\mathrm{d}x\hfill \\ & =& S\left(0\right)\sum _{k=1}^n\sum _{i=1}^m{\chi }_{i,k}^{\left(1\right)}\left[\frac{1}{1-b_{i,k}^1}+\frac{1}{a_{i,k}^{\left(1\right)}-1}\right]\hfill \\ & =& S\left(0\right)\sum _{k=1}^n\sum _{i=1}^m\frac{{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1)}{D\left(1-b_{i,k}^1\right)\left(a_{i,k}^{\left(1\right)}-1\right)}\hfill \\ & =& S\left(0\right)\sum _{k=1}^n\sum _{i=1}^m\frac{{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1)}{\delta +{\eta }_{{\lambda }_k,{\gamma }_i}(1,1)}\times \frac{-a_{i,k}^{\left(1\right)}{b}_{i,k}^{\left(1\right)}}{\left(1-b_{i,k}^1\right)\left(a_{i,k}^{\left(1\right)}-1\right)}\hfill \\ & =& S\left(0\right)\sum _{k=1}^n\sum _{i=1}^m\frac{{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,1)}{\delta +{\eta }_{{\lambda }_k,{\gamma }_i}(1,1)-\nu },\hfill \end{array}$$

where $D=\frac{{\sigma }^2}{2}$. □

If we consider the last survival, we get

$$\begin{array}{ccc}\hfill \mathbf{E}\left[e^{-\delta \overline{\tau }}b\left(S\left(\overline{\tau }\right)\right)\right]& =& \left[\sum _{i=1}^n\right.\frac{{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(1,0)}{\delta +{\eta }_{{\lambda }_i,{\gamma }_k}(1,0)-\nu }+\left(1-\theta \right)\sum _{i=1}^m\frac{{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(0,1)}{\delta +{\eta }_{{\lambda }_k,{\gamma }_i}(0,1)-\nu }\hfill \\ & +& \theta \left(\sum _{i=1}^m\frac{2{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(0,2)}{\delta +{\eta }_{{\lambda }_k,{\gamma }_i}(0,2)-\nu }+2\sum _{k=1}^{m-1}\sum _{l=k+1}^m\frac{{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(0,1,1)}{\delta +{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}(0,1,1)-\nu }\right)\hfill \\ & +& \theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,1)}{\delta +{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,1)-\nu }+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,1)}{\delta +{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,1)-\nu }\right)\hfill \\ & +& \theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{{\eta }_{{\lambda }_k,{\gamma }_i}^{\alpha ,\beta }(1,2)}{\delta +{\eta }_{{\lambda }_k,{\gamma }_i}(1,2)-\nu }+2\sum _{i=1}^n\sum _{k=1}^{m-1}\sum _{l=k+1}^m\frac{{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(1,1,1)}{\delta +{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}(1,1,1)-\nu }\right)\hfill \\ & -& \theta \left(\sum _{k=1}^m\sum _{j=1}^n\frac{{\eta }_{{\lambda }_i,{\gamma }_k}^{\alpha ,\beta }(2,2)}{\delta +{\eta }_{{\lambda }_i,{\gamma }_k}(2,2)-\nu }+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}^{\alpha ,\beta }(1,1,2)}{\delta +{\eta }_{{\lambda }_i,{\lambda }_j,{\gamma }_k}(1,1,2)-\nu }\right.\hfill \\ & +& 2\sum _{i=1}^n\sum _{k=1}^{m-1}\sum _{l=k+1}^m\frac{{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(2,1,1)}{\delta +{\eta }_{{\lambda }_i,{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }(2,1,1)-\nu }\hfill \\ \hfill & +& 4\left.\sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\left.\frac{{\eta }_{{\lambda }_i,{\lambda }_j{\gamma }_k,{\gamma }_l}^{\alpha ,\beta }}{\delta +{\eta }_{{\lambda }_i,{\lambda }_j{\gamma }_k,{\gamma }_l}-\nu }\right)\right]S\left(0\right).\hfill \end{array}$$

(16)

Remark 1.

(1)

An equivalent formula similar to Equations (15) and (16) can be obtained if $T_x$ and $T_y$ follow Weighted exponential distributions;

(2)

If $\nu =\delta$, it is straightforward to show that $\mathbf{E}\left[e^{-\tau }S\left(\tau \right)\right]=\mathbf{E}\left[e^{-\overline{\tau }}S\left(\overline{\tau }\right)\right]=S\left(0\right)$ which is the result in risk neutral pricing framework when δ represents the risk-free interest rate and the market is complete.

Hereafter, we denote $\mathbf{E}\left[e^{-\delta \tau }b\left(S\left(\tau \right)\right)\right]:={\pi }_1$ and $\mathbf{E}\left[e^{-\delta \overline{\tau }}b\left(S\left(\overline{\tau }\right)\right)\right]:={\pi }_2$

4.1. Out-of-Money All-or-Nothing Call Option

The payoff of this type of contract is given by

$$b\left(x\right)=x^{n_0}{\mathrm{𝟙}}_{\left(x>K\right)},$$

(17)

where K is the strike price.

If we consider this type of financial contract, the price of the joint life contract is given by

$$\begin{array}{ccc}\hfill {\pi }_1& =& \left(1+\theta \right)\sum _{k=1}^n\sum _{i=1}^m\frac{{\chi }_{i,k}^{\left(1\right)}}{a_{i,k}^{\left(1\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{i,k}^{\left(1\right)}}-\theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{{\chi }_{i,k}^{\left(2\right)}}{a_{i,k}^{\left(2\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{i,k}^{\left(2\right)}}\right.\hfill \\ & +& 2\left.\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\chi }_{k,i,j}^{\left(3\right)}}{a_{k,i,j}^{\left(3\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{k,i,j}^{\left(3\right)}}\right)+2\theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{{\chi }_{i,k}^{\left(4\right)}}{a_{i,k}^{\left(4\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{i,k}^{\left(4\right)}}\right.\hfill \\ & +& 2\left.\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{k,i,j}^{\left(5\right)}}{a_{k,i,j}^{\left(5\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{k,i,j}^{\left(5\right)}}\right)-2\theta \sum _{k=1}^m\sum _{i=1}^n\frac{{\chi }_{k,i}^{\left(6\right)}}{a_{k,i}^{\left(6\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{k,i}^{\left(6\right)}}\hfill \\ & -& 4\theta \left(\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{j,k,i}^{\left(7\right)}}{a_{j,k,i}^{\left(7\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{j,k,i}^{\left(7\right)}}+\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\chi }_{k,i,j}^{\left(8\right)}}{a_{k,i,j}^{\left(8\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{k,i,j}^{\left(8\right)}}\right)\hfill \\ \hfill & -& 8\theta \sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=+1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{i,j,k,l}^{\left(9\right)}}{a_{i,j,k,l}^{\left(9\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{i,j,k,l}^{\left(9\right)}}.\hfill \end{array}$$

(18)

This formula is valid if $n_0$ is less than the smallest non-negative root of Equation (A2).

Similarly, if we consider the last survival case and if $n_0$ does not exceed the smallest non-negative root of Equation (A3) then, the price of this financial contract becomes

$$\begin{array}{ccc}\hfill {\pi }_2& =& \sum _{i=1}^n\frac{{\kappa }_i^{\left(1\right)}}{c_i^{\left(1\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{c_i^{\left(1\right)}}+\left(1-\theta \right)\sum _{i=1}^m\frac{{\kappa }_i^{\left(2\right)}}{c_i^{\left(2\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{c_i^{\left(2\right)}}\hfill \\ & +& \theta \left(\sum _{i=1}^m\frac{{\kappa }_i^{\left(3\right)}}{c_i^{\left(3\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{c_i^{\left(3\right)}}+2\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\kappa }_{i,j}^{\left(4\right)}}{c_{i,j}^{\left(4\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{i,j}^{\left(4\right)}}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{{\kappa }_{k,i}^{\left(5\right)}}{c_{k,i}^{\left(5\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{k,i}^{\left(5\right)}}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\kappa }_{k,i,j}^{\left(6\right)}}{c_{k,i,j}^{\left(6\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{k,i,j}^{\left(6\right)}}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{{\kappa }_{k,i}^{\left(7\right)}}{c_{k,i}^{\left(7\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{c_i^{\left(7\right)}}+2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\kappa }_{k,i,j}^{\left(8\right)}}{c_{k,i,j}^{\left(8\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{k,i,j}^{\left(8\right)}}\right)\hfill \\ & -& \theta \left(\sum _{i=1}^m\sum _{j=1}^n\frac{{\kappa }_{i,j}^{\left(9\right)}}{c_{i,j}^{\left(9\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{i,j}^{\left(9\right)}}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\kappa }_{k,i,j}^{\left(10\right)}}{c_{k,i,j}^{\left(10\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{k,i,j}^{\left(10\right)}}\right.\hfill \\ & +& 2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\kappa }_{k,i,j}^{\left(11\right)}}{c_{k,i,j}^{\left(11\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{k,i,j}^{\left(11\right)}}\hfill \\ \hfill & +& 4\left.\sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\kappa }_{k,l,i,j}^{\left(12\right)}}{c_{k,l,i,j}^{\left(12\right)}-n_0}{K}^{n_0}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{k,l,i,j}^{\left(12\right)}}\right).\hfill \end{array}$$

(19)

4.2. At-the-Money All-or-Nothing Call Option

At-the-money option, the stock price at time 0 equal the strike price. Under this condition, Equations (18) and (19) become

$$\begin{array}{ccc}\hfill {\pi }_1& =& \left(1+\theta \right)\sum _{k=1}^n\sum _{i=1}^m\frac{{\chi }_{i,k}^{\left(1\right)}{K}^{n_0}}{a_{i,k}^{\left(1\right)}-n_0}-\theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{{\chi }_{i,k}^{\left(2\right)}{K}^{n_0}}{a_{i,k}^{\left(2\right)}-n_0}+2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\chi }_{k,i,j}^{\left(3\right)}}{a_{k,i,j}^{\left(3\right)}-n_0}\right)\hfill \\ & +& 2\theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{{\chi }_{i,k}^{\left(4\right)}{K}^{n_0}}{a_{i,k}^{\left(4\right)}-n_0}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{k,i,j}^{\left(5\right)}}{a_{k,i,j}^{\left(5\right)}-n_0}\right)-2\theta \sum _{k=1}^m\sum _{i=1}^n\frac{{\chi }_{k,i}^{\left(6\right)}{K}^{n_0}}{a_{k,i}^{\left(6\right)}-n_0}\hfill \\ & -& 4\theta \left(\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{j,k,i}^{\left(7\right)}{K}^{n_0}}{a_{j,k,i}^{\left(7\right)}-n_0}+\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\chi }_{k,i,j}^{\left(8\right)}{K}^{n_0}}{a_{k,i,j}^{\left(8\right)}-n_0}2\sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{i,j,k,l}^{\left(9\right)}{K}^{n_0}}{a_{i,j,k,l}^{\left(9\right)}-n_0}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\pi }_2& =& \sum _{i=1}^n\frac{{\kappa }_i^{\left(1\right)}{K}^{n_0}}{c_i^{\left(1\right)}-n_0}+\left(1-\theta \right)\sum _{i=1}^m\frac{{\kappa }_i^{\left(2\right)}{K}^{n_0}}{c_i^{\left(2\right)}-n_0}+\theta \left(\sum _{i=1}^m\frac{{\kappa }_i^{\left(3\right)}{K}^{n_0}}{c_i^{\left(3\right)}-n_0}+2\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\kappa }_{i,j}^{\left(4\right)}{K}^{n_0}}{c_{i,j}^{\left(4\right)}-n_0}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{{\kappa }_{k,i}^{\left(5\right)}{K}^{n_0}}{c_{k,i}^{\left(5\right)}-n_0}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\kappa }_{k,i,j}^{\left(6\right)}{K}^{n_0}}{c_{k,i,j}^{\left(6\right)}-n_0}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{{\kappa }_{k,i}^{\left(7\right)}{K}^{n_0}}{c_{k,i}^{\left(7\right)}-n_0}+2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\kappa }_{k,i,j}^{\left(8\right)}{K}^{n_0}}{c_{k,i,j}^{\left(8\right)}-n_0}\right)\hfill \\ & -& \theta \left(\sum _{i=1}^m\sum _{j=1}^n\frac{{\kappa }_{i,j}^{\left(9\right)}{K}^{n_0}}{c_{i,j}^{\left(9\right)}-n_0}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\kappa }_{k,i,j}^{\left(10\right)}{K}^{n_0}}{c_{k,i,j}^{\left(10\right)}-n_0}+2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\kappa }_{k,i,j}^{\left(11\right)}{K}^{n_0}}{c_{k,i,j}^{\left(11\right)}-n_0}\right.\hfill \\ & +& 4\left.\sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\kappa }_{k,l,i,j}^{\left(12\right)}{K}^{n_0}}{c_{k,l,i,j}^{\left(12\right)}-n_0}\right).\hfill \end{array}$$

4.3. Out-of-Money Call Option

The payoff of this type of contract is given by

$$b\left(x\right)={\left(x-K\right)}_{+}=x{\mathrm{𝟙}}_{(x>K)}-K{\mathrm{𝟙}}_{(x>K)}.$$

(20)

As the option is out-of money, $S\left(0\right)<K$. Then applying, Equation (18) yields the following when we consider the joint life contract.

$$\begin{array}{ccc}\hfill {\pi }_1& =& \left(1+\theta \right)\sum _{k=1}^n\sum _{i=1}^m\frac{K{\chi }_{i,k}^{\left(1\right)}}{a_{i,k}^{\left(1\right)}\left(a_{i,k}^{\left(1\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{i,k}^{\left(1\right)}}-\theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{K{\chi }_{i,k}^{\left(2\right)}}{a_{i,k}^{\left(2\right)}\left(a_{i,k}^{\left(2\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{i,k}^{\left(2\right)}}\right.\hfill \\ & +& 2\left.\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{K{\chi }_{k,i,j}^{\left(3\right)}}{a_{k,i,j}^{\left(3\right)}\left(a_{k,i,j}^{\left(3\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{k,i,j}^{\left(3\right)}}\right)+2\theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{K{\chi }_{i,k}^{\left(4\right)}}{a_{i,k}^{\left(4\right)}\left(a_{i,k}^{\left(4\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{i,k}^{\left(4\right)}}\right.\hfill \\ & +& 2\left.\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{K{\chi }_{k,i,j}^{\left(5\right)}}{a_{k,i,j}^{\left(5\right)}\left(a_{k,i,j}^{\left(5\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{k,i,j}^{\left(3\right)}}\right)-2\theta \sum _{k=1}^m\sum _{i=1}^n\frac{K{\chi }_{k,i}^{\left(6\right)}}{a_{k,i}^{\left(6\right)}\left(a_{k,i}^{\left(6\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{k,i}^{\left(6\right)}}\hfill \\ & -& 4\theta \left(\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{K{\chi }_{j,k,i}^{\left(7\right)}}{a_{j,k,i}^{\left(7\right)}\left(a_{j,k,i}^{\left(7\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{j,k,i}^{\left(7\right)}}+\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{K{\chi }_{k,i,j}^{\left(8\right)}}{a_{k,i,j}^{\left(8\right)}\left(a_{k,i,j}^{\left(8\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{k,i,j}^{\left(8\right)}}\right)\hfill \\ \hfill & -& 8\theta \sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=+1}^{n-1}\sum _{j=i+1}^n\frac{K{\chi }_{i,j,k,l}^{\left(9\right)}}{a_{i,j,k,l}^{\left(9\right)}\left(a_{i,j,k,l}^{\left(9\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{a_{i,j,k,l}^{\left(9\right)}}.\hfill \end{array}$$

(21)

Similarly, we consider the last survival contract, the price of out-of-money call option is

$$\begin{array}{ccc}\hfill {\pi }_2& =& \sum _{i=1}^n\frac{K{\kappa }_i^{\left(1\right)}}{c_i^{\left(1\right)}\left(c_i^{\left(1\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{c_i^{\left(1\right)}}+\left(1-\theta \right)\sum _{i=1}^m\frac{K{\kappa }_i^{\left(2\right)}}{c_i^{\left(2\right)}\left(c_i^{\left(2\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{c_i^{\left(2\right)}}\hfill \\ & +& \theta \left(\sum _{i=1}^m\frac{K{\kappa }_i^{\left(3\right)}}{c_i^{\left(3\right)}\left(c_i^{\left(3\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{c_i^{\left(3\right)}}+2\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{K{\kappa }_{i,j}^{\left(4\right)}}{c_{i,j}^{\left(4\right)}\left(c_{i,j}^{\left(4\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{i,j}^{\left(4\right)}}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{K{\kappa }_{k,i}^{\left(5\right)}}{c_{k,i}^{\left(5\right)}\left(c_{k,i}^{\left(5\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{k,i}^{\left(5\right)}}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{K{\kappa }_{k,i,j}^{\left(6\right)}}{c_{k,i,j}^{\left(6\right)}\left(c_{k,i,j}^{\left(6\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{k,i,j}^{\left(6\right)}}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{K{\kappa }_{k,i}^{\left(7\right)}}{c_{k,i}^{\left(7\right)}\left(c_{k,i}^{\left(7\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{c_i^{\left(7\right)}}+2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{K{\kappa }_{k,i,j}^{\left(8\right)}}{c_{k,i,j}^{\left(8\right)}\left(c_{k,i,j}^{\left(8\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{k,i,j}^{\left(8\right)}}\right)\hfill \\ & -& \theta \left(\sum _{i=1}^m\sum _{j=1}^n\frac{K{\kappa }_{i,j}^{\left(9\right)}}{c_{i,j}^{\left(9\right)}\left(c_{i,j}^{\left(9\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{i,j}^{\left(9\right)}}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{K{\kappa }_{k,i,j}^{\left(10\right)}}{c_{k,i,j}^{\left(10\right)}\left(c_{k,i,j}^{\left(10\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{k,i,j}^{\left(10\right)}}\right.\hfill \\ & +& 2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{K{\kappa }_{k,i,j}^{\left(11\right)}}{c_{k,i,j}^{\left(11\right)}\left(c_{k,i,j}^{\left(11\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{k,i,j}^{\left(11\right)}}\hfill \\ \hfill & +& 4\left.\sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{K{\kappa }_{k,l,i,j}^{\left(12\right)}}{c_{k,l,i,j}^{\left(12\right)}\left(c_{k,l,i,j}^{\left(12\right)}-1\right)}{\left(\frac{S\left(0\right)}{K}\right)}^{c_{k,l,i,j}^{\left(12\right)}}\right).\hfill \end{array}$$

(22)

4.4. At-the Money Call Option

At-the-money option, the stock price at time 0 equal the strike price. Under this condition, Equations (21) and (22) become

$$\begin{array}{ccc}\hfill {\pi }_1& =& \left(1+\theta \right)\sum _{k=1}^n\sum _{i=1}^m\frac{S\left(0\right){\chi }_{i,k}^{\left(1\right)}}{a_{i,k}^{\left(1\right)}\left(a_{i,k}^{\left(1\right)}-1\right)}-\theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{S\left(0\right){\chi }_{i,k}^{\left(2\right)}}{a_{i,k}^{\left(2\right)}\left(a_{i,k}^{\left(2\right)}-1\right)}\right.\hfill \\ & +& 2\left.\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{S\left(0\right){\chi }_{k,i,j}^{\left(3\right)}}{a_{k,i,j}^{\left(3\right)}\left(a_{k,i,j}^{\left(3\right)}-1\right)}\right)+2\theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{S\left(0\right){\chi }_{i,k}^{\left(4\right)}}{a_{i,k}^{\left(4\right)}\left(a_{i,k}^{\left(4\right)}-1\right)}\right.\hfill \\ & +& 2\left.\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{S\left(0\right){\chi }_{k,i,j}^{\left(5\right)}}{a_{k,i,j}^{\left(5\right)}\left(a_{k,i,j}^{\left(5\right)}-1\right)}\right)-2\theta \sum _{k=1}^m\sum _{i=1}^n\frac{S\left(0\right){\chi }_{k,i}^{\left(6\right)}}{a_{k,i}^{\left(6\right)}\left(a_{k,i}^{\left(6\right)}-1\right)}\hfill \\ & -& 4\theta \left(\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{S\left(0\right){\chi }_{j,k,i}^{\left(7\right)}}{a_{j,k,i}^{\left(7\right)}\left(a_{j,k,i}^{\left(7\right)}-1\right)}+\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{S\left(0\right){\chi }_{k,i,j}^{\left(8\right)}}{a_{k,i,j}^{\left(8\right)}\left(a_{k,i,j}^{\left(8\right)}-1\right)}\right)\hfill \\ & -& 8\theta \sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{S\left(0\right){\chi }_{i,j,k,l}^{\left(9\right)}}{a_{i,j,k,l}^{\left(9\right)}\left(a_{i,j,k,l}^{\left(9\right)}-1\right)}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\pi }_2& =& \sum _{i=1}^n\frac{S\left(0\right){\kappa }_i^{\left(1\right)}}{c_i^{\left(1\right)}\left(c_i^{\left(1\right)}-1\right)}+\left(1-\theta \right)\sum _{i=1}^m\frac{S\left(0\right){\kappa }_i^{\left(2\right)}}{c_i^{\left(2\right)}\left(c_i^{\left(2\right)}-1\right)}+\theta \left(\sum _{i=1}^m\frac{S\left(0\right){\kappa }_i^{\left(3\right)}}{c_i^{\left(3\right)}\left(c_i^{\left(3\right)}-1\right)}\right.\hfill \\ & +& 2\left.\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{S\left(0\right){\kappa }_{i,j}^{\left(4\right)}}{c_{i,j}^{\left(4\right)}\left(c_{i,j}^{\left(4\right)}-1\right)}\right)+\theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{S\left(0\right){\kappa }_{k,i}^{\left(5\right)}}{c_{k,i}^{\left(5\right)}\left(c_{k,i}^{\left(5\right)}-1\right)}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{S\left(0\right){\kappa }_{k,i,j}^{\left(6\right)}}{c_{k,i,j}^{\left(6\right)}\left(c_{k,i,j}^{\left(6\right)}-1\right)}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{S\left(0\right){\kappa }_{k,i}^{\left(7\right)}}{c_{k,i}^{\left(7\right)}\left(c_{k,i}^{\left(7\right)}-1\right)}+2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{S\left(0\right){\kappa }_{k,i,j}^{\left(8\right)}}{c_{k,i,j}^{\left(8\right)}\left(c_{k,i,j}^{\left(8\right)}-1\right)}\right)\hfill \\ & -& \theta \left(\sum _{i=1}^m\sum _{j=1}^n\frac{S\left(0\right){\kappa }_{i,j}^{\left(9\right)}}{c_{i,j}^{\left(9\right)}\left(c_{i,j}^{\left(9\right)}-1\right)}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{S\left(0\right){\kappa }_{k,i,j}^{\left(10\right)}}{c_{k,i,j}^{\left(10\right)}\left(c_{k,i,j}^{\left(10\right)}-1\right)}\right.\hfill \\ & +& 2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{S\left(0\right){\kappa }_{k,i,j}^{\left(11\right)}}{c_{k,i,j}^{\left(11\right)}\left(c_{k,i,j}^{\left(11\right)}-1\right)}+4\left.\sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{S\left(0\right){\kappa }_{k,l,i,j}^{\left(12\right)}}{c_{k,l,i,j}^{\left(12\right)}\left(c_{k,l,i,j}^{\left(12\right)}-1\right)}\right).\hfill \end{array}$$

4.5. Out-of-Money All-or-Nothing Put Option

For such a financial contingent, the function $b\left(\right)$ is defined by

$$b\left(x\right)=x^{n_0}{\mathrm{𝟙}}_{\left(x<K\right)},$$

(23)

where $n_0\in \mathbb{N}$ and $K<S\left(0\right)$.

By considering the joint live status, one can show that the price of such a contingent claim can be expressed as follows:

$$\begin{array}{ccc}\hfill {\pi }_1& =& \left(1+\theta \right)\sum _{k=1}^n\sum _{i=1}^m\frac{{\chi }_{i,k}^{\left(1\right)}{K}^{n_0}}{n_0-b_{i,k}^{\left(1\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{i,k}^{\left(1\right)}}-\theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{{\chi }_{i,k}^{\left(2\right)}{K}^{n_0}}{n_0-b_{i,k}^{\left(2\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{i,k}^{\left(2\right)}}\right.\hfill \\ & +& 2\left.\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\chi }_{k,i,j}^{\left(3\right)}{K}^{n_0}}{n_0-b_{k,i,j}^{\left(3\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{k,i,j}^{\left(3\right)}}\right)+2\theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{{\chi }_{i,k}^{\left(4\right)}{K}^{n_0}}{n_0-b_{i,k}^{\left(4\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{i,k}^{\left(4\right)}}\right.\hfill \\ & +& 2\left.\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{k,i,j}^{\left(5\right)}{K}^{n_0}}{n_0-b_{k,i,j}^{\left(5\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{k,i,j}^{\left(5\right)}}\right)-2\theta \sum _{k=1}^m\sum _{i=1}^n\frac{{\chi }_{k,i}^{\left(6\right)}{K}^{n_0}}{n_0-b_{k,i}^{\left(6\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{k,i}^{\left(6\right)}}\hfill \\ & -& 4\theta \left(\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{j,k,i}^{\left(7\right)}{K}^{n_0}}{n_0-b_{j,k,i}^{\left(7\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{j,k,i}^{\left(7\right)}}+\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\chi }_{k,i,j}^{\left(8\right)}{K}^{n_0}}{n_0-b_{k,i,j}^{\left(8\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{k,i,j}^{\left(8\right)}}\right)\hfill \\ \hfill & -& 8\theta \sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=+1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{i,j,k,l}^{\left(9\right)}{K}^{n_0}}{n_0-b_{i,j,k,l}^{\left(9\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{i,j,k,l}^{\left(9\right)}}.\hfill \end{array}$$

(24)

Proof. 

Without loss of generality, let us show the first part of the formula.

Let $k=ln\left[\frac{K}{S\left(0\right)}\right]$, since the contract is a put option, $k<0$. Hence we have:

$$\begin{array}{ccc}\hfill {\chi }_{i,k}^{\left(1\right)}{\int }_{-\infty }^k{\left(S\left(0\right)e^x\right)}^{n_0}{e}^{b_{i,k}^{\left(1\right)}x}\mathrm{d}x& =& {\chi }_{i,k}^{\left(1\right)}{\left(S\left(0\right)\right)}^{n_0}\frac{e^{-\left(b_{i,k}^{\left(1\right)}-n_0\right)k}}{-\left(b_{i,k}^{\left(1\right)}-n_0\right)}\hfill \\ & =& \frac{{\chi }_{i,k}^{\left(1\right)}{K}^{n_0}}{n_0-b_{i,k}^{\left(1\right)}}{\left[\frac{K}{S\left(0\right)}\right]}^{-b_{i,k}^{\left(1\right)}}.\hfill \end{array}$$

We complete the proof by using the linearity of the integral. □

Similarly, when the contract is written on the last survival status, we have

$$\begin{array}{ccc}\hfill {\pi }_2& =& \sum _{i=1}^n\frac{{\kappa }_i^{\left(1\right)}{K}^{n_0}}{n_0-d_i^{\left(1\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-d_i^{\left(1\right)}}+\left(1-\theta \right)\sum _{i=1}^m\frac{{\kappa }_i^{\left(2\right)}{K}^{n_0}}{n_0-d_i^{\left(2\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-d_i^{\left(2\right)}}\hfill \\ & +& \theta \left(\sum _{i=1}^m\frac{{\kappa }_i^{\left(3\right)}{K}^{n_0}}{n_0-d_i^{\left(3\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-d_i^{\left(3\right)}}+2\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\kappa }_{i,j}^{\left(4\right)}{K}^{n_0}}{n_0-d_{i,j}^{\left(4\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-d_{i,j}^{\left(4\right)}}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{{\kappa }_{k,i}^{\left(5\right)}{K}^{n_0}}{n_0-d_{k,i}^{\left(5\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-d_{k,i}^{\left(5\right)}}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\kappa }_{k,i,j}^{\left(6\right)}{K}^{n_0}}{n_0-d_{k,i,j}^{\left(6\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-d_{k,i,j}^{\left(6\right)}}\right)\hfill \\ & +& \theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{{\kappa }_{k,i}^{\left(7\right)}{K}^{n_0}}{n_0-d_{k,i}^{\left(7\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{c_i^{\left(7\right)}}+2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\kappa }_{k,i,j}^{\left(8\right)}{K}^{n_0}}{n_0-d_{k,i,j}^{\left(8\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-d_{k,i,j}^{\left(8\right)}}\right)\hfill \\ & -& \theta \left(\sum _{i=1}^m\sum _{j=1}^n\frac{{\kappa }_{i,j}^{\left(9\right)}{K}^{n_0}}{n_0-d_{i,j}^{\left(9\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-d_{i,j}^{\left(9\right)}}+2\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\kappa }_{k,i,j}^{\left(10\right)}{K}^{n_0}}{n_0-d_{k,i,j}^{\left(10\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-d_{k,i,j}^{\left(10\right)}}\right.\hfill \\ & +& 2\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\kappa }_{k,i,j}^{\left(11\right)}{K}^{n_0}}{n_0-d_{k,i,j}^{\left(11\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-d_{k,i,j}^{\left(11\right)}}\hfill \\ \hfill & +& 4\left.\sum _{k=1}^{m-1}\sum _{l=k+1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\kappa }_{k,l,i,j}^{\left(12\right)}{K}^{n_0}}{n_0-d_{k,l,i,j}^{\left(12\right)}}{\left(\frac{K}{S\left(0\right)}\right)}^{-d_{k,l,i,j}^{\left(12\right)}}\right).\hfill \end{array}$$

(25)

Remark 2.

(1)

When we let $K=S\left(0\right)$ in Equations (24) and (25), we obtain the price when the option is at-the money;

(2)

When $n=2$ and $m=2$, we obtain the formula for the option price when individual lifetime follows weighted exponential distribution.

4.6. Out-of-Money Put Option

For this specific financial contingent contract we have

$$b\left(x\right)={\left(K-x\right)}_{+}=K{\mathrm{𝟙}}_{(x<K)}-x{\mathrm{𝟙}}_{(x<K)}.$$

Applying Equation (24) for $n_0=0$ and $n_0=1$ twice, we have for joint life status,

$$\begin{array}{ccc}\hfill {\pi }_1& =& \left(1+\theta \right)\sum _{k=1}^n\sum _{i=1}^m\frac{{\chi }_{i,k}^{\left(1\right)}K}{b_{i,k}^{\left(1\right)}\left(b_{i,k}^{\left(1\right)}-1\right)}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{i,k}^{\left(1\right)}}-\theta \left(\sum _{k=1}^m\sum _{i=1}^n\frac{{\chi }_{i,k}^{\left(2\right)}K}{b_{i,k}^{\left(2\right)}\left(b_{i,k}^{\left(2\right)}-1\right)}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{i,k}^{\left(2\right)}}\right.\hfill \\ & +& 2\left.\sum _{k=1}^n\sum _{i=1}^{m-1}\sum _{j=i+1}^m\frac{{\chi }_{k,i,j}^{\left(3\right)}K}{b_{k,i,j}^{\left(3\right)}\left(b_{k,i,j}^{\left(3\right)}-1\right)}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{k,i,j}^{\left(3\right)}}\right)+2\theta \left(\sum _{k=1}^n\sum _{i=1}^m\frac{{\chi }_{i,k}^{\left(4\right)}K}{b_{i,k}^{\left(4\right)}\left(b_{i,k}^{\left(4\right)}-1\right)}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{i,k}^{\left(4\right)}}\right.\hfill \\ & +& 2\left.\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{k,i,j}^{\left(5\right)}K}{b_{k,i,j}^{\left(5\right)}\left(b_{k,i,j}^{\left(5\right)}-1\right)}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{k,i,j}^{\left(5\right)}}\right)-2\theta \sum _{k=1}^m\sum _{i=1}^n\frac{{\chi }_{k,i}^{\left(6\right)}K}{b_{k,i}^{\left(4\right)}\left(b_{k,i}^{\left(6\right)}-1\right)}{\left(\frac{K}{S\left(0\right)}\right)}^{-b_{k,i}^{\left(6\right)}}\hfill \\ & -& 4\theta \left(\sum _{k=1}^m\sum _{i=1}^{n-1}\sum _{j=i+1}^n\frac{{\chi }_{j,k,i}^{\left(7\right)}K}{b_{j,k,i}^{\left(7\right)}\left(b_{j,k,i}^{\left(7\right)}-1\right)}{\left(\frac{K}{}\right)}^{}\right.\hfill \end{array}$$