Abstract

We propose a model for valuing participating life insurance products under a generalized jump-diffusion model with a Markov-switching compensator. It also nests a number of important and popular models in finance, including the classes of jump-diffusion models and Markovian regime-switching models. The Esscher transform is employed to determine an equivalent martingale measure. Simulation experiments are conducted to illustrate the practical implementation of the model and to highlight some features that can be obtained from our model.

1. Introduction

In recent years, participating life insurance products become more and more popular in major insurance and finance markets around the world. These products can be regarded as investment plans with associated life insurance benefits, a specified benchmark return, a guarantee of an annual minimum rate of return, and a specified rule of the distribution of annual excess investment return above the guaranteed return. To enter the contract, policyholders pay annual premiums to an insurer, who will then manage and invest the funds in a specified reference portfolio. One key feature of these investment plans is the sharing of profits from an investment portfolio between the policyholders and the insurer. The specified rule of surplus distribution commonly used by insurers is known as reversionary bonus, which is employed to credit interest at or above a specified amount of guaranteed rate to the policyholders every year. The policyholders can receive an additional bonus at the maturity of the contract, namely, the terminal bonus, if the terminal surplus of the fund is positive at the maturity. If the insurer defaults at the maturity of the policy, the policyholders can only receive the outstanding assets. For more comprehensive discussion on various features of participating policies, refer to Grosen and Jørgensen [1]. Due to the internationally growing trend of adopting the market-based and fair valuation accountancy standards for the implementation of risk management practice for participating policies, it is practically important to develop appropriate, realistic, and objective models for valuing these policies.

Earlier works on exploring the use of the modern option pricing theory to value embedded options in with-profits life insurance policies go back to Brennan and Schwartz [2, 3] and Boyle and Schwartz [4]. Since then, there has been considerable interest on utilizing option pricing theory and its modern technologies to determine fair values of these policies. Grosen and Jørgensen [1] develop a flexible contingent claims model to incorporate the minimum rate guarantee, bonus distribution, and surrender risk. Prieul et al. [5] adopt a partial differential equation approach to value a participating policy and employ the method of similarity transformations of variables to reduce the dimension of the partial differential equation governing the value of the policy. Bacinello [6, 7] adopt binomial schemes for computing the numerical solutions to the fair valuation problems of participating policies with various contractual features. Bacinello [7] introduces a model for describing the feature of annual premiums. Grosen and Jørgensen [8] use a barrier option framework to study and document the effect of regulatory intervention rules on reducing the insolvency risk of the policies. Chu and Kwok [9] develop a flexible contingent claims model that describes rate guarantee, bonuses, and default risk. Siu [10] considers the pricing of a participating policy with surrender options when the market values of the reference portfolio are governed by a Markov-modulated geometric Brownian motion.

In this paper, we propose a model for valuing participating life insurance products under a generalized jump-diffusion model with a Markov-switching compensator. We suppose that the jump component is specified by the class of Markov-modulated kernel-biased completely random measures. The class of kernel-biased completely random measures is a wide class of jump-type processes. It has a very nice representation, which is a generalized kernel-based mixture of Poisson random measures (or, in general, random measures). The main idea of the kernel-biased completely random measure is to provide various forms of distortion of jump sizes of a completely random measure using the kernel function. This provides a great deal of flexibility in modeling different types of finite and infinite jump activities compared with some existing models in the literature. We also provide additional flexibility to incorporate the impact of structural changes in macroeconomic conditions and business cycles on the valuation of participating policies by introducing an observable, continuous-time and finite-state Markov chain. Here the states of the Markov chain may be interpreted as proxies of some observable macroeconomic indicators, such as gross domestic product and retail price index. They might also be considered economic ratings of a region or sovereign ratings. The model we considered here is general enough to nest a number of important and popular models for asset price dynamics in finance, including the two important classes of models, namely, the jump-diffusion models and the Markovian regime-switching models. These models are justified empirically in the literature and are shown to be practically useful for pricing and hedging derivatives. Our model can also be related to other important and popular classes of financial models, namely, the VG model pioneered by Madan et al. [11] and the CGMY model pioneered by Carr et al. [12].

For valuing participating products under the generalized jump-diffusion model, we employ a well-known tool in actuarial science, namely, the Esscher transform, which provides a convenient and flexible way to determine an equivalent martingale measure under the incomplete market setting. We consider various special cases of the Markov-modulated kernel-biased completely random measure for the jump component, namely, the Markov-modulated generalized Gamma (MGG) process, the scale-distorted version of the MGG process, and the power-distorted version of the MGG process. The MGG process encompasses the Markov-modulated weighted Gamma (MWG) process and the Markov-modulated inverse Gaussian (MIG) process as special cases. We compare the fair values of the options embedded in the participating products implied by our generalized jump-diffusion models with those obtained from other existing models in the literature via simulation experiments and highlight some features of the qualitative behavior of the fair values that can be obtained from different parametric specifications of our model. The paper is outlined as follows.

Section 2 presents the generalized jump-diffusion model for the market value of the reference asset and the Esscher transform for valuation. We also provide some discussion for the hedging and risk management issues. In Section 3, we consider three important parametric cases of the Markov-modulated kernel-biased completely random measures, namely, the MGG, the scale-distorted and power-distorted versions of the MGG process. The simulation procedure and the simulation results of the fair values of the options embedded in the policy are presented and discussed in Section 4. The final section summarizes this paper. The proofs of the lemmas and propositions are presented in the appendix.

2. The Valuation Model

In this section, we consider a financial model consisting of a risk-free money market account and a reference risky asset or portfolio. We suppose that the market value of the reference asset is governed by a jump-diffusion model with the jump component being specified as a kernel-biased completely random measure with Markov-switching compensator. We assume that the market is frictionless and that the mortality risk and surrender option are absent. We further impose certain assumptions on the rule of bonus distribution in our valuation model. We aim at developing a fair valuation model for participating life insurance policies which can incorporate the impact of the switching behavior of the states of the economy on the market value of the reference asset and fair value of the policy. The market described by the model is incomplete in general (see [1316]). Hence, there are infinitely many equivalent martingale measures and there is a range of no-arbitrage prices for a policy. Here, we determine an equivalent martingale measure by the Esscher transform. In the sequel, we introduce the set up of our model.

2.1. The Price Dynamics

In this subsection, we describe the price dynamics of the reference portfolio underlying the participating policy. Firstly, we fix a complete probability space (Ω,,𝒫), where 𝒫 is the real-world probability measure. Let 𝒯 denote the time index set [0,𝑇] of the economy. We describe the states of the economy by a continuous-time Markov chain {𝑋𝑡}𝑡𝒯 on (Ω,,𝒫) with a finite state space 𝒮=(𝑠1,𝑠2,,𝑠𝑁). Without loss of generality, we can identify the state space of the process {𝑋𝑡}𝑡𝒯 to be a finite set of unit vectors {𝑒1,𝑒2,,𝑒𝑁}, where 𝑒𝑖=(0,,1,,0)𝑁.

Write 𝑄 for the generator or 𝑄-matrix [𝑞𝑖𝑗]𝑖,𝑗=1,2,,𝑁. Then, from Elliott et al. [17], we have the following semimartingale decomposition for the process {𝑋𝑡}𝑡𝒯: 𝑋𝑡=𝑋0+𝑡0𝑄𝑋𝑠𝑑𝑠+𝑀𝑡.(2.1) Here {𝑀𝑡}𝑡𝒯 is an 𝑁-valued martingale with respect to the filtration generated by {𝑋𝑡}𝑡𝒯.

Let {𝑟(𝑡,𝑋𝑡)}𝑡𝒯 be the instantaneous market interest rate of a bank account or a money market account, which depends on the state of the economy. That is, 𝑟𝑡,𝑋𝑡=𝐫,𝑋𝑡=𝑁𝑖=1𝑟𝑖𝑋𝑡,𝑒𝑖,𝑡𝒯,(2.2) where 𝐫=(𝑟1,𝑟2,,𝑟𝑁) with 𝑟𝑖>0 for each 𝑖=1,2,,𝑁 and , denotes the inner product in the space 𝑁.

For notational simplicity, we write 𝑟𝑡 for 𝑟(𝑡,𝑋𝑡). In this case, the dynamics of the price process {𝐵𝑡}𝑡𝒯 for the bank account is described by 𝑑𝐵𝑡=𝑟𝑡,𝑋𝑡𝐵𝑡𝐵𝑑𝑡,0=1.(2.3) In the sequel, we first describe a Markov-switching kernel-biased completely random measure. James [18, 19] propose a kernel-biased representation of completely random measures, which provides a great deal of flexibility in modeling different types of finite and infinite jump activities by choosing different kernel functions. Here we employ the kernel-biased representation of completely random measures proposed by James [18, 19] and adapt this representation to the Markov-modulated case in which the compensator of the underlying random measure switches over time according to the state of {𝑋𝑡}𝑡𝒯.

Let (𝒯,(𝒯)) denote a measurable space, where (𝒯) is the Borel 𝜎-field generated by the open subsets of 𝒯. Write 0 for the family of Borel sets 𝑈+, whose closure 𝑈 does not contain the point 0. Let 𝒳 denote 𝒯×+. The measurable space (𝒳,(𝒳)) is then given by (𝒯×+,(𝒯)0).

For each 𝑈0, let 𝑁𝑋𝑡(,𝑈) denote a Markov-switching Poisson random measure on the space 𝒳. Write 𝑁𝑋𝑡(𝑑𝑡,𝑑𝑧) for the differential form of the measure 𝑁𝑋𝑡(𝑡,𝑈). Let 𝜌𝑋𝑡(𝑑𝑧|𝑡) denote a Markov-switching Lévy measure on the space 𝒳 depending on 𝑡 and the state 𝑋𝑡;𝜂 is a 𝜎-finite (nonatomic) measure on 𝒯. Note that if 𝑋𝑡=𝑒𝑖(𝑖=1,2,,𝑁), write 𝜌𝑖=𝜌𝑒𝑖(𝑑𝑧|𝑡). To ensure the existence of the kernel-biased completely random measure to be defined in the sequel (see [1820]), we assume that for an arbitrary strictly positive function on +,,𝜌𝑖, and 𝜂 are selected in such a way that for each bounded set in 𝒯, 𝑁𝑖=1+𝜌min(𝑧),1𝑖||(𝑑𝑧𝑡)𝜂(𝑑𝑡)<.(2.4) We assume that the Markov-switching intensity measure 𝜈𝑋𝑡(𝑑𝑡,𝑑𝑧) for the Poisson random measure 𝑁𝑋𝑡(𝑑𝑡,𝑑𝑧) is given by 𝜈𝑋𝑡(𝑑𝑡,𝑑𝑧)=𝜌𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑑𝑡)=𝑁𝑖=1𝜌𝑖||𝑋(𝑑𝑧𝑡)𝑡,𝑒𝑖𝜂(𝑑𝑡.(2.5) By modifying the kernel-biased representation of James [18, 19], we define a Markov-modulated kernel-biased completely random measure 𝜇𝑋𝑡(𝑑𝑡) on 𝒯 as follows: 𝜇𝑋𝑡(𝑑𝑡)=+(𝑧)𝑁𝑋𝑡(𝑑𝑡,𝑑𝑧),(2.6) which is a kernel-based mixture of the Markov-modulated Poisson random measure 𝑁𝑋𝑡(𝑑𝑡,𝑑𝑧) over the state space of the jump size + with the mixing kernel function (𝑧). See also Perman et al. [20] for discussion on representations of completely random measures. In general, we can replace the Poisson random measure with a random measure and choose some quite exotic functions for (𝑧) to generate different types of finite and infinite jump activities.

Let 𝑚𝑋𝑡 denote the mean measure of 𝜇𝑋𝑡. That is, 𝑚𝑋𝑡(𝑑𝑡)=+(𝑧)𝜈𝑋𝑡(𝑑𝑡,𝑑𝑧)=𝑁𝑖=1+(𝑧)𝜌𝑖||𝑋(𝑑𝑧𝑡)𝑡,𝑒𝑖𝜂(𝑑𝑡).(2.7)

Let {𝑊𝑡}𝑡𝒯 denote a standard Brownian motion on (Ω,,𝒫) with respect to the 𝒫-augmentation of its natural filtration 𝑊={𝑊𝑡}𝑡𝒯. We suppose that 𝑊,𝑋, and 𝜇𝑋𝑡(𝑑𝑡) are independent. Let 𝑁𝑋𝑡(𝑑𝑡,𝑑𝑧) denote the compensated Poisson random measure defined by 𝑁𝑋𝑡(𝑑𝑡,𝑑𝑧)=𝑁𝑋𝑡(𝑑𝑡,𝑑𝑧)𝜌𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑑𝑡).(2.8) Let 𝜇𝑡 and 𝜎𝑡 denote the drift and volatility of the market value of the reference asset, respectively. We suppose that 𝜇𝑡 and 𝜎𝑡 are given by 𝜇𝑡=𝝁,𝑋𝑡=𝑁𝑖=1𝜇𝑖𝑋𝑡,𝑒𝑖,𝜎𝑡=𝝈,𝑋𝑡=𝑁𝑖=1𝜎𝑖𝑋𝑡,𝑒𝑖,(2.9) where 𝝁=(𝜇1,𝜇2,,𝜇𝑁) and 𝝈=(𝜎1,𝜎2,,𝜎𝑁);𝜇𝑖 and 𝜎𝑖>0, for each 𝑖=1,2,,𝑁.

Then, we assume that the dynamic of the market value 𝐴 of the reference portfolio is governed by the following general geometric jump-diffusion process with a Markov-switching kernel-biased completely random measure: 𝑑𝐴𝑡=𝐴𝑡𝜇𝑡𝑑𝑡𝑚𝑋𝑡(𝑑𝑡)+𝜎𝑡𝑑𝑊𝑡++(𝑒(𝑧)1)𝑁𝑋𝑡(𝑑𝑡,𝑑𝑧).(2.10) By convention, we suppose that 𝐴0=1, 𝒫-a.s. Similar to the Merton jump-diffusion model, the drift term of 𝐴 is given by the mean 𝜇𝑡𝑑𝑡 minus the Markov-switching compensator 𝑚𝑋𝑡(𝑑𝑡) of 𝜇𝑋𝑡(𝑑𝑡). We can then write the dynamic of 𝐴 as follows: 𝑑𝐴𝑡=𝐴𝑡𝜇𝑡𝑑𝑡++𝑒(𝑧)𝜌1(𝑧)𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑑𝑡)+𝜎𝑡𝑑𝑊𝑡++𝑒(𝑧)𝑁1𝑋𝑡.(𝑑𝑡,𝑑𝑧)(2.11) In general, one can consider the situation that the drift 𝜇𝑡 and the volatility 𝜎𝑡 depend not only on the current economic state 𝑋𝑡, but also other state variables or market information, such as the current value of the reference portfolio 𝐴𝑡, when dealing with a long-term maturity. This represents an interesting and practically relevant direction for further generalizing the model. To focus on modeling and examining the impact of transitions of economic states on the price dynamics of the reference portfolio and the fair value of the policy, we assume here that 𝜇𝑡 and 𝜎𝑡 depend on the current economic state 𝑋𝑡 only.

Let 𝑌𝑡=ln(𝐴𝑡). Note that 𝑌0=0, 𝒫-a.s., since 𝐴0=1. Then, by Itô's formula, 𝑑𝑌𝑡=𝜇𝑡12𝜎2𝑡𝑑𝑡+𝜎𝑡𝑑𝑊𝑡++𝑁(𝑧)𝑋𝑡(𝑑𝑡,𝑑𝑧).(2.12)

2.2. The Crediting Scheme

Now, we describe the scheme for evaluating the interest rate credited to the policy reserve. Let 𝑅𝑡 denote the book value of the policy reserve and 𝐷𝑡 the bonus reserve, at time 𝑡𝒯. Then, as in Chu and Kwok [9], we have the following accounting identity for 𝐴𝑡, 𝑅𝑡, and 𝐷𝑡: 𝐴𝑡=𝑅𝑡+𝐷𝑡,𝑡𝒯,(2.13) where 𝑅0=𝛼𝑝𝐴0,𝛼𝑝(0,1], and 𝑅0 is interpreted as the single initial premium paid by the policyholder for acquiring the contract and 𝛼𝑝 is the cost allocation parameter. In this case, the 𝛼𝑝-portion of the initial asset portfolio is financed by the policyholder.

Write 𝑐𝑅(𝐴,𝑅) for the interest rate credited to the policy reserve. Then we have 𝑑𝑅𝑡=𝑐𝑅(𝐴,𝑅)𝑅𝑡𝑑𝑡.(2.14)

In practice, the specification of 𝑐𝑅(𝐴,𝑅) depends on the rule of bonus distribution, which is decided by the management level of an insurance company. Typically, an insurer distributes to his/her policyholder a certain proportion, say 𝛿, of the excess of the ratio of bonus reserve 𝐷𝑡 to the policy reserve 𝑅𝑡 over the target ratio 𝛽, which is a long-term constant target ratio specified by the management. The proportional constant 𝛿 is called the reversionary bonus distribution rate and it is assumed that 𝛿(0,1]. For the crediting scheme of interest rate, it is also assumed that there is a specified guarantee rate 𝑟𝑔 for the minimum interest rate credited to the policyholder's account. This means that the interest rate 𝑐𝑅(𝐴,𝑅)𝑟𝑔. Here, we adopt the interest rate crediting scheme used in Chu and Kwok [9] as follows: 𝑐𝑅𝐴𝑡,𝑅𝑡𝑟=max𝑔,𝐴ln𝑡𝑅𝑡𝛽,(2.15) where the interest rate 𝑐𝑅(𝐴𝑡,𝑅𝑡) credited to the policyholder's account depends on both the reversionary bonus 𝛽 and the guaranteed rate 𝑟𝑔.

2.3. Pricing by the Esscher Transform

In this subsection, we describe how to determine an equivalent martingale measure by the Esscher transform in the incomplete market specified by the generalized jump-diffusion model. Firstly, we provide a short discussion on other existing approaches for option valuation in an incomplete market.

Different approaches have been proposed in the literature on how to pick an equivalent martingale pricing measure in an incomplete market. Föllmer and Sondermann [21], Föllmer and Schweizer [22] introduce the notion of a minimal martingale measure and select a unique equivalent martingale measure via risk-minimization. Duffie and Richardson [23] and Schweizer [24] propose the mean-variance criterion for determining an equivalent martingale measure. Davis [25] adopts the marginal rate of substitution, which is a sound equilibrium argument in economic theory, to pick a pricing measure by solving a utility maximization problem. The pioneering work by Gerber and Shiu [26] provides a pertinent solution to the option pricing problem in an incomplete market by the Esscher transform, a time-honored tool in actuarial science introduced by Esscher [27]. The Esscher transform provides market practitioners with a convenient and flexible way to value options. Here, we employ the regime-switching Esscher transform in the work of Elliott et al. [16] and present the idea of this transform in the sequel.

Firstly, we describe the information structure of the model. Let 𝑋={𝑋𝑡}𝑡𝒯 and 𝑌={𝑌𝑡}𝑡𝒯 denote the 𝒫-augmentation of the natural filtration generated by 𝑋 and 𝑌, respectively. For each 𝑖=1,2 and 𝑡𝒯, write 𝒢𝑡 for the 𝜎-algebra 𝑋𝑇𝑌𝑡. Let 𝐵𝑀(𝒯) denote the collection of (𝒯)-measurable and nonnegative functions with compact support on 𝒯. Write (𝒯) for the Borel 𝜎-field of 𝒯. For each process 𝜃𝐵𝑀(𝒯), write (𝜃𝑌)𝑡 for 𝑡0𝜃𝑢𝑑𝑌𝑢, for each 𝑡𝒯. Let 𝑌(𝜃)𝑡=𝐸𝒫[𝑒(𝜃𝑌)𝑡𝑋𝑇], where 𝐸𝒫 represents expectation under 𝒫.

Let {Λ𝑡}𝑡𝒯 denote a 𝒢-adapted stochastic process defined as below: Λ𝑡𝑒=(𝜃𝑌)𝑡𝑌(𝜃)𝑡,𝑡𝒯.(2.16) Applying Itô's differentiation rule for jump-diffusion processes (see, e.g., [28, 29]), we have 𝑒(𝜃𝑌)𝑡=1𝑡0𝑒(𝜃𝑌)𝑠𝜃𝑠𝜇𝑠12𝜎2𝑠𝑑𝑠𝑡0𝑒(𝜃𝑌)𝑠𝜃𝑠𝜎𝑠𝑑𝑊𝑠𝑡0+𝑒(𝜃𝑌)𝑠𝜃𝑠𝑁(𝑧)𝑋𝑠1(𝑑𝑠,𝑑𝑧)+2𝑡0𝑒(𝜃𝑌)𝑠𝜃2𝑠𝜎2𝑠+𝑑𝑠𝑡0+𝑒(𝜃𝑌)𝑠𝑒𝜃𝑠(𝑧)𝑁1𝑋𝑠+(𝑑𝑠,𝑑𝑧)𝑡0+𝑒(𝜃𝑌)𝑠𝑒𝜃𝑠(𝑧)1+𝜃𝑠𝜌(𝑧)𝑋𝑠||(𝑑𝑧𝑠)𝜂(𝑑𝑠).(2.17) Conditioning on 𝑋𝑇 for both sides of (2.17), 𝐸𝑒(𝜃𝑌)𝑡|||𝑋𝑇=1𝑡0𝐸𝑒(𝜃𝑌)𝑠|||𝑋𝑇𝜃𝑠𝜇𝑠12𝜎2𝑠1𝑑𝑠+2𝑡0𝐸𝑒(𝜃𝑌)𝑠|||𝑋𝑇𝜃2𝑠𝜎2𝑠+𝑑𝑠𝑡0+𝐸𝑒(𝜃𝑌)𝑠|||𝑋𝑇𝑒𝜃𝑠(𝑧)1+𝜃𝑠𝜌(𝑧)𝑋𝑠||(𝑑𝑧𝑠)𝜂(𝑑𝑠).(2.18) Hence, 𝑌(𝜃)𝑡=exp𝑡0𝜃𝑠𝜇𝑠12𝜎2𝑠1𝑑𝑠+2𝑡0𝜃2𝑠𝜎2𝑠𝑑𝑠+𝑡0+𝑒𝜃𝑠(𝑧)1+𝜃𝑠𝜌(𝑧)𝑋𝑠||.(𝑑𝑧𝑠)𝜂(𝑑𝑠)(2.19) Therefore, Λ𝑡=exp𝑡0𝜃𝑠𝜎𝑠𝑑𝑊𝑠12𝑡0𝜃2𝑠𝜎2𝑠𝑑𝑠𝑡0+𝜃𝑠𝑁(𝑧)𝑋𝑠(𝑑𝑠,𝑑𝑧)𝑡0+𝑒𝜃𝑠(𝑧)1+𝜃𝑠𝜌(𝑧)𝑋𝑠||.(𝑑𝑧𝑠)𝜂(𝑑𝑠)(2.20)

Lemma 2.1. Λ is a (𝒢,𝒫)-martingale.

Then, the Esscher transform 𝒬𝒫 on 𝒢𝑡 with respect to {𝜃𝑡𝑡𝒯} is defined as 𝑑𝒬|||𝑑𝒫𝒢𝑡=Λ𝑡,𝑡𝒯.(2.21) Harrison and Kreps [30] and Harrison and Pliska [31, 32] establish the relationship between the absence of arbitrage opportunities and the existence of an equivalent martingale measure. This is called the fundamental theorem of asset pricing. Delbaen and Schachermayer [33] point out that the equivalent relationship does not hold “true” in general and show that the absence of arbitrage is “essentially” equivalent to the existence of an equivalent martingale measure under which the discounted stock price process is a martingale. Write 𝐴𝑡=exp(𝑡0𝑟𝑢𝑑𝑢)𝐴𝑡. In our setting, the martingale condition is given by 𝐴𝑠=𝐸𝒬𝐴𝑡||𝒢𝑠,forany𝑡,𝑠𝒯with𝑡𝑠,(2.22) where 𝐸𝒬 represents expectation under 𝒬.

Proposition 2.2. Suppose there exists a function 𝜂()(0,𝑇)+ such that 𝜂(𝑑𝑡)=𝜂(𝑡)𝑑𝑡. Then, the martingale condition is satisfied if and only if 𝜃𝑡 satisfies 𝜇𝑡𝑟𝑡𝜃𝑡𝜎2𝑡++𝑒𝜃𝑡(𝑧)𝑒(𝑧)𝜌1(𝑧)𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑡)=0,(2.23)for each 𝑡𝒯.

Proposition 2.3. Let 𝑊𝑊={𝑡𝑇} denote a standard Brownian motion and let 𝑁𝒬𝑋𝑡(𝑑𝑡,𝑑𝑧) denote a compensated Markov-modulated Poisson random measure with compensator 𝜌𝒬𝑋𝑡(𝑑𝑧|𝑡)𝜂(𝑡)𝑑𝑡 under 𝒬, where 𝜌𝒬𝑋𝑡(𝑑𝑧|𝑡)=𝑒𝜃𝑡(𝑧)𝜌𝑋𝑡(𝑑𝑧|𝑡). Then, under 𝒬, 𝑑𝑌𝑡=𝑟𝑡12𝜎2𝑡𝑑𝑡++1𝑒(𝑧)𝜌+(𝑧)𝒬𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑡)𝑑𝑡+𝜎𝑡𝑑𝑊𝑡++𝑁(𝑧)𝒬𝑋𝑡(𝑑𝑡,𝑑𝑧),(2.24)where 𝑋 is governed by (2.1).

2.4. Fair Valuation

Here, we present the procedure for the fair valuation based on an equivalent martingale measure chosen by the regime-switching Esscher transform in the last subsection.

Let 𝑉(𝐴𝑇,𝑅𝑇,𝑋𝑇) denote the terminal payoff of the participating policy on the policy's maturity date 𝑇, when the state of the economy 𝑋𝑇 at time 𝑇 is 𝑋. Then, 𝑉𝐴𝑇,𝑅𝑇,𝑋𝑇=𝐴𝑇,if𝐴𝑇<𝑅𝑇,𝑅𝑇,if𝑅𝑇𝐴𝑇𝑅𝑇𝛼𝑝,𝑅𝑇+𝛾𝑃1𝑇,if𝐴𝑇>𝑅𝑇𝛼𝑝,(2.25) where 𝛾 is the terminal bonus distribution rate and 𝑃1𝑇=max(𝛼𝑝𝐴𝑇𝑅𝑇,0) is the terminal bonus option.

Let 𝑃2𝑇=max(𝑅𝑇𝐴𝑇,0), where 𝑃2𝑇 represents the terminal default option on the policy's maturity date 𝑇. Then, the terminal payoff 𝑉(𝐴𝑇,𝑅𝑇,𝑋𝑇) can be written in the following form: 𝑉𝐴𝑇,𝑅𝑇,𝑋𝑇=𝑅𝑇+𝛾𝑃1𝑇𝑃2𝑇.(2.26) Note that the bonus option can be viewed as a standard European call option that grants the policyholder the right to pay the policy value as a strike price to receive 𝛼𝑝-portion of the asset portfolio. Instead of evaluating the fair value of the terminal payoff of the policy, we consider the fair valuation for each of the components of the terminal payoff of the policy, namely, the guaranteed benefit 𝑅𝑇, the terminal bonus option 𝑃1𝑇, and the terminal default option 𝑃2𝑇. Given knowledge of 𝒢𝑡, the conditional fair values of the guaranteed benefit, the terminal bonus option, and the terminal default option at time 𝑡 are, respectively, 𝐺(𝑡)=𝐸𝒬exp𝑇𝑡𝑟𝑠𝑅𝑑𝑠𝑇|||𝒢𝑡,𝑃1(𝑡)=𝐸𝒬exp𝑇𝑡𝑟𝑠𝑃𝑑𝑠1𝑇|||𝒢𝑡,𝑃2(𝑡)=𝐸𝒬exp𝑇𝑡𝑟𝑠𝑃𝑑𝑠2𝑇|||𝒢𝑡.(2.27) Note that the discount factor here is stochastic and switches over time according to the states of the Markov chain.

As in Buffington and Elliott [14, 15], given that 𝐴𝑡=𝐴,𝑅𝑡=𝑅 and 𝑋𝑡=𝑋, the fair values of the guaranteed benefit, the terminal bonus option, and the terminal default option at time 𝑡 are, respectively, 𝐺𝐴𝑡,𝑅𝑡,𝑋𝑡=𝐸𝒬exp𝑇𝑡𝑟𝑠𝑅𝑑𝑠𝑇𝐴𝑡=𝐴,𝑅𝑡=𝑅,𝑋𝑡,𝑃=𝑋(2.28)1𝐴𝑡,𝑅𝑡,𝑋𝑡=𝐸𝒬exp𝑇𝑡𝑟𝑠𝑃𝑑𝑠1𝑇𝐴𝑡=𝐴,𝑅𝑡=𝑅,𝑋𝑡,𝑃=𝑋(2.29)2𝐴𝑡,𝑅𝑡,𝑋𝑡=𝐸𝒬exp𝑇𝑡𝑟𝑠𝑃𝑑𝑠2𝑇𝐴𝑡=𝐴,𝑅𝑡=𝑅,𝑋𝑡.=𝑋(2.30)

2.5. Hedging and Risk Management

Besides fair valuation of the options embedded in the participating policy, it is interesting to investigate how the risks inherent in these options can be hedged once the policy has been sold from a risk management perspective. The main focus of the current paper is the fair valuation issue of the policy. In practice, the hedging and risk management issues of the policy are also important. So, we provide some discussion for the hedging and risk management issues of the policy here. The hedging and risk management issues of the policy are certainly interesting and important topics for future research.

There are different ways to hedge the risks inherent in the options embedded in the policy. Hedging via the Greeks and the risk-minimizing hedging represent two popular approaches to hedging these risks. However, due to the fact that the market model considered here is incomplete, perfect hedging cannot be achieved. Here we discuss the use of the Greeks to hedge the risks inherent in the options, namely, the guaranteed benefit, the terminal bonus option, and the default option, embedded in the policy. Note that hedging using the Greeks is only an approximating hedging strategy and that it cannot provide a perfect hedging result due to the market incompleteness. There are different approaches to compute the Greeks based on the Monte Carlo simulation of the price paths. The basic method is the Monte Carlo finite-difference approach. The key idea of this method is to compute the finite difference approximation of the differentials using the Monte Carlo simulation. For illustration, we consider the use of this method to compute the Delta. Suppose 𝑉(𝐴) and 𝑉(𝐴+𝜖) denote Monte Carlo estimators of “true” prices 𝑉(𝐴) and 𝑉(𝐴+𝜖), respectively, where 𝐴 represents the initial value of the reference portfolio and 𝜖 is a (small) positive constant. Then, the Delta Δ(𝐴) of an option evaluated at the initial value 𝐴 can be estimated by the finite-difference estimator as follows: Δ(𝐴)=𝑉(𝐴+𝜖)𝑉(𝐴)𝜖.(2.31) Glynn [34] shows that if the simulations of the two estimators 𝑉(𝐴) and 𝑉(𝐴+𝜖) are drawn independently, the best possible convergence rate is 𝑛1/4, where 𝑛 is the number of simulation runs. The convergence rate can be improved using the central difference (𝑉(𝐴+𝜖)𝑉(𝐴𝜖))/2𝜖. In this case, the best possible convergence rate is 𝑛1/3. The convergence rate can further be improved using common random numbers for both Monte Carlo estimators. The optimal convergence rate one can achieve in this case is 𝑛1/2, which is the same as the best possible convergence rate of a crude Monte Carlo method.

Other approaches that enhance the efficiency of the computation of the Greeks based on the Monte Carlo simulation include the simple differentiation approach proposed by Broadie and Glasserman [35] and the Malliavin calculus approach discussed by Fournié et al. [36, 37]. Chen and Glasserman [38] nvestigate the connection between the Malliavin calculus approach and the traditional approach based on the pathwise method and likelihood method. Recently, the Malliavin calculus approach for the Monte Carlo computation of the Greeks for jump-diffusion models and Lévy processes has been developed by several authors, including León et al. [39], El-Khatib and Privault [40], Davis and Johansson [41], and others. It is interesting to explore how the Malliavin calculus approach for jump-diffusion processes and Lévy processes can be extended to deal with the hedging of the risks inherent in the options embedded in the participating policy under the Markovian regime-switching jump-diffusion model considered here. Besides using the Malliavin calculus approach, one may also consider the use of an extended Clark-Haussman-Ocone formula using the white noise analysis in the work of Aase et al. [42] to hedge the risks of the options embedded in the policy under the Markovian regime-switching jump-diffusion model. This also represents an interesting topic for further research.

3. Various Parametric Specifications to the Jump Component

In the previous section, we have defined a general jump-diffusion process with the jump component specified by a kernel-biased Markov-modulated completely random measure. Here, we consider some parametric cases of the general jump process by specifying some particular forms of the kernel function and the Markovian regime-switching intensity measure. These parametric cases include the MGG process, the scale-distorted and power-distorted versions of the MGG process, and their special cases. We also derive the risk-neutral dynamics for the logarithmic return process {𝑌𝑡}𝑡𝒯 under 𝒬 for various parametric specifications which will be used for computing the fair values of the policies in Section 4. It is interesting to note that the kernel-biased completely random measure has some connections to some important Lévy processes in the literature including the VG process by Madan et al. [11] and the CGMY model of Carr et al. [12]. We also discuss these connections in this section.

3.1. Markov-Modulated Generalized Gamma (MGG) Process

The generalized Gamma (GG) process is a wide class of jump-type processes, which consists of the weighted Gamma (WG) process and the inverse Gaussian (IG) process as special cases. The GG process is a special case of the kernel-biased completely random measure and can be obtained by setting the kernel function (𝑧)=𝑧 and choosing a particular parametric form of the compensator measure. To provide more flexibility in describing the impact of the states of an economy on the jump component, we consider a Markov-modulated GG process, called the MGG process, whose compensator switches over time according to the states of the economy. We first describe the MGG process in the sequel.

Let 𝛼0 denote a constant shape parameter of the MGG process. We suppose that the scale parameter of the MGG process 𝑏(𝑡)=𝑏(𝑡,𝑋𝑡) switches over time according to the states of the Markov chain 𝑋 and is given by 𝑏(𝑡)=𝐛,𝑋𝑡=𝑁𝑖=1𝑏𝑖𝑋𝑡,𝑒𝑖,(3.1) where 𝐛=(𝑏1,𝑏2,,𝑏𝑁)𝑁 and 𝑏𝑖0, for each 𝑖=1,2,,𝑁.

Then, the Markov-switching intensity process of the MGG process is 𝜌𝑋𝑡||(𝑑𝑧𝑡)𝜂1(𝑡)𝑑𝑡=𝑒Γ(1𝛼)𝐛,𝑋𝑡𝑧𝑧𝛼1𝑑𝑧𝜂(𝑡)𝑑𝑡=𝑁𝑖=11𝑒Γ(1𝛼)𝑏𝑖𝑧𝑧𝛼1𝑋𝑡,𝑒𝑖𝑑𝑧𝜂(𝑡)𝑑𝑡.(3.2) In this case, the martingale condition becomes 𝜇𝑡𝑟𝑡𝜃𝑡𝜎2𝑡++𝑒𝜃𝑡𝑧𝑒𝑧𝜌1𝑧𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑡)=0,(3.3) where 𝜌𝑋𝑡(𝑑𝑧𝑡)𝜂(𝑡) is given by (3.2).

Write 𝜌𝒬𝑋𝑡(𝑑𝑧|𝑡)=𝑒𝜃𝑡𝑧𝜌𝑋𝑡(𝑑𝑧|𝑡), where 𝜃𝑡 satisfies (3.3). Let 𝑁𝒬𝑋𝑡(𝑑𝑡,𝑑𝑧) denote a Poisson random measure with Markov-switching compensator 𝜌𝒬𝑋𝑡(𝑑𝑧|𝑡)𝜂(𝑡)𝑑𝑡 under 𝒬. Then, under 𝒬, the dynamic of 𝑌 is 𝑑𝑌𝑡=𝑟𝑡12𝜎2𝑡𝑑𝑡++1𝑒(𝑧)𝜌𝒬𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑡)𝑑𝑡+𝜎𝑡𝑑𝑊𝑡++(𝑧)𝑁𝒬𝑋𝑡(𝑑𝑡,𝑑𝑧).(3.4) When 𝛼=0, the MGG process reduces to a Markov-modulated WG (MWG) process. That is, the Markov-switching intensity of the MWG process is 𝜌𝑋𝑡||(𝑑𝑧𝑡)𝜂𝑒(𝑡)𝑑𝑡=𝐛,𝑋𝑡𝑧𝑧𝑑𝑧𝜂(𝑡)𝑑𝑡=𝑁𝑖=1𝑒𝑏𝑖𝑧𝑧𝑋𝑡,𝑒𝑖𝑑𝑧𝜂(𝑡)𝑑𝑡.(3.5) In this case, the martingale condition becomes 𝜇𝑡𝑟𝑡𝜃𝑡𝜎2𝑡++𝑒𝜃𝑡𝑧𝑒𝑧𝑒1𝑧𝐛,𝑋𝑡𝑧𝑧𝑑𝑧𝜂(𝑡)=0.(3.6) Under 𝒬, 𝑑𝑌𝑡=𝑟𝑡12𝜎2𝑡𝑑𝑡++1𝑒(𝑧)𝑒(𝜃𝑡+𝐛,𝑋𝑡)𝑧𝑧𝑑𝑧𝜂(𝑡)𝑑𝑡+𝜎𝑡𝑑𝑊𝑡++(𝑧)𝑁𝒬𝑋𝑡(𝑑𝑡,𝑑𝑧),(3.7) where 𝑁𝒬𝑋𝑡(𝑑𝑡,𝑑𝑧) is a Poisson random measure with Markov-switching compensator, 𝜌𝒬𝑋𝑡||(𝑑𝑧𝑡)𝜂𝑒(𝑡)𝑑𝑡=(𝜃𝑡+𝐛,𝑋𝑡)𝑧𝑧𝑑𝑧𝜂(𝑡)𝑑𝑡,(3.8)and 𝜃𝑡 satisfies (3.6).

When 𝛼=1/2, the MGG becomes a Markov-modulated IG (MIG) process. In this case, the martingale condition becomes 𝜇𝑡𝑟𝑡𝜃𝑡𝜎2𝑡++𝑒𝜃𝑡𝑧𝑒𝑧11𝑧Γ(1/2𝑧𝑒𝐛,𝑋𝑡𝑧𝑑𝑧𝑑𝑧𝜂(𝑡)=0.(3.9) Under 𝒬, the dynamic of 𝑌 is 𝑑𝑌𝑡=𝑟𝑡12𝜎2𝑡𝑑𝑡++1𝑒(𝑧)1Γ(1/2)𝑧𝑒(𝜃𝑡+𝐛,𝑋𝑡)𝑧𝑑𝑧𝜂(𝑡)𝑑𝑡+𝜎𝑡𝑑𝑊𝑡++(𝑧)𝑁𝒬𝑋𝑡(𝑑𝑡,𝑑𝑧),(3.10) where the intensity process for 𝑁𝒬𝑋𝑡(𝑑𝑡,𝑑𝑧) is 𝜌𝒬𝑋𝑡||(𝑑𝑧𝑡)𝜂1(𝑡)𝑑𝑡=Γ(1/2)𝑧𝑒(𝜃𝑡+𝐛,𝑋𝑡)𝑧𝑑𝑧𝜂(𝑡)𝑑𝑡,(3.11)and 𝜃𝑡 satisfies (3.9).

3.2. The Scale and Power Distortions of the MGG Process

In this section, we consider the scale-distorted and power-distorted versions of the MGG process. The scale-distorted and power-distorted versions of the MGG process provide additional flexibility for describing various jump-type behaviors. They can describe the overstate and understate of jump amplitudes due to overreaction and underreaction of market participants to extraordinary events, respectively. For the scale-distorted version of the MGG process, the kernel function (𝑧)=𝑐𝑧, where c is a positive constant. When 𝑐>1, jump sizes are overstated. When 0<𝑐<1, jump sizes are understated. For the power-distorted version of the MGG process, the kernel function (𝑧)=𝑧𝑞, where 𝑞>0. When 𝑞>1, small jump sizes (i.e., 0<𝑧<1) are understated and large jump sizes (i.e., 𝑧>1) are overstated. When 0<𝑞<1, small jump sizes are overstated and large jump sizes are understated. The scale-distorted and power-distorted versions of the MGG process with different scale and power-distorted parameters can generate different types of behaviors of market participants when they react to extraordinary events. They can shed lights on understanding the impact of these market participants' behaviors on the price dynamic of the reference asset from a behavioral finance perspective. In general, one can also assume that the scale 𝑐 for the scale-distorted version of the MGG process and the power 𝑞 for the power-distorted version of the MGG process also switch over time according to the state of the Markov chain. In this case, the overstate and understate of the jump amplitudes also depend on the state of the economy. Here, we consider the case that both 𝑐 and 𝑞 are constants for illustration.

For both the scale-distorted and power-distorted versions of the MGG process, the Markov-switching intensity processes are the same as that of the MGG process. For the scale-distorted version of the MGG process, the martingale condition is given by 𝜇𝑡𝑟𝑡𝜃𝑡𝜎2𝑡++𝑒𝜃𝑡𝑐𝑧𝑒𝑐𝑧𝜌1𝑐𝑧𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑡)=0,(3.12) where 𝜌𝑋𝑡(𝑑𝑧|𝑡)𝜂(𝑡) is given by (3.2).

Under 𝒬, the dynamic of 𝑌 is 𝑑𝑌𝑡=𝑟𝑡12𝜎2𝑡𝑑𝑡++1𝑒𝑐𝑧𝜌𝒬𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑡)𝑑𝑡+𝜎𝑡𝑑𝑊𝑡++𝑐𝑧𝑁𝒬𝑋𝑡(𝑑𝑡,𝑑𝑧),(3.13) where 𝜌𝒬𝑋𝑡(𝑑𝑧|𝑡)=𝑒𝜃𝑡𝑧𝜌𝑋𝑡(𝑑𝑧|𝑡)𝜂(𝑑𝑡) and 𝜌𝑋𝑡(𝑑𝑧|𝑡)𝜂(𝑡)𝑑𝑡 is given by (3.2).

For the power-distorted version of the MGG process, the martingale condition is 𝜇𝑡𝑟𝑡𝜃𝑡𝜎2𝑡++𝑒𝜃𝑡𝑧𝑞𝑒𝑧𝑞1𝑧𝑞𝜌𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑡)=0,(3.14) where 𝜌𝑋𝑡(𝑑𝑧|𝑡)𝜂(𝑡) is specified by (3.2).

Under 𝒬, the dynamic of 𝑌 is 𝑑𝑌𝑡=𝑟𝑡12𝜎2𝑡𝑑𝑡++1𝑒𝑧𝑞𝜌𝒬𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑡)𝑑𝑡+𝜎𝑡𝑑𝑊𝑡++𝑧𝑞𝑁𝒬𝑋𝑡(𝑑𝑡,𝑑𝑧),(3.15) where 𝜌𝒬𝑋𝑡(𝑑𝑧|𝑡)=𝑒𝜃𝑡𝑧𝜌𝑋𝑡(𝑑𝑧|𝑡) and 𝜌𝑋𝑡(𝑑𝑧|𝑡)𝜂(𝑡) is given by (3.2).

When 𝛼=0, the scale-distorted version of the MGG process becomes the scale-distorted version of the MWG process and the power-distorted version of the MGG process reduces to the power-distorted version of the MWG process. When 𝛼=1/2, the scale-distorted and power-distorted versions of the MGG process become the scale-distorted and power-distorted versions of the MIG process, respectively.

3.3. Connections to the VG and CGMY Processes

Now, we outline some connections of a modified version of the Markov-modulated kernel-biased completely random measure to the VG and CGMY processes. We first provide some discussions on the VG model. The VG process can be represented in a number of equivalent ways, namely, the representation based on the time-changed Brownian motion, the difference between two gamma processes, the Lévy measure representation, the predictable compensator representation, where the predictable compensator representation is closely related to the Lévy measure representation in a fundamental way. Madan et al. [11] and Elliott and Royal [43] provide detailed discussion for different representations of the VG process. It has been shown by Elliott and Royal [43] that the predictable compensator of the VG process is the same as the Lévy measure. The Lévy measure of the VG process is given by (see [11]) 𝜈VG(𝑑𝑧,𝑑𝑡)=𝑘VG(𝑧)𝑑𝑧𝑑𝑡=𝐶exp(𝑀𝑧)𝑧𝐼𝑧>0𝐶exp(𝐺𝑧)𝑧𝐼𝑧<0𝑑𝑧𝑑𝑡,(3.16) where 𝐶,𝑀,𝐺+ are parameters of the VG process.

We consider a Markov-modulated version of the VG process with the following Markov-switching compensator: 𝜈VG𝑋𝑡(𝑑𝑧,𝑑𝑡)=𝑘VG𝑋𝑡=(𝑧)𝑑𝑧𝑑𝑡𝐶exp𝐌,𝑋𝑡𝑧𝑧𝐼𝑧>0𝐶exp𝐆,𝑋𝑡𝑧𝑧𝐼𝑧<0𝑑𝑧𝑑𝑡,(3.17) where 𝐌=(𝑀1,𝑀2,,𝑀𝑁)𝑁 and 𝐆=(𝐺1,𝐺2,,𝐺𝑁)𝑁 with 𝑀𝑖>0 and 𝐺𝑖>0, for each 𝑖=1,2,,𝑁.

The Markov-modulated VG process can be related to a modified version of the Markov-switching kernel-biased completely random measure by suitable matching of the model parameters. Consider two Markov-switching Poisson random measures 𝑁𝑘𝑋𝑡(𝑑𝑧,𝑑𝑡),𝑘=1,2, with the following intensity processes: 𝜌𝑘𝑋𝑡||(𝑑𝑧𝑡)𝜂𝑒(𝑡)𝑑𝑡=(1)𝑘𝐛𝑘,𝑋𝑡𝑧𝑧𝑑𝑧𝜂(𝑡)𝑑𝑡,(3.18)where 𝐛𝑘=(𝑏𝑘1,𝑏𝑘2,,𝑏𝑘𝑁) with 𝑏𝑘𝑖>0, for each 𝑖=1,2,,𝑁.

Write 𝑁𝑋𝑡(𝑑𝑧,𝑑𝑡)=𝑁1𝑋𝑡(𝑑𝑧,𝑑𝑡)𝐼{𝑧>0}+𝑁2𝑋𝑡(𝑑𝑧,𝑑𝑡)𝐼{𝑧<0}. Then, the intensity process of 𝑁𝑋𝑡(𝑑𝑧,𝑑𝑡) is 𝜌𝑋𝑡||(𝑑𝑧𝑡)𝜂𝑒(𝑡)𝑑𝑡=𝐛1,𝑋𝑡𝑧𝑧𝐼𝑧>0𝑒𝐛2,𝑋𝑡𝑧𝑧𝐼𝑧<0𝑑𝑧𝜂(𝑡)𝑑𝑡.(3.19) Suppose, for each 𝑘=1,2, that 𝑘()++ satisfies the following condition: 𝑁𝑖=1+min𝑘𝜌(𝑧),1𝑘𝑖||(𝑑𝑧𝑡)𝜂(𝑑𝑡)<.(3.20)Let () be a real-valued function defined as follows: (𝑧)=1(𝑧)𝐼𝑧>02(𝑧)𝐼𝑧<0.(3.21) Then, define a process 𝜇(𝑡) as follows: 𝜇(𝑡)=𝑡0(𝑧)𝑁𝑋𝑢(𝑑𝑧,𝑑𝑢).(3.22) Let 𝜂(𝑑𝑡)=𝑑𝑡 (i.e., 𝜂() is a uniform density). In this case, 𝜂(𝑡)=1. We further assume that 𝑏1𝑖=𝑀𝑖;𝑏2𝑖=𝐺𝑖;𝐶=1. Then, the Markov-modulated VG process with Markov-switching compensator 𝜈VG𝑋𝑡(𝑑𝑧,𝑑𝑡) coincides with the process 𝜇(𝑡), which is a modified version of the Markov-switching kernel-biased completely random measure.

In the sequel, we consider the CGMY process. From Carr et al. [12], the Lévy measure of the CGMY process is 𝜈CGMY(𝑑𝑧,𝑑𝑡)=𝑘CGMY=𝐶||𝑧||(𝑧)𝑑𝑧𝑑𝑡exp𝐺||𝑧||1+𝑌𝐼𝑧<0||𝑧||+𝐶exp𝑀||𝑧||1+𝑌𝐼𝑧>0𝑑𝑧𝑑𝑡,(3.23) where 𝐶>0,𝐺0,𝑀0, and 𝑌<2.

We consider a Markov-modulated version of the CGMY process with the following Markov-switching compensator: 𝜈CGMY𝑋𝑡(𝑑𝑧,𝑑𝑡)=𝑘CGMY𝑋𝑡=𝐶(𝑧)𝑑𝑧𝑑𝑡exp𝐆,𝑋𝑡||𝑧||||𝑧||1+𝑌𝐼𝑧<0+𝐶exp𝐌,𝑋𝑡||𝑧||||𝑧||1+𝑌𝐼𝑧>0𝑑𝑧𝑑𝑡,(3.24) where 𝐌=(𝑀1,𝑀2,,𝑀𝑁)𝑁 and 𝐆=(𝐺1,𝐺2,,𝐺𝑁)𝑁 with 𝑀𝑖>0 and 𝐺𝑖>0, for each 𝑖=1,2,,𝑁.

For each 𝑘=1,2, let 𝑁𝑘𝑋𝑡(𝑑𝑧,𝑑𝑡) denote a Markov-switching Poisson random measure with the following intensity process: 𝜌𝑘𝑋𝑡||(𝑑𝑧𝑡)𝜂1(𝑡)𝑑𝑡=𝑒Γ(1𝛼)𝐛𝑘,𝑋𝑡𝑧𝑧𝛼1𝑑𝑧𝜂=1(𝑡)𝑑𝑡𝑒Γ(1𝛼)𝐛𝑘,𝑋𝑡||𝑧||||𝑧||1+𝛼𝑑𝑧𝜂(𝑡)𝑑𝑡.(3.25) Let 𝑁𝑋𝑡(𝑑𝑧,𝑑𝑡)=𝑁1𝑋𝑡(𝑑𝑧,𝑑𝑡)𝐼{𝑧>0}+𝑁2𝑋𝑡(𝑑𝑧,𝑑𝑡)𝐼{𝑧<0}. Then, the Markov-switching intensity process for 𝑁𝑋𝑡(𝑑𝑧,𝑑𝑡) is 𝜌𝑋𝑡||(𝑑𝑧𝑡)𝜂1(𝑡)𝑑𝑡=𝑒Γ(1𝛼)𝐛1,𝑋𝑡||𝑧||||𝑧||1+𝛼𝐼𝑧>0+1𝑒Γ(1𝛼)𝐛2,𝑋𝑡||𝑧||||𝑧||1+𝛼𝐼𝑧<0𝑑𝑧𝜂(𝑡)𝑑𝑡.(3.26)

In this case, the kernel function is given by (3.21). Define a process 𝜇(𝑡) as below: 𝜇(𝑡)=𝑡0(𝑧)𝑁𝑋𝑢(𝑑𝑧,𝑑𝑢).(3.27) Let 𝜂(𝑑𝑡)=𝑑𝑡;𝑏1𝑖=𝑀𝑖;𝑏2𝑖=𝐺𝑖;𝛼=𝑌;𝐶=1/Γ(1𝛼). Then, the Markov-modulated CGMY process coincides with the process 𝜇(𝑡), which is a modified version of the Markov-switching kernel-biased completely random measure.

4. Simulation Experiments and Comparisons

In this section, we conduct simulation experiments to compare the fair values of the guaranteed benefit, the terminal bonus option, and the default option embedded in the participating policy implied by various parametric specifications of our generalized jump-type model described in Section 3 with those obtained from other existing models in the literature, such as the Merton jump-diffusion model, the VG process, and the geometric Brownian motion (GBM). We also document the impact of the regime-switching effect in the price dynamic of the reference portfolio on the fair values of the embedded options. We highlight some features of the qualitative behavior of the fair values of the embedded options that can be obtained from different parametric specifications of our model. Besides investigating the implications for the fair values of the embedded options, we also compare the default probabilities of the embedded options implied by various specifications of our jump-diffusion process with those implied by other models.

For simulating various parametric cases of our generalized jump-type process, we adopt the Poisson weighted algorithm by Lee and Kim [44] to simulate completely random measures with Markov-switching compensator. The Poisson weighted algorithm is applicable for a wide class of completely random measures, which are very difficult, if not impossible, to simulate directly in practice. The main idea of the Poisson weighted algorithm is that instead of generating jump sizes of a completely random measure directly from a nonstandard density function, one can first generate jump sizes from a proposed density function, which is a standard density function, like a gamma density, and then adjust the simulated jump sizes by the corresponding Poisson weights. The Poisson weights are simulated from a Poisson distribution with intensity parameter given by the odd ratio of the compensator of the completely random measure and the compensator corresponding to the proposed density.

In the sequel, we describe the modified Poisson weighted algorithm. For generality, we consider the full jump-diffusion model in Section 2, which is a Markov-modulated kernel-biased completely random measure. Suppose we wish to sample from the following process under the risk-neutral probability measure 𝒬: 𝜇𝒬(𝑡)=𝑡0+(𝑧)𝑁𝒬𝑋𝑢(𝑑𝑧,𝑑𝑢),(4.1) where the Markov-switching compensator for the Poisson random measure 𝑁𝒬𝑋𝑡(𝑑𝑧,𝑑𝑡) under 𝒬 is 𝜈𝒬𝑋𝑡(𝑑𝑧,𝑑𝑡)=𝜌𝒬𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑡)𝑑𝑡=𝑁𝑖=1𝜌𝒬𝑖||𝑋(𝑑𝑧𝑡)𝑡,𝑒𝑖𝜂(𝑡)𝑑𝑡.(4.2) Suppose we divide the time horizon [0,𝑇] of 𝑇 years into 𝑛𝑇 subintervals with equal length of Δ=1/𝑛, where 𝑇 is a positive integer. That is, one year is divided into 𝑛 subintervals. If 𝑛=252, each subinterval represents one trading day. Write [𝑡𝑗,𝑡𝑗+1] for the (𝑗+1)st subinterval, for each 𝑗=0,1,,𝑛1. Let 𝑀 denote the number of jumps of the completely random measure over a one-year time horizon. Here, 𝑀 controls the degree of accuracy of the approximation by the Poisson weighted algorithm and can be set as different values according to the desirable degree of accuracy. The larger 𝑀 is, the more accurate the approximation is. Here, we set 𝑀=100 and 𝑛=252. In general, they can be set as any positive numbers. We take 𝜂(𝑡) to be a proper density (i.e., 𝜂(𝑇)=𝑇0𝜂(𝑡)𝑑𝑡<). For illustration, we assume here that 𝜂(𝑡) is an unnormalized density function of a uniform density on [0,𝑇]. Then, the Poisson weighted algorithm is described as follows.

(1)Simulate the Markov chain process {𝑋𝑗𝑗=1,2,,𝑛𝑇}.(2)Generate i.i.d. positive random variables 𝑇1,,𝑇𝑀𝑇, which represent jump times, from the normalized density function 𝜂(𝑡)/𝜂(𝑇), where 𝜂(𝑇)=𝑇0𝜂(𝑡)𝑑𝑡.(3)Generate the jump size 𝑊𝑖 from the proposed density function 𝑔𝑇𝑖, where 𝑔𝑇𝑖 is defined as a conditional density function knowledge of 𝑇𝑖 , for each 𝑖=1,2,𝑀𝑇. Here, 𝑔𝑇𝑖 is assumed to be a gamma density function.(4)For each fixed 𝑖=1,2,,𝑀𝑇, if 𝑇𝑖[𝑡𝑗,𝑡𝑗+1), calculate 𝜆𝑖=𝜂(𝑇)𝜌𝒬𝑋𝑗(𝑊𝑖|𝑇𝑖)/𝑀𝑇𝑔𝑇𝑖(𝑊𝑖), for each 𝑖=1,,𝑀.(5)Generate the Poisson weights 𝑍𝑖 from a Poisson distribution with intensity parameter 𝜆𝑖.(6)𝜇(𝑡)=𝑀𝑇𝑖=1(𝑍𝑖𝑊𝑖)𝐼(𝑇𝑖𝑡), where () is the kernel function of the kernel-biased completely random measure. The process generated by the Poisson weighting algorithm converges in distribution to the completely random measure 𝜇(𝑡) on the space 𝐷[0,𝑇] of real-valued functions defined on the compact domain [0,𝑇] with the Skorohod topology as 𝑀. For the proof of the convergence, interested readers may refer to Lee and Kim [44].

In our simulation experiments, we compute the fair values for the embedded options, including the guaranteed benefit 𝐺(0), the terminal bonus option 𝑃1(0), and the default option 𝑃2(0), all with maturity 𝑇 equal to 20 years. Each fair value is computed by using 10 000 simulation paths over 20 years. All computations here were done by C++ codes. Note that 𝑛=252 (i.e., Δ=1/252). In other words, we simulate daily observations for the dynamic of the reference asset 𝐴. We consider a two-state Markov chain model 𝑋 with 𝑁=2, where “𝑋𝑡=1” represents “Good” economy while “𝑋𝑡=2” represents “Bad” economy. We take the transition probability matrix for 𝑋 as follows: 0.60.40.40.6.(4.3) We generate 10 000 simulation paths for 𝑋 over 20 years (i.e., {𝑋𝑗}𝑗=1,2,,5040) and suppose that 𝑋0=1. For each simulation path of 𝑋, we generate the log return series {𝑌𝑗}𝑗=1,2,,5040 for the full model in Section 2 from the following discretized version of the risk-neutral log return process based on the forward Euler discretization scheme: 𝑌𝑗+1=𝑌𝑗+𝑟𝑋𝑗12𝜎2𝑋𝑗Δ++1𝑒(𝑧)𝜌+(𝑧)𝒬𝑋𝑗||𝑡(𝑑𝑧𝑗)Δ+𝜎𝑋𝑗×𝜖×Δ+𝜇𝒬𝑋𝑗𝑡𝑗+1𝜇𝒬𝑋𝑗𝑡𝑗,(4.4) where 𝜖𝑁(0,1) and 𝜇𝒬𝑋𝑗(𝑡)=𝜇𝒬𝑋𝑗(𝑡)𝑡0+𝜌𝒬𝑋𝑗||(𝑑𝑧𝑢)𝑑𝑢.(4.5) We take the initial value 𝑌0 to be zero.

The forward Euler discretization scheme is a popular method to approximate the paths of a continuous-time process when performing Monte Carlo simulation and/or estimation of the process. It provides a natural, intuitive, and convenient way to discretize a continuous-time process. It has widely been adopted in approximating the paths of continuous-time asset price dynamics when performing Monte Carlo simulations in financial engineering (see, e.g., [45]). Under some conditions, it can be shown that the Euler approximation converges weakly to the target continuous-time process when the number of discretization intervals tends to infinity. Kloeden and Platen [46] provided an excellent discussion for the convergence of the forward Euler discretization scheme. They also presented numerous higher-order discretization schemes, which are more efficient than the Euler scheme, and discussed their convergence. These higher-order schemes include the Milstein scheme and the Platen-Wagner scheme. To illustrate the practical implementation of our model, we decide to use the forward Euler discretization scheme for being computationally convenient.

Let Δ𝑌𝑗+1=𝑌𝑗+1𝑌𝑗, for each 𝑗=0,1,,5039; then, Δ𝑌𝑗+1𝑟=𝑋𝑗12𝜎2𝑋𝑗Δ++1𝑒(𝑧)𝜌+(𝑧)𝒬𝑋𝑗|||𝑡𝑑𝑧𝑗Δ+𝜎𝑋𝑗×𝜖×Δ+𝜇𝒬𝑋𝑗𝑡𝑗+1𝜇𝒬𝑋𝑗𝑡𝑗.(4.6) Given each simulated path of {Δ𝑌𝑗}𝑗=1,2,,5039, {𝐴𝑗}𝑗=1,2,,5040 is calculated as 𝐴𝑗=𝐴0𝑒𝑗𝑘=1Δ𝑌𝑘,(4.7) where it is assumed that 𝐴0=100.

We calculate the value of the policy reserve {𝑅𝑡}𝑡=1,2,,20 annually over 20 years by applying the following forward Euler discretization scheme iteratively: 𝑅𝑡+1=𝑅𝑡exp[252𝑗=1𝑟max𝑔𝐴,ln252𝑡+𝑗𝑅𝑡𝛽Δ],(4.8) where 𝑡 represents the 𝑡th year and the initial value 𝑅0=𝛼𝑝𝐴0.

We assume some specimen values for the parameters for the participating policy: 𝑟𝑔=0.04;𝛽=0.5;𝛾=0.7;𝛼𝑝=0.6;𝐴0=100;𝑇=20years.(4.9) We consider some specimen values for the model parameters: 𝜇1=0.10;𝜇2=0.05;𝑟1=0.035;𝑟2=0.015;𝜎1=0.2;𝜎2𝑏=0.4;1=200.0,𝑏2=500.0.(4.10) We choose these specimen values to illustrate the practical implementation of our model. These values are in the reasonable ranges of magnitudes from a practical perspective and are consistent with the magnitudes of model parameters estimated in some empirical literature on jump-diffusion models. The estimation issue of our model, in particular, the Bayesian nonparametric estimation, is an interesting topic for future research. We plan to pursue this direction in our future research.

4.1. The MGG Processes

In this subsection, we consider the MGG process, the GG process, the Merton jump-diffusion model, the VG process, and the GBM. We suppose that the shape parameter 𝛼 for the (M)GG processes from 0.0 to 0.9, with an increment of 0.1. When 𝛼=0.0, the (M)GG process becomes the (M)WG process. When 𝛼=0.5, the (M)GG process becomes the (M)IG process. Other values of 𝛼 generate different parametric forms of the (M)GG processes. We assume that the parameter values of the no-regime-switching versions of these processes match with those in the corresponding regime-switching processes when the economy is in “State 1.” For the Merton jump diffusion model, we consider the following parameter values: 𝜇=0.1;𝑟=0.035;𝜎=0.2;𝜇𝑋=0.05;𝜎𝑋=0.07;𝜆=0.6,(4.11) where the jump size 𝑋 of the compound Poisson process follows a normal distribution with mean 𝜇𝑋 and variance 𝜎2𝑋;𝜆 is the intensity parameter of the Poisson process.

We consider a VG process for the log return process {𝑌𝑡}𝑡𝒯 with the following time-changed or subordinated Brownian motion representation: 𝑌𝑡=𝑍𝐿(𝑡,𝑣),𝑡𝒯,(4.12) where 𝑍𝑡=𝜃𝑡+𝜎VG𝑊𝑡;𝑊𝑡 is a standard Brownian motion; the subordinated process 𝐿(𝑡,𝑣) is a Gamma process with unit mean rate and variance 𝑣.

We assume the following specimen values for the parameters of the VG process: 𝜃=0.0;𝑣=0.01;𝜎VG=0.2.(4.13) The values of the parameters 𝜇,𝑟, and 𝜎 in this case are the same as those in the case of the Merton jump-diffusion model.

The fair values for the guaranteed benefit, the terminal bonus option, and the default option under the Merton jump-diffusion model and the VG model are evaluated under the Esscher transform. These fair values are computed using Monte Carlo simulations. In all figures, “with Markov switching” refers to the models with both the jump component and the model parameters being modulated by the two-state Markov chain; “without Markov switching” refers to the models with the jump component and constant model parameters; “no jump with Markov switching” refers to the Markovian regime-switching geometric Brownian motion; “no jump without Markov switching” refers to the geometric Brownian motion; “Merton jump” and “Variance Gamma” refer to the Merton jump-diffusion model and the variance Gamma process with constant model parameters, respectively.

Figures 14 display the numerical results for the probabilities that 𝐴𝑇<𝑅𝑇 under 𝒫 (i.e., the ruin probabilities) and the fair values for each of the embedded options. The numerical results for various parametric cases, namely, the MGG and GG processes with different values of the shape parameter 𝛼, the Merton jump-diffusion model, the VG process, and the GBM, are displayed here.

From Figure 1, we see that the probabilities that 𝐴(𝑇)<𝑅(𝑇) increase significantly as 𝛼 does in both the regime-switching and no-regime-switching cases. When 𝛼<0.5, the probabilities that 𝐴(𝑇)<𝑅(𝑇) in the regime-switching case are significantly larger than their corresponding values in the no-regime-switching case. From Figures 2 to 4, the impact of 𝛼 on the fair values of three embedded options is significant. We can also see that the effect of switching regimes on the fair values is also significant.

4.2. The Scale-Distorted Version of the MGG and GG Processes

We consider the scale-distorted version of the MGG and GG processes, the Merton jump-diffusion model, the VG process, and the GBM. We focus on investigating the impact of different values of the scale distortion parameter 𝑐 on the underlying price behaviors, the payoff structures, and the fair values of the participating policy. In particular, we suppose that 𝑐 takes values 0.5, 1.0 (i.e., no scale distortion), 2.0, and 3.0. Throughout this subsection, we suppose that the shape parameter 𝛼=0.5 for the scale-distorted version of the MGG and GG processes. The parameter values for the Merton jump-diffusion model and the VG model are given by those in Section 4.1.

Figures 58 display the numerical results for the ruin probabilities and the fair values of the three embedded options for various parametric cases, namely, the scale-distorted version of the MGG and GG processes with different values of the scale distortion parameter 𝑐, the Merton jump-diffusion model, the VG process, and the GBM.

From Figures 58, the impact of the scale distortion parameter 𝑐 on the the ruin probabilities and fair values of the embedded options is much less significant compared with that of the shape parameter 𝛼. From Figures 68, the effect of switching regimes on the fair values of the embedded options is still significant.

4.3. The Power-Distorted Version of the MGG and GG Processes

Now, we consider the power-distorted version of the MGG and GG processes, the Merton jump-diffusion model, the VG process, and the GBM. We suppose that the power distortion parameter 𝑞 takes values 0.6, 0.8, 1.0 (i.e., no scale distortion), 1.2, and 1.4. Throughout this subsection, we suppose that the shape parameter 𝛼=0.5 for the power-distorted version of the MGG and GG processes. The Merton jump diffusion model and the VG process for fair valuation are the same as before.

Figures 912 display the numerical results for the ruin probabilities and the fair values of the three embedded options for various parametric cases, namely, the power-distorted version of the MGG and GG processes with different values of the power distortion parameter 𝑞, the Merton jump-diffusion model, the VG process, and the GBM.

From Figures 912, the impact of the power distortion parameter 𝑞 on the ruin probabilities and fair values of the embedded options is less significant than that of the shape parameter 𝛼, but more significant than that of the scale distortion parameter 𝑐. The effect of switching regimes on the fair values of the embedded options is still significant in this case (see Figures 1012).

The market consistent valuation and the risk management of modern insurance products are important issues as highlighted by Solvency II and the International Accounting Standards Board (IASB). Solvency II refers to an amended set of regulatory requirements for insurance companies operating in the European Union region. Like the well-known regulatory requirements for banks and financial institutions, namely, Basel II, Solvency II operates as a three-pillar system, which consists of the quantitative evaluation of risk capitals, the review of the models by supervisors, and the disclosure of the risk information. So, Solvency II is also referred to as Basel II for insurance companies. Under Solvency II, insurance companies need to build their internal models for market consistent valuation and risk management and these models are then sent to supervisors for assessment and review. So, an important question for the insurance companies, perhaps the supervisors as well, is how to develop or build appropriate models for fair valuation and risk management of the insurance policies. An appropriate stochastic model for modeling the asset price dynamics plays a key role to answer this important question.

The numerical results have some implications for the market consistent valuation and the risk management practice of participating life insurance policies. They shed some lights on the importance of the correct specification of stochastic models for the long-term price movements of the reference portfolio. In particular, our study reveals that the specification of the parametric distribution of the jump component (i.e., the specification of the parameter 𝛼) and the incorporation of the regime-switching effect in the stochastic models for the long-term movements of the reference portfolio play a significant role in the market consistent valuation and the risk management via hedging of participating policies.

Besides providing some implications for the stochastic modeling of the asset price dynamics, the results of our studies also have some implications for the design of the products, for example, the choices of the target ratio 𝛽 and the guaranteed rate 𝑟𝑔. From Figure 1, we see that the ruin probabilities increase substantially as 𝛼 increases. Also, the ruin probabilities implied by the model without switching regimes are substantially lower than the corresponding values arising from the model with switching regimes. In other words, the mis-specification of the parametric distribution of the jump component in the stochastic model of the reference portfolio and/or neglecting the regime-switching effect may lead to underestimation of the ruin probabilities. So, the model risk is quite substantially here. One possible remedy for the model risk is to set the target ratio 𝛽 and the guaranteed rate 𝑟𝑔 in a more prudent way. For example, 𝛽 can be set higher and 𝑟𝑔 can be set lower, so that the interest rate credited in the scheme becomes lower and a lower terminal policy reserve may result. However, these two rates cannot be set as lower (higher) as we wish. It is important to consider the marketing/sale issue and the competition of other insurance companies which offer similar products when setting these rates.

5. Summary

We considered the pricing of participating life insurance policies when the market value of the reference asset is governed by a generalized jump-diffusion model with a Markov-switching compensator, where the jump component is specified by a Markov-modulated kernel-biased completely random measure. The Esscher transform was adopted to determine an equivalent martingale measure under the incomplete market setting. Various parametric cases of the Markov-modulated kernel-biased completely random measure were considered. We conducted simulation experiments using the Poisson weighted algorithm to compare the fair values of participating products implied by our model with those obtained from other existing models in the literature and to highlight some features that can be obtained from our model. The simulation results reveal that the impacts of various specifications of jump component and the switching regimes on the fair values of the embedded options in participating products are significant. The results of our studies highlight the importance of the specification of the parametric distribution of the jump component and the incorporation of the regime-switching effect in the stochastic model for the reference portfolio underlying a participating policy. They also shed some lights on the design of the contractual structure of the policy.

Appendix

Proofs

Proof of Lemma 2.1. First, for any 𝑡,𝑠𝒯 with 𝑡𝑠, 𝐸Λ𝑡Λ𝑠|||𝒢𝑠=𝐸exp𝑡𝑠𝜃𝑢𝜎𝑢𝑑𝑊𝑢12𝑡0𝜃2𝑢𝜎2𝑢𝑑𝑢𝑡𝑠+𝜃𝑢𝑁(𝑧)𝑋𝑢(𝑑𝑢,𝑑𝑧)𝑡𝑠+𝑒𝜃𝑢(𝑧)1+𝜃𝑢𝜌(𝑧)𝑋𝑢|||||𝒢(𝑑𝑧𝑢)𝜂(𝑑𝑢)𝑠.(A.1) Note that 𝐸exp𝑡𝑠𝜃𝑢𝜎𝑢𝑑𝑊𝑢|||𝒢𝑠1=exp2𝑡𝑠𝜃2𝑢𝜎2𝑢𝑑𝑢,(A.2) and, by James [18, 19], 𝐸exp𝑡𝑠+𝜃𝑢𝑁(𝑧)𝑋𝑢|||𝒢(𝑑𝑢,𝑑𝑧)𝑠=exp𝑡𝑠+𝑒𝜃𝑢(𝑧)1+𝜃𝑢𝜌(𝑧)𝑋𝑢||.(𝑑𝑧𝑢)𝜂(𝑑𝑢)(A.3) Hence, 𝐸Λ𝑡Λ𝑠|||𝒢𝑠=1,𝒫-a.s.(A.4)

Proof of Proposition 2.2. First, by Bayes' rule, 𝐸𝒬exp𝑡0𝑟𝑢𝐴𝑑𝑢𝑡|||𝒢0=exp𝑡0𝑟𝑢𝐸𝑑𝑢𝒫Λ𝑡exp𝑡0𝑑𝑌𝑢||𝒢0=exp𝑡0𝑟𝑢𝐸𝑑𝑢𝒫exp𝑡0(𝜃𝑢1)𝑑𝑌𝑢||𝒢0𝑌(𝜃)𝑡=exp𝑡0𝑟𝑢𝑑𝑢𝑌(𝜃1)𝑡𝑌(𝜃)𝑡=exp𝑡0𝜇𝑠𝑟𝑠12𝜎2𝑠1𝑑𝑠+2𝑡012𝜃𝑠𝜎2𝑠+𝑑𝑠𝑡0+𝑒(𝜃𝑠1)(𝑧)𝑒𝜃𝑠(𝑧)𝜌(𝑧)𝑋𝑠||(𝑑𝑧𝑠)𝜂.(𝑠)𝑑𝑠(A.5) Then, by setting 𝑠=0, the martingale condition implies that 1=𝐸𝒬𝐴𝑡||𝒢0.(A.6) This implies that 𝑡0𝜇𝑠𝑟𝑠12𝜎2𝑠1𝑑𝑠+2𝑡012𝜃𝑠𝜎2𝑠𝑑𝑠+𝑡0+𝑒(𝜃𝑠1)(𝑧)𝑒𝜃𝑠(𝑧)𝜌(𝑧)𝑋𝑠||(𝑑𝑧𝑠)𝜂(𝑠)𝑑𝑠=0,(A.7) for each 𝑡𝒯. Hence (2.28) is proved. Given (2.28), the martingale condition is satisfied.

Proof of Proposition 2.3. Let 𝑍𝑢𝐵𝑀(𝒯). Then, by Bayes' rule, 𝜃𝑌(𝑍)𝑡=𝐸𝒬𝑒(𝑍𝑌)𝑡𝒢0Λ=𝐸𝑡𝑒(𝑍𝑌)𝑡𝒢0=exp𝑡0𝑍𝑠𝜇𝑠𝜃𝑠𝜎2𝑠12𝜎2𝑠1𝑑𝑠+2𝑡0𝑍2𝑠𝜎2𝑠+𝑑𝑠𝑡0+𝑒𝜃𝑠(𝑧)𝑒𝑍𝑠(𝑧)1𝑍𝑠𝜌(𝑧)𝑋𝑠||(𝑑𝑧𝑠)𝜂.(𝑠)𝑑𝑠(A.8) From the martingale condition, 𝜇𝑡𝜃𝑡𝜎2𝑡=𝑟𝑡+𝑒𝜃𝑡(𝑧)𝑒(𝑧)𝜌1(𝑧)𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑡)𝑑𝑡.(A.9) Hence, 𝜃𝑌(𝑍)𝑡=exp𝑡0𝑍𝑠𝑟𝑠12𝜎2𝑠𝑑𝑠+𝑡0+𝑍𝑠𝑒(𝑧)𝑒1(𝑧)𝜃𝑠(𝑧)𝜌𝑋𝑠||(𝑑𝑧𝑠)𝜂+1(𝑠)𝑑𝑠2𝑡0𝑍2𝑠𝜎2𝑠𝑑𝑠+𝑡0+𝑒𝑍𝑠(𝑧)1+𝑍𝑠𝑒(𝑧)𝜃𝑠(𝑧)𝜌𝑋𝑠||(𝑑𝑧𝑠)𝜂(𝑠)𝑑𝑠=exp𝑡0𝑍𝑠𝑟𝑠12𝜎2𝑠𝑑𝑠𝑡0+𝑍𝑠1𝑒(𝑧)𝜌+(𝑧)𝒬𝑋𝑠||(𝑑𝑧𝑠)𝜂+1(𝑠)𝑑𝑠2𝑡0𝑍2𝑠𝜎2𝑠𝑑𝑠+𝑡0+𝑒𝑍𝑠(𝑧)1+𝑍𝑠𝜌(𝑧)𝒬𝑋𝑠||(𝑑𝑧𝑠)𝜂.(𝑠)𝑑𝑠(A.10) Then, under 𝒬, 𝑑𝑌𝑡=𝑟𝑡12𝜎2𝑡𝑑𝑡++1𝑒(𝑧)𝜌+(𝑧)𝒬𝑋𝑡||(𝑑𝑧𝑡)𝜂(𝑡)𝑑𝑡+𝜎𝑡𝑑𝑊𝑡++𝑁(𝑧)𝒬𝑋𝑡(𝑑𝑡,𝑑𝑧).(A.11) Since 𝑋 is independent with 𝑊 and 𝑁, the probability law of 𝑋 remains unchanged under the change of measures from 𝒫 to 𝒬.

Acknowledgments

The authors would like to thank the referees for their many helpful and valuable comments and suggestions. Hailiang Yang wishes to thank for the support from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no. HKU 7426/06H).