# Pricing Participating Products under a Generalized Jump-Diffusion Model

Authors: Tak Kuen Siu; John W. Lau; Hailiang Yang
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2008)

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## 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.


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## Body

## 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.Section2 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 [13–16]). 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,ℱ,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}t∈𝒯 on (Ω,ℱ,𝒫) with a finite state space 𝒮:=(s1,s2,…,sN).
Without loss of generality, we can identify the state space of the process {Xt}t∈𝒯 to be a finite set of unit vectors {e1,e2,…,eN},
where ei=(0,…,1,…,0)∈ℛN.WriteQ 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}t∈𝒯: (2.1)Xt=X0+∫0tQXsds+Mt. Here {Mt}t∈𝒯 is an ℛN-valued martingale with
respect to the filtration generated by {Xt}t∈𝒯.Let{r(t,Xt)}t∈𝒯 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, (2.2)r(t,Xt)=〈r,Xt〉=∑i=1Nri〈Xt,ei〉,t∈T, where r:=(r1,r2,…,rN) with ri>0 for each i=1,2,…,N and 〈⋅,⋅〉 denotes the inner product in the space ℛN.For notational simplicity, we writert for r(t,Xt).
In this case, the dynamics of the price process {Bt}t∈𝒯 for the bank account is described
by (2.3)dBt=r(t,Xt)Btdt,B0=1. 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}t∈𝒯.Let(𝒯,ℬ(𝒯)) denote a measurable space, where ℬ(𝒯) is the Borel σ-field generated by the open subsets of 𝒯.
Write ℬ0 for the family of Borel sets U∈ℛ+,
whose closure U¯ does not contain the point 0.
Let 𝒳 denote 𝒯×ℛ+.
The measurable space (𝒳,ℬ(𝒳)) is then given by (𝒯×ℛ+,ℬ(𝒯)⊗ℬ0).For eachU∈ℬ0,
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 [18–20]), we assume that for an arbitrary strictly positive
function on ℛ+,h,ρi, and η are selected in such a way that for each
bounded set ℬ in 𝒯, (2.4)∑i=1N∫B∫R+min⁡(h(z),1)ρi(dz|t)η(dt)<∞. We assume that the
Markov-switching intensity measure νXt(dt,dz) for the Poisson random measure NXt(dt,dz) is given by (2.5)νXt(dt,dz):=ρXt(dz|t)η(dt)=∑i=1N(ρi(dz|t)〈Xt,ei〉)η(dt). 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: (2.6)μXt(dt):=∫R+h(z)NXt(dt,dz), which is a kernel-based mixture
of the Markov-modulated Poisson random measure NXt(dt,dz) over the state space of the jump size ℛ+ with the mixing kernel function h(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 h(z) to generate different types of finite and
infinite jump activities.LetmXt denote the mean measure of μXt.
That is, (2.7)mXt(dt)=∫R+h(z)νXt(dt,dz)=∑i=1N(∫R+h(z)ρi(dz|t)〈Xt,ei〉η(dt)).Let{Wt}t∈𝒯 denote a standard Brownian motion on (Ω,ℱ,𝒫) with respect to the 𝒫-augmentation of its natural filtration ℱW:={ℱtW}t∈𝒯.
We suppose that W,X, and μXt(dt) are independent. Let NXt(dt,dz) denote the compensated Poisson random measure
defined by (2.8)NXt(dt,dz):=NXt(dt,dz)−ρXt(dz|t)η(dt). 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 (2.9)μt:=〈μ,Xt〉=∑i=1Nμi〈Xt,ei〉,σt:=〈σ,Xt〉=∑i=1Nσi〈Xt,ei〉, where μ:=(μ1,μ2,…,μN) and σ:=(σ1,σ2,…,σN);μi∈ℛ and σi>0,
for each i=1,2,…,N.Then, we assume that the dynamic of the market valueA of the reference portfolio is governed by the
following general geometric jump-diffusion process with a Markov-switching
kernel-biased completely random measure: (2.10)dAt=At−[μtdt−mXt(dt)+σtdWt+∫R+(eh(z)−1)NXt(dt,dz)]. 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: (2.11)dAt=At−[μtdt+∫R+(eh(z)−1−h(z))ρXt(dz|t)η(dt)+σtdWt+∫R+(eh(z)−1)NXt(dt,dz)]. 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.LetYt:=ln(At).
Note that Y0=0, 𝒫-a.s., since A0=1.
Then, by Itô's formula, (2.12)dYt=(μt−12σt2)dt+σtdWt+∫R+h(z)NXt(dt,dz).


### 2.2. The Crediting Scheme

Now, we describe the scheme for evaluating the
interest rate credited to the policy reserve. LetRt denote the book value of the policy reserve
and Dt the bonus reserve, at time t∈𝒯.
Then, as in Chu and Kwok [9], we have the following accounting identity for At, Rt, and Dt: (2.13)At=Rt+Dt,t∈T, 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.WritecR(A,R) for the interest rate credited to the policy
reserve. Then we have (2.14)dRt=cR(A,R)Rtdt.In practice, the specification ofcR(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: (2.15)cR(At,Rt)=max⁡(rg,(ln⁡AtRt−β)), 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. LetℱX:={ℱtX}t∈𝒯 and ℱY:={ℱtY}t∈𝒯 denote the 𝒫-augmentation of the natural filtration
generated by X and Y,
respectively. For each i=1,2 and t∈𝒯,
write 𝒢t for the σ-algebra ℱTX∨ℱtY.
Let BM(𝒯) denote the collection of ℬ(𝒯)-measurable and nonnegative functions with
compact support on 𝒯.
Write ℬ(𝒯) for the Borel σ-field of 𝒯.
For each process θ∈BM(𝒯),
write (θ⋅Y)t for ∫0tθudYu,
for each t∈𝒯.
Let ℳY(θ)t:=E𝒫[e−(θ⋅Y)t∣ℱTX],
where E𝒫 represents expectation under 𝒫.Let{Λt}t∈𝒯 denote a 𝒢-adapted stochastic process defined as
below: (2.16)Λt:=e−(θ⋅Y)tMY(θ)t,t∈T. Applying Itô's differentiation
rule for jump-diffusion processes (see, e.g., [28, 29]), we
have (2.17)e−(θ⋅Y)t=1−∫0te−(θ⋅Y)sθs(μs−12σs2)ds−∫0te−(θ⋅Y)sθsσsdWs−∫0t∫R+e−(θ⋅Y)s−θs−h(z)NXs(ds,dz)+12∫0te−(θ⋅Y)sθs2σs2ds+∫0t∫R+e−(θ⋅Y)s−(e−θsh(z)−1)NXs(ds,dz)+∫0t∫R+e−(θ⋅Y)s−(e−θsh(z)−1+θsh(z))ρXs(dz|s)η(ds). Conditioning on ℱTX for both sides of (2.17),
(2.18)E[e−(θ⋅Y)t|FTX]=1−∫0tE[e−(θ⋅Y)s|FTX]θs(μs−12σs2)ds+12∫0tE[e−(θ⋅Y)s|FTX]θs2σs2ds+∫0t∫R+E[e−(θ⋅Y)s−|FTX](e−θsh(z)−1+θsh(z))ρXs(dz|s)η(ds). Hence, (2.19)MY(θ)t=exp⁡[−∫0tθs(μs−12σs2)ds+12∫0tθs2σs2ds+∫0t∫R+(e−θsh(z)−1+θsh(z))ρXs(dz|s)η(ds)].
Therefore, (2.20)Λt=exp⁡[−∫0tθsσsdWs−12∫0tθs2σs2ds−∫0t∫R+θsh(z)NXs(ds,dz)−∫0t∫R+(e−θsh(z)−1+θsh(z))ρXs(dz|s)η(ds)].Lemma 2.1.
Λ is a (𝒢,𝒫)-martingale.Then, the
Esscher transform𝒬~𝒫 on 𝒢t with respect to {θt∣t∈𝒯} is defined as (2.21)dQdP|Gt=Λt,t∈T. 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(−∫0trudu)At.
In our setting, the martingale condition is given by (2.22)As=EQ[At|Gs],foranyt,s∈Twitht≥s, where E𝒬 represents expectation under 𝒬.Proposition 2.2.
Suppose there exists a
functionη′(⋅):(0,T)→ℛ+ such that η(dt)=η′(t)dt.
Then, the martingale condition is satisfied if and only if θt satisfies (2.23)μt−rt−θtσt2+∫R+[e−θth(z)(eh(z)−1)−h(z)]ρXt(dz|t)η′(t)=0,for each t∈𝒯.Proposition 2.3.
LetW:={Wt∈T} denote a standard Brownian motion and let NXt𝒬(dt,dz) denote a compensated Markov-modulated Poisson
random measure with compensator ρXt𝒬(dz|t)η′(t)dt under 𝒬,
where ρXt𝒬(dz|t):=e−θth(z)ρXt(dz|t).
Then, under 𝒬,(2.24)dYt=(rt−12σt2)dt+∫R+(1−eh(z)+h(z))ρXtQ(dz|t)η′(t)dt+σtdWt+∫R+h(z)NXtQ(dt,dz),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.LetV(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, (2.25)V(AT,RT,XT)={AT,ifAT<RT,RT,ifRT≤AT≤RTαp,RT+γP1T,ifAT>RTαp, where γ is the terminal bonus distribution rate and P1T:=max(αpAT−RT,0) is the terminal bonus option.LetP2T:=max(RT−AT,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: (2.26)V(AT,RT,XT)=RT+γP1T−P2T. 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, (2.27)G(t)=EQ[exp⁡(−∫tTrsds)RT|Gt],P1(t)=EQ[exp⁡(−∫tTrsds)P1T|Gt],P2(t)=EQ[exp⁡(−∫tTrsds)P2T|Gt]. 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, (2.28)G(At,Rt,Xt)=EQ[exp⁡(−∫tTrsds)RT∣At=A,Rt=R,Xt=X],(2.29)P1(At,Rt,Xt)=EQ[exp⁡(−∫tTrsds)P1T∣At=A,Rt=R,Xt=X],(2.30)P2(At,Rt,Xt)=EQ[exp⁡(−∫tTrsds)P2T∣At=A,Rt=R,Xt=X].


### 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. SupposeV^(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: (2.31)Δ^(A)=V^(A+ϵ)−V^(A)ϵ. 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 n−1/4,
where n is the number of simulation runs. The
convergence rate can be improved using the central difference (V^(A+ϵ)−V^(A−ϵ))/2ϵ.
In this case, the best possible convergence rate is n−1/3.
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 n−1/2,
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.


## 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,ℱ,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}t∈𝒯 on (Ω,ℱ,𝒫) with a finite state space 𝒮:=(s1,s2,…,sN).
Without loss of generality, we can identify the state space of the process {Xt}t∈𝒯 to be a finite set of unit vectors {e1,e2,…,eN},
where ei=(0,…,1,…,0)∈ℛN.WriteQ 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}t∈𝒯: (2.1)Xt=X0+∫0tQXsds+Mt. Here {Mt}t∈𝒯 is an ℛN-valued martingale with
respect to the filtration generated by {Xt}t∈𝒯.Let{r(t,Xt)}t∈𝒯 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, (2.2)r(t,Xt)=〈r,Xt〉=∑i=1Nri〈Xt,ei〉,t∈T, where r:=(r1,r2,…,rN) with ri>0 for each i=1,2,…,N and 〈⋅,⋅〉 denotes the inner product in the space ℛN.For notational simplicity, we writert for r(t,Xt).
In this case, the dynamics of the price process {Bt}t∈𝒯 for the bank account is described
by (2.3)dBt=r(t,Xt)Btdt,B0=1. 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}t∈𝒯.Let(𝒯,ℬ(𝒯)) denote a measurable space, where ℬ(𝒯) is the Borel σ-field generated by the open subsets of 𝒯.
Write ℬ0 for the family of Borel sets U∈ℛ+,
whose closure U¯ does not contain the point 0.
Let 𝒳 denote 𝒯×ℛ+.
The measurable space (𝒳,ℬ(𝒳)) is then given by (𝒯×ℛ+,ℬ(𝒯)⊗ℬ0).For eachU∈ℬ0,
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 [18–20]), we assume that for an arbitrary strictly positive
function on ℛ+,h,ρi, and η are selected in such a way that for each
bounded set ℬ in 𝒯, (2.4)∑i=1N∫B∫R+min⁡(h(z),1)ρi(dz|t)η(dt)<∞. We assume that the
Markov-switching intensity measure νXt(dt,dz) for the Poisson random measure NXt(dt,dz) is given by (2.5)νXt(dt,dz):=ρXt(dz|t)η(dt)=∑i=1N(ρi(dz|t)〈Xt,ei〉)η(dt). 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: (2.6)μXt(dt):=∫R+h(z)NXt(dt,dz), which is a kernel-based mixture
of the Markov-modulated Poisson random measure NXt(dt,dz) over the state space of the jump size ℛ+ with the mixing kernel function h(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 h(z) to generate different types of finite and
infinite jump activities.LetmXt denote the mean measure of μXt.
That is, (2.7)mXt(dt)=∫R+h(z)νXt(dt,dz)=∑i=1N(∫R+h(z)ρi(dz|t)〈Xt,ei〉η(dt)).Let{Wt}t∈𝒯 denote a standard Brownian motion on (Ω,ℱ,𝒫) with respect to the 𝒫-augmentation of its natural filtration ℱW:={ℱtW}t∈𝒯.
We suppose that W,X, and μXt(dt) are independent. Let NXt(dt,dz) denote the compensated Poisson random measure
defined by (2.8)NXt(dt,dz):=NXt(dt,dz)−ρXt(dz|t)η(dt). 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 (2.9)μt:=〈μ,Xt〉=∑i=1Nμi〈Xt,ei〉,σt:=〈σ,Xt〉=∑i=1Nσi〈Xt,ei〉, where μ:=(μ1,μ2,…,μN) and σ:=(σ1,σ2,…,σN);μi∈ℛ and σi>0,
for each i=1,2,…,N.Then, we assume that the dynamic of the market valueA of the reference portfolio is governed by the
following general geometric jump-diffusion process with a Markov-switching
kernel-biased completely random measure: (2.10)dAt=At−[μtdt−mXt(dt)+σtdWt+∫R+(eh(z)−1)NXt(dt,dz)]. 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: (2.11)dAt=At−[μtdt+∫R+(eh(z)−1−h(z))ρXt(dz|t)η(dt)+σtdWt+∫R+(eh(z)−1)NXt(dt,dz)]. 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.LetYt:=ln(At).
Note that Y0=0, 𝒫-a.s., since A0=1.
Then, by Itô's formula, (2.12)dYt=(μt−12σt2)dt+σtdWt+∫R+h(z)NXt(dt,dz).


## 2.2. The Crediting Scheme

Now, we describe the scheme for evaluating the
interest rate credited to the policy reserve. LetRt denote the book value of the policy reserve
and Dt the bonus reserve, at time t∈𝒯.
Then, as in Chu and Kwok [9], we have the following accounting identity for At, Rt, and Dt: (2.13)At=Rt+Dt,t∈T, 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.WritecR(A,R) for the interest rate credited to the policy
reserve. Then we have (2.14)dRt=cR(A,R)Rtdt.In practice, the specification ofcR(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: (2.15)cR(At,Rt)=max⁡(rg,(ln⁡AtRt−β)), 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. LetℱX:={ℱtX}t∈𝒯 and ℱY:={ℱtY}t∈𝒯 denote the 𝒫-augmentation of the natural filtration
generated by X and Y,
respectively. For each i=1,2 and t∈𝒯,
write 𝒢t for the σ-algebra ℱTX∨ℱtY.
Let BM(𝒯) denote the collection of ℬ(𝒯)-measurable and nonnegative functions with
compact support on 𝒯.
Write ℬ(𝒯) for the Borel σ-field of 𝒯.
For each process θ∈BM(𝒯),
write (θ⋅Y)t for ∫0tθudYu,
for each t∈𝒯.
Let ℳY(θ)t:=E𝒫[e−(θ⋅Y)t∣ℱTX],
where E𝒫 represents expectation under 𝒫.Let{Λt}t∈𝒯 denote a 𝒢-adapted stochastic process defined as
below: (2.16)Λt:=e−(θ⋅Y)tMY(θ)t,t∈T. Applying Itô's differentiation
rule for jump-diffusion processes (see, e.g., [28, 29]), we
have (2.17)e−(θ⋅Y)t=1−∫0te−(θ⋅Y)sθs(μs−12σs2)ds−∫0te−(θ⋅Y)sθsσsdWs−∫0t∫R+e−(θ⋅Y)s−θs−h(z)NXs(ds,dz)+12∫0te−(θ⋅Y)sθs2σs2ds+∫0t∫R+e−(θ⋅Y)s−(e−θsh(z)−1)NXs(ds,dz)+∫0t∫R+e−(θ⋅Y)s−(e−θsh(z)−1+θsh(z))ρXs(dz|s)η(ds). Conditioning on ℱTX for both sides of (2.17),
(2.18)E[e−(θ⋅Y)t|FTX]=1−∫0tE[e−(θ⋅Y)s|FTX]θs(μs−12σs2)ds+12∫0tE[e−(θ⋅Y)s|FTX]θs2σs2ds+∫0t∫R+E[e−(θ⋅Y)s−|FTX](e−θsh(z)−1+θsh(z))ρXs(dz|s)η(ds). Hence, (2.19)MY(θ)t=exp⁡[−∫0tθs(μs−12σs2)ds+12∫0tθs2σs2ds+∫0t∫R+(e−θsh(z)−1+θsh(z))ρXs(dz|s)η(ds)].
Therefore, (2.20)Λt=exp⁡[−∫0tθsσsdWs−12∫0tθs2σs2ds−∫0t∫R+θsh(z)NXs(ds,dz)−∫0t∫R+(e−θsh(z)−1+θsh(z))ρXs(dz|s)η(ds)].Lemma 2.1.
Λ is a (𝒢,𝒫)-martingale.Then, the
Esscher transform𝒬~𝒫 on 𝒢t with respect to {θt∣t∈𝒯} is defined as (2.21)dQdP|Gt=Λt,t∈T. 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(−∫0trudu)At.
In our setting, the martingale condition is given by (2.22)As=EQ[At|Gs],foranyt,s∈Twitht≥s, where E𝒬 represents expectation under 𝒬.Proposition 2.2.
Suppose there exists a
functionη′(⋅):(0,T)→ℛ+ such that η(dt)=η′(t)dt.
Then, the martingale condition is satisfied if and only if θt satisfies (2.23)μt−rt−θtσt2+∫R+[e−θth(z)(eh(z)−1)−h(z)]ρXt(dz|t)η′(t)=0,for each t∈𝒯.Proposition 2.3.
LetW:={Wt∈T} denote a standard Brownian motion and let NXt𝒬(dt,dz) denote a compensated Markov-modulated Poisson
random measure with compensator ρXt𝒬(dz|t)η′(t)dt under 𝒬,
where ρXt𝒬(dz|t):=e−θth(z)ρXt(dz|t).
Then, under 𝒬,(2.24)dYt=(rt−12σt2)dt+∫R+(1−eh(z)+h(z))ρXtQ(dz|t)η′(t)dt+σtdWt+∫R+h(z)NXtQ(dt,dz),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.LetV(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, (2.25)V(AT,RT,XT)={AT,ifAT<RT,RT,ifRT≤AT≤RTαp,RT+γP1T,ifAT>RTαp, where γ is the terminal bonus distribution rate and P1T:=max(αpAT−RT,0) is the terminal bonus option.LetP2T:=max(RT−AT,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: (2.26)V(AT,RT,XT)=RT+γP1T−P2T. 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, (2.27)G(t)=EQ[exp⁡(−∫tTrsds)RT|Gt],P1(t)=EQ[exp⁡(−∫tTrsds)P1T|Gt],P2(t)=EQ[exp⁡(−∫tTrsds)P2T|Gt]. 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, (2.28)G(At,Rt,Xt)=EQ[exp⁡(−∫tTrsds)RT∣At=A,Rt=R,Xt=X],(2.29)P1(At,Rt,Xt)=EQ[exp⁡(−∫tTrsds)P1T∣At=A,Rt=R,Xt=X],(2.30)P2(At,Rt,Xt)=EQ[exp⁡(−∫tTrsds)P2T∣At=A,Rt=R,Xt=X].


## 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. SupposeV^(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: (2.31)Δ^(A)=V^(A+ϵ)−V^(A)ϵ. 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 n−1/4,
where n is the number of simulation runs. The
convergence rate can be improved using the central difference (V^(A+ϵ)−V^(A−ϵ))/2ϵ.
In this case, the best possible convergence rate is n−1/3.
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 n−1/2,
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}t∈𝒯 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 functionh(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.Letα≥0 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 (3.1)b(t):=〈b,Xt〉=∑i=1Nbi〈Xt,ei〉, where b:=(b1,b2,…,bN)∈ℛN and bi≥0,
for each i=1,2,…,N.Then, the Markov-switching intensity process of the
MGG process is(3.2)ρXt(dz|t)η'(t)dt=1Γ(1−α)e−〈b,Xt〉zz−α−1dzη′(t)dt=∑i=1N1Γ(1−α)e−bizz−α−1〈Xt,ei〉dzη′(t)dt. In this case, the martingale
condition becomes (3.3)μt−rt−θtσt2+∫R+[e−θtz(ez−1)−z]ρXt(dz|t)η′(t)=0, where ρXt(dz∣t)η′(t) is given by (3.2).WriteρXt𝒬(dz|t):=e−θtzρXt(dz|t),
where θt satisfies (3.3). Let NXt𝒬(dt,dz) denote a Poisson random measure with
Markov-switching compensator ρXt𝒬(dz|t)η′(t)dt under 𝒬.
Then, under 𝒬,
the dynamic of Y is (3.4)dYt=(rt−12σt2)dt+∫R+(1−eh(z))ρXtQ(dz|t)η′(t)dt+σtdWt+∫R+h(z)NXtQ(dt,dz). 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 (3.5)ρXt(dz|t)η'(t)dt=e−〈b,Xt〉zzdzη′(t)dt=∑i=1Ne−bizz〈Xt,ei〉dzη′(t)dt. In this case, the martingale
condition becomes (3.6)μt−rt−θtσt2+∫R+[e−θtz(ez−1)−z](e−〈b,Xt〉zz)dzη′(t)=0. Under 𝒬, (3.7)dYt=(rt−12σt2)dt+∫R+(1−eh(z))(e−(θt+〈b,Xt〉)zz)dzη'(t)dt+σtdWt+∫R+h(z)NXtQ(dt,dz), where NXt𝒬(dt,dz) is a Poisson random measure with
Markov-switching compensator, (3.8)ρXtQ(dz|t)η′(t)dt=(e−(θt+〈b,Xt〉)zz)dzη′(t)dt,and θt satisfies (3.6).Whenα=1/2,
the MGG becomes a Markov-modulated IG (MIG) process. In this case, the
martingale condition becomes (3.9)μt−rt−θtσt2+∫R+[e−θtz(ez−1)−z](1Γ(1/2)ze−〈b,Xt〉zdz)dzη′(t)=0. Under 𝒬,
the dynamic of Y is (3.10)dYt=(rt−12σt2)dt+∫R+(1−eh(z))(1Γ(1/2)ze−(θt+〈b,Xt〉)z)dzη'(t)dt+σtdWt+∫R+h(z)NXtQ(dt,dz), where the intensity process for NXt𝒬(dt,dz) is (3.11)ρXtQ(dz|t)η′(t)dt=(1Γ(1/2)ze−(θt+〈b,Xt〉)z)dzη′(t)dt,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 functionh(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 h(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(3.12)μt−rt−θtσt2+∫R+[e−θtcz(ecz−1)−cz]ρXt(dz|t)η′(t)=0, where ρXt(dz|t)η′(t) is given by (3.2).Under𝒬,
the dynamic of Y is (3.13)dYt=(rt−12σt2)dt+∫R+(1−ecz)ρXtQ(dz|t)η′(t)dt+σtdWt+∫R+czNXtQ(dt,dz), where ρXt𝒬(dz|t)=e−θtzρXt(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(3.14)μt−rt−θtσt2+∫R+[e−θtzq(ezq−1)−zq]ρXt(dz|t)η′(t)=0, where ρXt(dz|t)η′(t) is specified by (3.2).Under𝒬,
the dynamic of Y is (3.15)dYt=(rt−12σt2)dt+∫R+(1−ezq)ρXtQ(dz|t)η′(t)dt+σtdWt+∫R+zqNXtQ(dt,dz), where ρXt𝒬(dz|t)=e−θtzρXt(dz|t) and ρXt(dz|t)η′(t) is given by (3.2).Whenα=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]) (3.16)νVG(dz,dt)=kVG(z)dzdt=(Cexp⁡(−Mz)zI{z>0}−Cexp⁡(Gz)zI{z<0})dzdt, where C,M,G∈ℛ+ are parameters of the VG process.We consider a Markov-modulated version of the VG
process with the following Markov-switching compensator:(3.17)νXtVG(dz,dt)=kXtVG(z)dzdt=(Cexp⁡(−〈M,Xt〉z)zI{z>0}−Cexp⁡(〈G,Xt〉z)zI{z<0})dzdt, where M:=(M1,M2,…,MN)∈ℛN and G:=(G1,G2,…,GN)∈ℛN 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 measuresNXtk(dz,dt),k=1,2, with the following intensity
processes: (3.18)ρXtk(dz|t)η′(t)dt=e(−1)k〈bk,Xt〉zzdzη′(t)dt,where bk:=(b1k,b2k,…,bNk)∈ℛ with bik>0,
for each i=1,2,…,N.WriteNXt(dz,dt):=NXt1(dz,dt)I{z>0}+NXt2(dz,dt)I{z<0}.
Then, the intensity process of NXt(dz,dt) is (3.19)ρXt(dz|t)η′(t)dt=(e−〈b1,Xt〉zzI{z>0}−e〈b2,Xt〉zzI{z<0})dzη′(t)dt. Suppose, for each k=1,2,
that hk(⋅):ℛ+→ℛ+ satisfies the following
condition: (3.20)∑i=1N∫B∫R+min⁡(hk(z),1)ρik(dz|t)η(dt)<∞.Let h(⋅):ℛ→ℛ be a real-valued function defined as follows: (3.21)h(z)=h1(z)I{z>0}−h2(−z)I{z<0}. Then, define a process μ(t) as follows: (3.22)μ(t):=∫0t∫Rh(z)NXu(dz,du). Let η(dt)=dt (i.e., η(⋅) is a uniform density). In this case, η′(t)=1.
We further assume that bi1=Mi;bi2=Gi;C=1.
Then, the Markov-modulated VG process with Markov-switching compensator νXtVG(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 (3.23)νCGMY(dz,dt)=kCGMY(z)dzdt=(Cexp⁡(−G|z|)|z|1+YI{z<0}+Cexp⁡(−M|z|)|z|1+YI{z>0})dzdt, where C>0,G≥0,M≥0, and Y<2.We consider a Markov-modulated version of the CGMY
process with the following Markov-switching compensator:(3.24)νXtCGMY(dz,dt)=kXtCGMY(z)dzdt=(Cexp⁡(−〈G,Xt〉|z|)|z|1+YI{z<0}+Cexp⁡(−〈M,Xt〉|z|)|z|1+YI{z>0})dzdt, where M:=(M1,M2,…,MN)∈ℛN and G:=(G1,G2,…,GN)∈ℛN with Mi>0 and Gi>0,
for each i=1,2,…,N.For eachk=1,2,
let N¯Xtk(dz,dt) denote a Markov-switching Poisson random
measure with the following intensity process: (3.25)ρXtk(dz|t)η′(t)dt=1Γ(1−α)e−〈bk,Xt〉zz−α−1dzη′(t)dt=1Γ(1−α)e−〈bk,Xt〉|z||z|1+αdzη′(t)dt. Let N¯Xt(dz,dt):=N¯Xt1(dz,dt)I{z>0}+N¯Xt2(dz,dt)I{z<0}.
Then, the Markov-switching intensity process for N¯Xt(dz,dt) is (3.26)ρXt(dz|t)η′(t)dt=(1Γ(1−α)e−〈b1,Xt〉|z||z|1+αI{z>0}+1Γ(1−α)e−〈b2,Xt〉|z||z|1+αI{z<0})dzη′(t)dt.In this case, the kernel functionh is given by (3.21). Define a process μ¯(t) as below: (3.27)μ¯(t):=∫0t∫Rh(z)N¯Xu(dz,du). Let η(dt)=dt;bi1=Mi;bi2=Gi;α=Y;C=1/Γ(1−α).
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.


## 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 functionh(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.Letα≥0 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 (3.1)b(t):=〈b,Xt〉=∑i=1Nbi〈Xt,ei〉, where b:=(b1,b2,…,bN)∈ℛN and bi≥0,
for each i=1,2,…,N.Then, the Markov-switching intensity process of the
MGG process is(3.2)ρXt(dz|t)η'(t)dt=1Γ(1−α)e−〈b,Xt〉zz−α−1dzη′(t)dt=∑i=1N1Γ(1−α)e−bizz−α−1〈Xt,ei〉dzη′(t)dt. In this case, the martingale
condition becomes (3.3)μt−rt−θtσt2+∫R+[e−θtz(ez−1)−z]ρXt(dz|t)η′(t)=0, where ρXt(dz∣t)η′(t) is given by (3.2).WriteρXt𝒬(dz|t):=e−θtzρXt(dz|t),
where θt satisfies (3.3). Let NXt𝒬(dt,dz) denote a Poisson random measure with
Markov-switching compensator ρXt𝒬(dz|t)η′(t)dt under 𝒬.
Then, under 𝒬,
the dynamic of Y is (3.4)dYt=(rt−12σt2)dt+∫R+(1−eh(z))ρXtQ(dz|t)η′(t)dt+σtdWt+∫R+h(z)NXtQ(dt,dz). 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 (3.5)ρXt(dz|t)η'(t)dt=e−〈b,Xt〉zzdzη′(t)dt=∑i=1Ne−bizz〈Xt,ei〉dzη′(t)dt. In this case, the martingale
condition becomes (3.6)μt−rt−θtσt2+∫R+[e−θtz(ez−1)−z](e−〈b,Xt〉zz)dzη′(t)=0. Under 𝒬, (3.7)dYt=(rt−12σt2)dt+∫R+(1−eh(z))(e−(θt+〈b,Xt〉)zz)dzη'(t)dt+σtdWt+∫R+h(z)NXtQ(dt,dz), where NXt𝒬(dt,dz) is a Poisson random measure with
Markov-switching compensator, (3.8)ρXtQ(dz|t)η′(t)dt=(e−(θt+〈b,Xt〉)zz)dzη′(t)dt,and θt satisfies (3.6).Whenα=1/2,
the MGG becomes a Markov-modulated IG (MIG) process. In this case, the
martingale condition becomes (3.9)μt−rt−θtσt2+∫R+[e−θtz(ez−1)−z](1Γ(1/2)ze−〈b,Xt〉zdz)dzη′(t)=0. Under 𝒬,
the dynamic of Y is (3.10)dYt=(rt−12σt2)dt+∫R+(1−eh(z))(1Γ(1/2)ze−(θt+〈b,Xt〉)z)dzη'(t)dt+σtdWt+∫R+h(z)NXtQ(dt,dz), where the intensity process for NXt𝒬(dt,dz) is (3.11)ρXtQ(dz|t)η′(t)dt=(1Γ(1/2)ze−(θt+〈b,Xt〉)z)dzη′(t)dt,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 functionh(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 h(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(3.12)μt−rt−θtσt2+∫R+[e−θtcz(ecz−1)−cz]ρXt(dz|t)η′(t)=0, where ρXt(dz|t)η′(t) is given by (3.2).Under𝒬,
the dynamic of Y is (3.13)dYt=(rt−12σt2)dt+∫R+(1−ecz)ρXtQ(dz|t)η′(t)dt+σtdWt+∫R+czNXtQ(dt,dz), where ρXt𝒬(dz|t)=e−θtzρXt(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(3.14)μt−rt−θtσt2+∫R+[e−θtzq(ezq−1)−zq]ρXt(dz|t)η′(t)=0, where ρXt(dz|t)η′(t) is specified by (3.2).Under𝒬,
the dynamic of Y is (3.15)dYt=(rt−12σt2)dt+∫R+(1−ezq)ρXtQ(dz|t)η′(t)dt+σtdWt+∫R+zqNXtQ(dt,dz), where ρXt𝒬(dz|t)=e−θtzρXt(dz|t) and ρXt(dz|t)η′(t) is given by (3.2).Whenα=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]) (3.16)νVG(dz,dt)=kVG(z)dzdt=(Cexp⁡(−Mz)zI{z>0}−Cexp⁡(Gz)zI{z<0})dzdt, where C,M,G∈ℛ+ are parameters of the VG process.We consider a Markov-modulated version of the VG
process with the following Markov-switching compensator:(3.17)νXtVG(dz,dt)=kXtVG(z)dzdt=(Cexp⁡(−〈M,Xt〉z)zI{z>0}−Cexp⁡(〈G,Xt〉z)zI{z<0})dzdt, where M:=(M1,M2,…,MN)∈ℛN and G:=(G1,G2,…,GN)∈ℛN 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 measuresNXtk(dz,dt),k=1,2, with the following intensity
processes: (3.18)ρXtk(dz|t)η′(t)dt=e(−1)k〈bk,Xt〉zzdzη′(t)dt,where bk:=(b1k,b2k,…,bNk)∈ℛ with bik>0,
for each i=1,2,…,N.WriteNXt(dz,dt):=NXt1(dz,dt)I{z>0}+NXt2(dz,dt)I{z<0}.
Then, the intensity process of NXt(dz,dt) is (3.19)ρXt(dz|t)η′(t)dt=(e−〈b1,Xt〉zzI{z>0}−e〈b2,Xt〉zzI{z<0})dzη′(t)dt. Suppose, for each k=1,2,
that hk(⋅):ℛ+→ℛ+ satisfies the following
condition: (3.20)∑i=1N∫B∫R+min⁡(hk(z),1)ρik(dz|t)η(dt)<∞.Let h(⋅):ℛ→ℛ be a real-valued function defined as follows: (3.21)h(z)=h1(z)I{z>0}−h2(−z)I{z<0}. Then, define a process μ(t) as follows: (3.22)μ(t):=∫0t∫Rh(z)NXu(dz,du). Let η(dt)=dt (i.e., η(⋅) is a uniform density). In this case, η′(t)=1.
We further assume that bi1=Mi;bi2=Gi;C=1.
Then, the Markov-modulated VG process with Markov-switching compensator νXtVG(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 (3.23)νCGMY(dz,dt)=kCGMY(z)dzdt=(Cexp⁡(−G|z|)|z|1+YI{z<0}+Cexp⁡(−M|z|)|z|1+YI{z>0})dzdt, where C>0,G≥0,M≥0, and Y<2.We consider a Markov-modulated version of the CGMY
process with the following Markov-switching compensator:(3.24)νXtCGMY(dz,dt)=kXtCGMY(z)dzdt=(Cexp⁡(−〈G,Xt〉|z|)|z|1+YI{z<0}+Cexp⁡(−〈M,Xt〉|z|)|z|1+YI{z>0})dzdt, where M:=(M1,M2,…,MN)∈ℛN and G:=(G1,G2,…,GN)∈ℛN with Mi>0 and Gi>0,
for each i=1,2,…,N.For eachk=1,2,
let N¯Xtk(dz,dt) denote a Markov-switching Poisson random
measure with the following intensity process: (3.25)ρXtk(dz|t)η′(t)dt=1Γ(1−α)e−〈bk,Xt〉zz−α−1dzη′(t)dt=1Γ(1−α)e−〈bk,Xt〉|z||z|1+αdzη′(t)dt. Let N¯Xt(dz,dt):=N¯Xt1(dz,dt)I{z>0}+N¯Xt2(dz,dt)I{z<0}.
Then, the Markov-switching intensity process for N¯Xt(dz,dt) is (3.26)ρXt(dz|t)η′(t)dt=(1Γ(1−α)e−〈b1,Xt〉|z||z|1+αI{z>0}+1Γ(1−α)e−〈b2,Xt〉|z||z|1+αI{z<0})dzη′(t)dt.In this case, the kernel functionh is given by (3.21). Define a process μ¯(t) as below: (3.27)μ¯(t):=∫0t∫Rh(z)N¯Xu(dz,du). Let η(dt)=dt;bi1=Mi;bi2=Gi;α=Y;C=1/Γ(1−α).
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 Section3 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 Section2, 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: (4.1)μQ(t)=∫0t∫R+h(z)NXuQ(dz,du), where the Markov-switching
compensator for the Poisson random measure NXt𝒬(dz,dt) under 𝒬 is (4.2)νXtQ(dz,dt)=ρXtQ(dz|t)η′(t)dt=∑i=1N(ρiQ(dz|t)〈Xt,ei〉)η′(t)dt. 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,…,n−1.
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):=∫0Tη′(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{Xj∣j=1,2,…,nT}.(2)
Generate i.i.d. positive random variablesT1,…,TMT,
which represent jump times, from the normalized density function η′(t)/η(T),
where η(T):=∫0Tη′(t)dt.(3)
Generate the jump sizeWi 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 fixedi=1,2,…,MT,
if Ti∈[tj,tj+1),
calculate λi=η(T)ρXj𝒬(Wi|Ti)/MTgTi(Wi),
for each i=1,…,M.(5)
Generate the Poisson weightsZi from a Poisson distribution with intensity
parameter λi.(6)
μ(t)=∑i=1MTh(ZiWi)I(Ti≤t),
where h(⋅) 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 benefitG(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: (4.3)[0.60.40.40.6]. 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: (4.4)Yj+1=Yj+(rXj−12σXj2)Δ+∫R+(1−eh(z)+h(z))ρXjQ(dz|tj)Δ+σXj×ϵ×Δ+μXjQ(tj+1)−μXjQ(tj), where ϵ~N(0,1) and (4.5)μXjQ(t)=μXjQ(t)−∫0t∫R+ρXjQ(dz|u)du. 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.LetΔYj+1:=Yj+1−Yj,
for each j=0,1,…,5039;
then, (4.6)ΔYj+1:=(rXj−12σXj2)Δ+∫R+(1−eh(z)+h(z))ρXjQ(dz|tj)Δ+σXj×ϵ×Δ+μXjQ(tj+1)−μXjQ(tj). Given each simulated path of {ΔYj}j=1,2,…,5039, {Aj}j=1,2,…,5040 is calculated as (4.7)Aj=A0e∑k=1jΔYk, 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: (4.8)Rt+1=Rtexp⁡[∑j=1252max⁡(rg,ln⁡(A252t+jRt)−β)Δ], where t represents the tth year and the initial value R0=αpA0.We assume some
specimen values for the parameters for the participating
policy:(4.9)rg=0.04;β=0.5;γ=0.7;αp=0.6;A0=100;T=20years. We consider some specimen values
for the model parameters: (4.10)μ1=0.10;μ2=0.05;r1=0.035;r2=0.015;σ1=0.2;σ2=0.4;b1=200.0,b2=500.0. 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
parameterα 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: (4.11)μ=0.1;r=0.035;σ=0.2;μX=−0.05;σX=0.07;λ=0.6, where the jump size X of the compound Poisson process follows a
normal distribution with mean μX and variance σX2;λ is the intensity parameter of the Poisson
process.We consider a
VG process for the log return process{Yt}t∈𝒯 with the following time-changed or
subordinated Brownian motion representation: (4.12)Yt=ZL(t,v),t∈T, 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:(4.13)θ=0.0;v=0.01;σVG=0.2. 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.Figures1–4 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.Figure 1
Probabilities that the terminal asset
valueA(T) is less than the terminal policy reserve R(T) under 𝒫.Figure 2
Fair value of the guaranteed benefit.Figure 3
Fair value of the terminal bonus
option.Figure 4
Fair value of the default option.From Figure1, 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 parameterc 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.Figures5–8 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.Figure 5
Probabilities that the terminal asset
valueA(T) is less than the terminal policy reserve R(T) under 𝒫.Figure 6
Fair value of the guaranteed benefit.Figure 7
Fair value of the terminal bonus
option.Figure 8
Fair value of the default option.From Figures5–8, 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 6–8, 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 parameterq 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.Figures9–12 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.Figure 9
Probabilities that the terminal asset
valueA(T) is less than the terminal policy reserve R(T) under 𝒫.Figure 10
Fair value of the guaranteed benefit.Figure 11
Fair value of the terminal bonus
option.Figure 12
Fair value of the default option.From Figures9–12, 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 10–12).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
parameterα) 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 ratioβ 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.


## 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
parameterα 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: (4.11)μ=0.1;r=0.035;σ=0.2;μX=−0.05;σX=0.07;λ=0.6, where the jump size X of the compound Poisson process follows a
normal distribution with mean μX and variance σX2;λ is the intensity parameter of the Poisson
process.We consider a
VG process for the log return process{Yt}t∈𝒯 with the following time-changed or
subordinated Brownian motion representation: (4.12)Yt=ZL(t,v),t∈T, 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:(4.13)θ=0.0;v=0.01;σVG=0.2. 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.Figures1–4 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.Figure 1
Probabilities that the terminal asset
valueA(T) is less than the terminal policy reserve R(T) under 𝒫.Figure 2
Fair value of the guaranteed benefit.Figure 3
Fair value of the terminal bonus
option.Figure 4
Fair value of the default option.From Figure1, 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 parameterc 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.Figures5–8 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.Figure 5
Probabilities that the terminal asset
valueA(T) is less than the terminal policy reserve R(T) under 𝒫.Figure 6
Fair value of the guaranteed benefit.Figure 7
Fair value of the terminal bonus
option.Figure 8
Fair value of the default option.From Figures5–8, 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 6–8, 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 parameterq 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.Figures9–12 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.Figure 9
Probabilities that the terminal asset
valueA(T) is less than the terminal policy reserve R(T) under 𝒫.Figure 10
Fair value of the guaranteed benefit.Figure 11
Fair value of the terminal bonus
option.Figure 12
Fair value of the default option.From Figures9–12, 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 10–12).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
parameterα) 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 ratioβ 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.



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*Source: 474623-2008-07-13.xml*