Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level

In this paper, we investigate the valuation of dynamic fund protections under the assumption that the market value of the basic fund and the protection level follow regime-switching processes with jumps. The price of the dynamic fund protection (DFP) is associated with the Laplace transform of the first passage time. We derive the explicit formula for the Laplace transform of the DFP under the regime-switching, hyper-exponential jump-diffusion process. By using the Gaver-Stehfest algorithm, we present some numerical results for the price of the DFP.


1.
Introduction. Sales of equity-indexed annuities (EIAs) have grown considerably in recent years. An equity-indexed annuity (EIA) is an insurance product issued by an insurance company whose earning rate is closely related to an equity index. EIAs offer a minimum guaranteed interest rate combined with an interest rate linked to a market index. In general, investing directly in the stock market involves high risk, while investing in a fixed deposit usually earns a very low return rate. An equity-indexed annuity is a hybrid between a fixed and variable annuity. Due to the minimum guaranteed return, EIAs have less market risk than variable annuities. EIAs can also earn returns better than traditional fixed annuities when the stock market is advancing. Therefore, they are appealing to moderately conservative investors who usually hesitate to take high risks but want a moderate investment growth.
Regime switches are often interpreted as structural changes in macro-economic conditions and in different stages of business cycles. By using monthly returns data from the Standard and Poor's 500 and the Toronto Stock Exchange 300 indices, Hardy (2001) [14] finds that the regime-switching lognormal model fits to the monthly returns data much better than other econometric models, such as the independent lognormal model and the ARCH type models.
Motivated by the previous works, in this paper we investigate the valuation of DFP under a regime-switching process. Recently, Siu et al. (2015) [21] investigate the valuation of various equity-linked benefits including DFP under a regimeswitching, double exponential jump-diffusion process by using Laplace transform. Jin et al. (2016) [16] study the value of dynamic fund protections under a generalized regime-switching jump-diffusion model. Instead of investigating the Laplace transform of the first passage time in [21], they derive the coupled system of integrodifferential equations for the value of DFP. Since it is impossible to find the analytic solutions, they focus on designing a numerical scheme to approximate the value of DFP. In general, the protection level is often set to be a constant or an exponential function with time t. However, the protection level is not necessarily chosen to be deterministic. See for example, Dong (2013) [3] proposes a stochastic protection level with the guaranteed rate matching the return of a government bond and derives the fair value of DFP under a Vasicek interest rate environment. Gerber and Shiu (2003) [10] model the protection by a Black-Scholes model. In fact, DFP can be beneficial in that it protects downside risk for investors. If the protection level is modeled by a stochastic process with positive jumps, it can help to provide a higher protection for the investors since the value of the protection level will increase during economic growth.
There is rather abundant research on the valuation of DFP with a deterministic protection level. However, relatively few studies focus on the pricing models using stochastic protection level with jumps. Recently, Xu and Dong (2018) [24] model the protection level by a jump-diffusion process. In this paper, we shall price DFP under a regime-switching jump-diffusion model with a stochastic protection level by using a Laplace transform method, which is similar to [21]. However, comparing with [21], where the authors focus on a regime-switching, double exponential jump-diffusion process and a constant protection level, we consider a more general jump distribution and a stochastic protection level with jumps. Furthermore, we derive the integro-differential equations satisfied by the Laplace transforms of the first passage time. In particular, we present explicit formula for the Laplace transform of the value of the DFP at time 0 by solving integro-differential system when the size of jumps has a regime-switching hyper-exponential distribution, which is more extensive than a regime-switching double exponential distribution in [21]. It is well known that empirically asset return distributions have high peaks and heavy tails. The hyper-exponential distribution is rich enough to approximate many other distributions, including any discrete distribution, the normal distribution, and various heavy-tailed distributions such as Gamma, Weibull and Pareto distributions in the sense of weak convergence. Furthermore, the stochastic protection level with jumps can produce a higher protection level, so that it may be more attracted to the investors.
The main contribution of this paper is that we have proposed a regime-switching jump-diffusion model with a stochastic protection level for pricing DFP, provided a method to value DFP, and characterized explicitly the Laplace transform of the value of the DFP at time 0 when the size of jumps has a regime-switching hyperexponential distribution.
The rest of the paper is organized as follows. Section 2 introduces the models for the fund price and the protection level. In Section 3, we derive a system of integro-differential equations for the Laplace transforms of the first passage time. In particular, closed-form solution for the Laplace transform of the value of the DFP at time 0 under the regime-switching hyper-exponential jump-diffusion model is obtained. Section 4 carries out some calculations. Section 5 concludes the paper. The proofs are presented in the appendix. } 0≤t≤T , Q} be a filtered complete probability space, where Q is the risk neutral measure such that the discounted value process of F (t) is a Q-martingale, and { t } 0≤t≤T is a filtration satisfying the usual conditions. Throughout the paper, it is assumed that all random variables are well defined on this probability space and T −measurable. Let The states of the Markov chain X are interpreted as different states of an macro-economy or different stages of a business cycle. Without loss of generality, we identify the states of the Markov chain X with a set of standard unit vectors E = {e 1 , e 2 , · · · , e N }, where e i = (0, · · · , 0, 1, 0, · · · , 0) * ∈ R N , * denotes the transpose of a vector or a matrix. Let ., . denote a scalar product in R N , that is, for any Let X := { X t |t ∈ T } be a filtration generated by the Markov chain X. Elliott et al. (1994) [5] provide the following semi-martingale decomposition for X: where M(t) is an R N -valued martingale with respect to the filtration X under the measure Q. Gerber and Shiu (2003)[10] model the basic fund price and the protection level by a bivariate Winer process. Motivated by them, we shall model the processes of the basic fund price and the protection level by two regime-switching processes with jumps. Consider two regime-switching compound Poisson processes L 1 := {L 1 (t)|t ∈ T } and L 2 := {L 2 (t)|t ∈ T } such that where N i = {N i (t)|t ∈ T } is a regime-switching Poisson process with intensity given by λ i (t) = λ i , X(t) for a constant vector λ i = (λ i1 , · · · , λ iN ) * with λ ij > 0 for each i = 1, 2, j = 1, 2, · · · , N ; the non-negative sequences {Y 1 1 , Y 1 2 , · · · }, {Y 2 1 , Y 2 2 , · · · } and {Y 3 1 , Y 3 2 , · · · }, are independent and they are independent of N 1 , N 2 , given the path of the Markov chain X. Furthermore, given X, the random variables Y i j , j = 1, 2, · · · , are assumed to be independent and identically distributed with the common conditional distribution F i t (y). Assume F i t (.) = F i (.), X(t) , where F i (.) = (F i1 (.), · · · , F iN (.)) and the distribution functions F ij (.), i = 1, 2, 3, j = 1, · · · , N, satisfy some suitable integrability conditions. That is to say, when X(s) = e j for s ∈ (0, t], the process N i follows a Poisson distribution with parameter λ ij , and the corresponding jump amounts have a common distribution F ij .
We now introduce the DFP. Denote the time-t account value with dynamic guarantee byF (t). In Gerber and Pafumi (2000) [8], the relation between the two pro-cessesF (t) and F (t) is given bŷ Hence, the terminal payoff for DFP with a maturity T is given by Remark 2.1. Note that, the protection level and the basic fund price process have common upward jumps so that the insurer can provide a higher protection against downside risk for the investors. In fact, DFP can be beneficial in that it protects downside risk for investors. However, in the view of insurers, this implies that they must settle for downside risk that they originally did not face. Therefore, if the protection level is modeled by a stochastic process with downward jumps, then it can help to reduce such an unsatisfying risk for insurers since the value of the protection level will decrease during economic recession. So we can also model the protection level by a regime-switching two-sided jump-diffusion process. But this choice does not bring any essential extension of mathematics.
Then under the new measureQ, the processS(t) is a regime-switching compound Poisson process, whereÑ (t) is a regime-switching Poisson process with intensity given byλ(t) = λ 1 (t) +λ 2 (t), and given X, the common conditional distribution of the random variables Z j , j = 1, 2, · · · is given bỹ The proof is presented in the appendix.
From Lemmas 3.1, 3.2, we can conclude that the process Z(t) is also a regimeswitching jump-diffusion process underQ. Although it is difficult to give the distribution of the running minimum process m Z (t), we can obtain the integro-differential system satisfied by the Laplace transforms of the first passage time under the regimeswitching jump-diffusion model.
Let u > 0 be a given constant and define the first passage time by with the convention inf{φ} = ∞. Then we havẽ For u > 0, δ > 0, define It is easy to see To simplify the notation, we drop δ in the parameters. Firstly, we derive the integrodifferential equations for Φ i (u), i = 1, 2, · · · , N.
Proof. The proof is presented in the appendix.
In order to obtain a closed form expression for the Laplace transform Φ i (u), in what follows, we consider a continuous-time Markov chain X with two states (i.e., N = 2). We suppose that state e 1 (state e 2 ) represents a "bad" ("good") economic state. Therefore, the intensity matrix can be written as where a 12 > 0, a 21 > 0. Furthermore, for each j = 1, 2, we assume the distributions F 1j and F 2j follow the hyper-exponential distributions with density functions given by The condition α i1 < α i2 , β i1 < β i2 holds due to the fact that the expectation of the jump size corresponding to the "bad" economic state should be greater than that corresponding to the "good" economic state. Then From Lemma 3.1, we can obtain that the distributionsF 1j andF 2j have the density functions given byf For each j = 1, 2, we assume the distribution F 3j follows an exponential distribution with density function given by f 3j (y) =β 0j e −β0j y , y > 0, whereβ m2 <β 01 <β 02 .
For simplicity, we only consider the case when all x ij 's are distinct and all y ij 's are also distinct since the analysis of the other case is more tedious. Lemma 3.3. For δ > 0, the equation (3.18) has 2m + 2 district positive real roots, and 2m + 4 district negative real roots. Proof. The proof is presented in the appendix.
Parallel to the first passage time distribution under the regime-switching Brownian motion framework in Guo (2001) [12] and under the regime-switching doubleexponential jump-diffusion model in Siu et al. (2015) [21], the uniqueness and existence of the solution to Similarly, d j 's satisfy = a 22 , i = 0, 1, · · · , m.
By using the same arguments as in the proof of the non-singularity of the matrix H, we can prove the matrix G is invertible. Then we have d j = (detG) −1 (detG j ), j = 1, 2, · · · , 2m + 4. Let π 1 = Q(X(0) = e 1 ) and π 2 = Q(X(0) = e 2 ) with π 1 + π 2 = 1. Note that, the distribution of the Markov chain X underQ remains the same. We have Proof. The proof is presented in the appendix. Theorem 3.2. For any δ > 0, the Laplace transform of DF P 0 is given by  [4]. For the details of the implementation of the Gaver-Stehfest algorithm, we refer to Section 5 in [18] or Section 4 in [4]. For all the computations, the values of certain parameters are held fixed except otherwise indicated: we take T = 5, F 0 = 100, K 0 = 70, a 11 = a 22 = −0.1, r 1 = 0.02, r 2 = 0.05, σ 1 = 0.6, σ 2 = 0.3, λ 11 = λ 21 = 2, λ 12 = λ 22 = 1, m = 1, α 11 = β 11 = 25, α 12 = β 12 = 50, β 01 = 30, β 02 = 60. For the distribution of X(0), we consider two choices: π 1 = 1, π 2 = 0 and π 1 = 0, π 2 = 1, which correspond to the cases that the chain starts from e 1 ("bad" economic state) and e 2 ("good" economic state), respectively. Figures 1-6 present the effects of maturity T and model parameters on the values of DF P 0 for π 1 = 1 and π 1 = 0. From them we can see the value of DF P 0 is much larger for π 1 = 1. That is to say, the value of DF P 0 is much higher when we start at the state e 1 ("bad" economic state) at time t = 0.   Figure 1 displays the relationship between DF P 0 and T for the stochastic protection level and the constant protection level. From it we can observe that DF P 0 increases with T. This is because a longer protection period can lead to a larger protection cost. We can also see that the value of DF P 0 for the stochastic protection level with upward jumps is much larger than that of the constant protection level. This is because the insurer provides a higher protection against the downside risk for the investors. Figure 2 plots DF P 0 as a function of F 0 . It can be seen that DF P 0 is a decreasing function of F 0 . The reason is that a higher value of F 0 leads to a decreasing probability of the asset price hitting the protection level.  Figure  4. DF P 0 versus a 12 Figure 3 represents the relationship between K 0 and DP F 0 . It can be seen that DF P 0 increases with K 0 , since a higher value of K 0 leads to a higher protection level. Figure 4 presents the impact of transition intensity a 12 on DP F 0 with a 21 = a 12 . From it we can conclude that a higher a 12 with a 12 = a 21 results in a higher value of DP F 0 if we start at the "good" economy at time t = 0. This is because higher a 21 leads to an increasing probability of switching to the "bad" economy. On the other hand, if we start at the "bad" economy at time t = 0, the value of DP F 0 decreases with a 12 . This is due to an increasing probability of switching to the "good" economy.  Figure  6. DF P 0 versus σ 1 Figure 5 plots the impact of jump intensity λ 11 on the values of DP F 0 with λ 11 = 2λ 12 . It can be seen that DF P 0 increases with jump intensity, which implies that a higher jump risk can lead to a larger protection cost. Figure 6 graphs DF P 0 as a function of the basic fund's volatility σ 1 with σ 1 = 2σ 2 . From it we can conclude that DF P 0 is an increasing function of the volatility, in line with stylized features and financial intuition: the option price increases as the volatility increases.
To sum up, numerical results indicate that changes of market regimes and the stochastic protection level have material effects on DF P 0 .

5.
Conclusions. In this paper, we investigate the valuation of DFP under a regimeswitching jump-diffusion model. Furthermore, we consider a stochastic protection level with positive jumps, which provide a higher protection, so that it may be more appealing to the investors. Since the Laplace transform of the price of DFP is associated with the Laplace transform of the first passage time, we derive integrodifferential system for the Laplace transforms of the first passage time. Explicit solutions for the system are obtained when the jumps follow a regime-switching hyper-exponential distribution. Based on the results, we give numerical calculations for the value of DFP by inverting Laplace transforms via Gaver-Stehfest algorithm. Numerical results illustrate the regime-switching effects have a significant effect on the value of DFP. Therefore, we should incorporate changes of market regimes into models for pricing long term insurance products.
There remain many open questions for the valuation of DFP under regimeswitching models. For example, the results obtained under a regime-switching hyper-exponential jump-diffusion model are useful for the valuation of DFP, and can also provide a useful benchmark for more complicated regime-switching models. Based on the derived results, we may investigate the pricing of DFP under a general regime-switching jump-diffusion model by resorting to some numerical procedures. Note that the pricing measure in this paper is fixed. We may investigate how to choose a risk-neutral measure since there are many risk-neutral probability measures. We leave these and other questions for future research. 6. Appendix. Proof of Lemma 3.2. For y ∈ R, let Since the path of the Markov chain (X s ) s≤t is known to us, we denote the jump times in the interval [0, t] of the Markov chain X s by 0 = T 0 < T 1 < · · · < T k = t.

Then we have that
From [1], we obtain where diag(θ) denotes a diagonal matrix with the diagonal entries given by the vector θ and 1 = (1, · · · , 1) * . Similarly, we can also prove that Conditioning on the event occurs in the interval [0, h], we have By Taylor's expansion, we have and Plugging the above formulas into (A.5), dividing both sides by h and letting h → 0 we obtain (3.5).