2. Range Value-at-Risk

Let $(\Omega,\mathcal{F},P)$ be an atomless probability space and $\mathcal{S}$ be the set of real-valued random variables denoting loss amounts (a negative value represents gains). A risk measure $\rho$ is a mapping from $\mathcal{S}$ to $\mathbb{R}$ . For instance, VaR and CVaR are commonly used risk measures in the financial industry which are defined as

(1) \begin{equation} VaR_\alpha(L)=\inf\{v\in \mathbb{R}\;:\;P(L>v)\leq 1-\alpha\} \end{equation}

and

(2) \begin{equation} CVaR_\alpha (L)=\frac{1}{1-\alpha}\int_{\alpha}^{1}VaR_s(L) ds \end{equation}

where $\alpha\in (0,1)$ is the risk level and the random variable $L\in \mathcal{S}$ with finite mean.

In this paper, we would like to investigate a measure called Range Value-at-Risk (RVaR) that can be seen as a bridge between VaR and CVaR.

The RVaR at the level $(\alpha,\beta)$ such that $0<\alpha\leq \alpha+\beta< 1$ is

(3) \begin{equation} RVaR_{\alpha,\beta}(L)=\bigg\{\begin{array}{ll} \frac{1}{\beta}\int_\alpha^{\alpha+\beta}VaR_s(L)ds, & if\:\:\beta>0, \\[5pt] VaR_\alpha(L), & if\:\:\beta=0 \end{array}\end{equation}

which is the average value of VaR among specific risk levels $[\alpha, \alpha+\beta]$ .

A risk measure is called coherent if it satisfies (see Artzner et al. Reference Artzner, Delbaen, Eber and Heath1999 for details):

1. (1) Monotonicity: If $L_1$ , $L_2\in \mathcal{S}$ such that $L_1\leq L_2$ , then $\rho(L_1)\leq \rho(L_2)$ ;

2. (2) Positive homogeneity: If $L\in \mathcal{S}$ and $s\in \mathbb{R}^+$ , then $\rho(sL)=s\rho(L)$ ;

3. (3) Translation invariance: If $L\in \mathcal{S}$ and $s\in \mathbb{R}$ , then $\rho(L+s)=\rho(L)+s$ ;

4. (4) Subadditivity: If $L_1$ , $L_2\in \mathcal{S}$ , then $\rho(L_1+L_2)\leq\rho(L_1)+\rho(L_2)$ .

CVaR is a coherent risk measure. However, Embrechts et al. (Reference Embrechts, Liu and Wang2018) have proved that both VaR and RVaR only satisfy properties (1)–(3) listed above while they do not satisfy subadditivity. Instead, for $L_1$ , $L_2\in \mathcal{S}$ , the following inequalities hold true:

(4) \begin{equation} VaR_{\alpha_1+\alpha_2}(L_1+L_2)\leq VaR_{\alpha_1}(L_1)+VaR_{\alpha_2}(L_2),\end{equation}

(5) \begin{equation} RVaR_{\tilde{\alpha}+\tilde{\beta}}(L_1+L_2)\leq RVaR_{\alpha_1,\beta_1}(L_1)+RVaR_{\alpha_2,\beta_2}(L_2)\end{equation}

where $\tilde{\alpha}=\alpha_1+\alpha_2$ and $\tilde{\beta}=\max\{\beta_1,\beta_2\}$ . (4) and (5) are called the special form of subadditivity.

Desired properties of RVaR are that it is robust, and it is a general tail risk measure which includes VaR as well as CVaR as two limiting cases, that is,

(6) \begin{equation} RVaR_{\alpha,0}(L)=VaR_\alpha(L)=\underset{\beta\rightarrow{0^+}}{\lim}RVaR_{\alpha,\beta}(L),\end{equation}

(7) \begin{equation} RVaR_{\alpha,1-\alpha}(L)=CVaR_\alpha(L)\end{equation}

Additionally, all three measures belong to a wide class of distortion risk measures, such that

(8) \begin{equation} \rho(L)=\int_0^1VaR_s(L)dh(s)\end{equation}

where h(s) called a distortion function is a non-decreasing and left-continuous function from [0,1] to [0,1] satisfying $h(0)=0$ and $h(1)=1$ . Specifically, for $\alpha\in (0,1)$ , $\beta\in [0,1)$ , the distortion function of $RVaR_{\alpha,\beta}$ is

(9) \begin{equation} h_{\alpha,\beta}(s)=\bigg\{\begin{array}{ll} \min\big\{I_{\{s>\alpha\}}\frac{s-\alpha}{\beta},1\big\}, & if\:\:\beta>0, \\[5pt] I_{\{s>\alpha\}}, & if\:\:\beta=0 \end{array}\end{equation}

As we know that while VaR and CVaR have some common properties, they are still quite different and consequently, investigating RVaR which is a bridge between them helps us to understand their connections and to gain a more integrated picture regarding to risk criteria.

3. Optimal RVaR-based Hedging and Connection with CVaR and VaR Hedging

3.1 Optimal RVaR-based hedging

A European contingent claim X with maturity T is a non-negative $\mathcal{F}$ measurable random variable which can be thought of as a contract or agreement that pays X at the maturity T. Assume that a financial market is arbitrage free but does not have to be complete. In this case, arbitrage free price of a claim may not be unique and a popular option pricing approach in incomplete markets is the utility-based indifference pricing (UBIP). The idea behind this pricing method is that all investors’ risk appetite can be fully described by a utility function, and all investors are presumed to maximise their expected utility of wealth. Then, the utility-based indifference price of a claim is the amount that makes no difference to investors’ expected utility no matter whether they include the claim into their portfolio or not. One important property of utility-based indifference price is that it does not depend on the completeness of the market, and in the special case that a market is complete, it coincides with the fair price of that claim (see Henderson & Hobson Reference Henderson and Hobson2004 for more properties regarding UBIP).

Suppose a hedger is exposed to a future obligation X at time T. Option pricing theorem points out that a claim can be hedged perfectly if the initial hedging capital is no less than its minimal super-hedging price (or fair price in complete markets). However, if the initial hedging capital $\tilde{v}_0$ is not enough, the hedger is exposed to some downside risk. The main goal of this paper is to find an optimal hedging strategy that achieves the minimal risk under the measure RVaR subject to an initial wealth constraint.

We start with a partition of the contingent claim X such that $X=f(X)+R_f(X)$ . f(X) named the hedged loss function represents the part that would be hedged with the initial capital $\tilde{v}_0$ and $R_f(X)$ called the retained loss function is the payout the hedger retains. Thus, the problem of optimal hedging can be solved in two steps: first, find the optimal partition $f^*(X)$ , $R_{f^*}(X)$ satisfying the initial capital constraint and some assumptions regarding to hedged loss functions; second, hedge the payout $f^\ast(X)$ perfectly. Cong et al. (Reference Cong, Tan and Weng2013, Reference Cong, Tan and Weng2014) considered VaR and CVaR hedging problems with some restrictive assumptions regarding to the hedged loss functions. Assumptions they imposed are

1. (a) Not globally over-hedged: $f(x)\leq x$ for all $x\geq 0$ ;

2. (b) Not locally over-hedged: $f(x_2)-f(x_1)\leq x_2-x_1$ for all $0\leq x_1\leq x_2$ ;

3. (c) Non-negativity of the hedged loss: $f(x)\geq 0$ for all $x\geq 0$ .

Assumption (a) ensures that the hedged loss would be bounded from above by the original risk to be hedged. Assumption (b) indicates that the increment of the hedged part should be no more than the increment of the risk itself. Although adding assumption (b) makes hedged loss functions less general, it indeed brings advantages. For instance, as we will show later, the resulting optimal hedged loss function does not depend on the underlying market model if we have the assumption (b). Assumption (c) is commonly imposed that ensures hedging strategies take positive values, and Cong et al. (Reference Cong, Tan and Weng2013) mentioned that without the assumption (c), the optimal partial hedging problem might be ill-posed.

Hence, the admissible set $\mathcal{D}$ of the hedged loss function is represented as

(10) \begin{align} \mathcal{D}=\big\{&0\leq f(x)\leq x\;:\:\:\;R_f(x)=x-f(x)\\[5pt] \nonumber& \text{is a non-decreasing and left-continuous function} \big\} \end{align}

Based on the above considerations, the RVaR hedging problem in our setting is

(11) \begin{equation} \Big\{\begin{array}{ll} \underset{f\in \mathcal{D}}{\min}\:\: RVaR_{\alpha,\beta}(R_f(X)), \\[5pt] s.t \:\: \Pi \big(f(X)\big)\leq \tilde{v}_0< \Pi \big(X\big) \end{array} \end{equation}

where $\Pi(X)$ denotes the utility indifference price of X at time 0. Conclusions in this paper can be applied for all forms of utility indifference price.

Depending on the size of the initial capital $\tilde{v}_0$ , we represent optimal hedged loss functions of the problem (11) in Theorems 1 and 4 correspondingly.

Theorem 1. If $\Pi \big(XI_{\{X\leq VaR_{\alpha+\beta}(X)\}}\big)\leq \tilde{v}_0$ , the optimal hedged loss function satisfies

(12) \begin{align} f^*(x)=xI_{\{x\leq v_{\alpha+\beta}\}}\end{align}

where $v_{\alpha+\beta}=VaR_{\alpha+\beta}(X)$ .

Moreover, the minimal value of RVaR is zero, that is,

\begin{equation*} RVaR_{\alpha,\beta}(R_{f^*}(X))=0\end{equation*}

Proof. For $f^*(x)$ defined as (12), the retained loss function is

\begin{equation*} R_{f^*}(x)=x-f^*(x)=x-xI_{\{x\leq v_{\alpha+\beta}\}}=xI_{\{x> v_{\alpha+\beta}\}}\end{equation*}

which is non-decreasing and left continuous regarding to x, and hence, the hedged loss function $f^*(x)\in \mathcal{D}$ and $f^*(X)$ satisfies the wealth constraint.

Notice that, for any $s\leq \alpha+\beta$ , we have

\begin{align*} P\Big(XI_{\{X> v_{\alpha+\beta}\}}>0\Big)&\leq P\Big(X>v_{\alpha+\beta}\Big)\\[5pt] &\leq \big(1-(\alpha+\beta)\big)\leq 1-s\end{align*}

and consequently we get

\begin{equation*} VaR_s\Big(XI_{\{X> v_{\alpha+\beta}\}}\Big)=0,\:\:\:\forall s\leq \alpha+\beta\end{equation*}

which implies

\begin{align*} RVaR_{\alpha,\beta}\big(R_{f^*}(X)\big)&=RVaR_{\alpha,\beta}\big(XI_{\{X> v_{\alpha+\beta}\}}\big)\\[5pt] &=\frac{1}{\beta}\int_\alpha^{\alpha+\beta}VaR_s\big(XI_{\{X> v_{\alpha+\beta}\}}\big)ds\\[5pt] &=0\end{align*}

Furthermore, for any $f\in \mathcal{D}$ , we have $RVaR_{\alpha+\beta}\big(R_f(X)\big)\geq 0=RVaR_{\alpha+\beta}\big(R_{f^\ast}(X)\big),$ and hence, we conclude that $f^*$ is the optimal hedged loss function.

Let us now consider the second case: $ \tilde{v}_0<\Pi \big(XI_{\{X\leq VaR_{\alpha+\beta}(X)\}}\big)$ . To begin with, we need to recall the definition of stop-loss ordering between two random variables $X_1$ and $X_2$ :

Suppose $X_1$ and $X_2$ are two random variables with finite means under a probability measure P. We say $X_1$ is smaller than $X_2$ in stop-loss order under P, if

(13) \begin{equation} E\big((X_1-m)^+\big)\leq E\big((X_2-m)^+\big), \:\:\:\forall m\in \mathbb{R}\end{equation}

Such a relationship is denoted as $X_1\leq _{sl}^P X_2$ .

Lemma 2. For a given random variable X and any function $f\in \mathcal{D}$ , let

(14) \begin{equation} g_{f}(x)=\min\big\{(x-d)^+,\bar{u}\big\}I_{\{x\leq v_{\alpha+\beta}\}}\end{equation}

where $v_{\alpha}=VaR_\alpha(X)$ , $d=v_{\alpha}-f(v_{\alpha})$ and $\bar{u}$ is chosen such that

(15) \begin{equation} RVaR_{\alpha,\beta}(R_f(X))=RVaR_{\alpha,\beta}(R_{g_f}(X))\end{equation}

Then, we have

1. (1) $g_f$ is well defined and $g_f\in \mathcal{D}$ ;

2. (2) $g_f(X)$ is smaller than f(X) in stop-loss order under the measure P, that is, $g_f(X)\leq_{sl}^P f(X).$

Remark 3. Let us denote $u=d+\bar{u}$ . Since $\bar{u}\in [0,v_{\alpha+\beta}-d]$ , and $d=v_{\alpha}-f(v_{\alpha})\geq 0$ , we have $d\leq u\leq v_{\alpha+\beta}$ . Consequently, $g_f(x)$ can be rewritten as

(16) \begin{align} g_f(x)&=\min\big\{(x-d)^+,u-d\big\}I_{\{x\leq v_{\alpha+\beta}\}}\nonumber\\[5pt] &=\Big[(x-d)^+-(x-u)^+\Big]I_{\{x\leq v_{\alpha+\beta}\}}\end{align}

where $0\leq d\leq v_{\alpha}$ and $d\leq u\leq v_{\alpha+\beta}$ .

According to Cong et al. (Reference Cong, Tan and Weng2014), the utility-based indifference price preserves the stop-loss order, that is, $\Pi(X_1)\leq \Pi(X_2)$ , if $X_1\leq _{sl}^P X_2,$ and therefore, Lemma 2 indicates that for any $f\in \mathcal{D}$ , there is a $g_f\in \mathcal{D}$ such that

\begin{equation*} RVaR_{\alpha,\beta}\big(R_f(X)\big)=RVaR_{\alpha,\beta}\big(R_{g_f}(X)\big);\end{equation*}

\begin{equation*} \Pi\big(g_f(X)\big)\leq \Pi\big(f(X)\big)\end{equation*}

Thereby, we can focus on hedged loss functions with the form (16). Let us formulate another theorem about the problem (11).

Theorem 4. If $ \tilde{v}_0<\Pi\big(XI_{\{X\leq v_{\alpha+\beta}\}}\big)$ , the optimal hedged loss function that solves the RVaR minimisation problem (11) is

(17) \begin{equation} f^\ast(x)=\Big[(x-d^\ast)^+-(x-u^\ast)^+\Big]I_{\{x\leq v_{\alpha+\beta}\}} \end{equation}

where $(d^\ast, u^\ast)$ is the solution to the following 2-dimensional optimisation problem

(18) \begin{equation} \bigg\{\begin{array}{ll} \underset{0\leq d\leq v_\alpha,d \leq u\leq v_{\alpha+\beta}}{\min}d+\frac{1}{\beta}\int _\alpha^{\alpha+\beta} (VaR_s(X)-u)^+ ds, \\[5pt] s.t.\:\: \Pi\Big(\big[(X-d)^+-(X-u)^+\big]I_{\{X\leq v_{\alpha+\beta}\}}\Big)\leq \tilde{v}_0 \end{array} \end{equation}

Proof. By Remark 3, we know that the optimal hedged loss function admits the form (16), and hence, the objective function in (11) becomes

\begin{align*} RVaR_{\alpha+\beta}\big(R_{g_f}(X)\big)&=\frac{1}{\beta}\int _\alpha^{\alpha+\beta}VaR_s\big(R_{g_f}(X)\big)ds\\[5pt] &=\frac{1}{\beta}\int _\alpha^{\alpha+\beta}R_{g_f}\big(VaR_s(X)\big)ds\\[5pt] &=\frac{1}{\beta}\int _\alpha^{\alpha+\beta}VaR_s(X)-g_f(VaR_s(X))ds\\[5pt] &=\frac{1}{\beta}\int _\alpha^{\alpha+\beta} VaR_s(X)-(VaR_s(X)-d)+(VaR_s(X)-u)^+ ds \\[5pt] &=d+\frac{1}{\beta}\int _\alpha^{\alpha+\beta} (VaR_s(X)-u)^+ ds \end{align*}

such that $d\in[0,v_\alpha]$ and $u\in [d,v_{\alpha+\beta}]$ . Hence, (11) is rewritten as (18) and Theorem 4 is proved.

Theorem 4 shows that the optimal strategy is to long a knock-out call option on the payout X with a strike price $d^*$ and a barrier $v_{\alpha+\beta}$ while short another knock-out call option with a higher strike price $u^*$ but with the same barrier. If those knock-out call options are available in the market, our optimal partial hedging is to buy (sell) and hold those options. Otherwise, it is to dynamically hedge those knock-out call options with the initial capital $\tilde{v}_0$ . It is worth to mention that both Theorem 1 and Theorem 4 indicate that the optimal strategy does not cover the extreme loss that exceeds $v_{\alpha+\beta}$ . This is because we only focus on the risk between levels $[\alpha,\alpha+\beta]$ .

3.2 Connection among RVaR, CVaR, and VaR optimal hedging

As discussed in section 2, VaR and CVaR can be seen as two limiting cases of RVaR. In this subsection, we want to discuss connections among optimal RVaR hedging, optimal VaR hedging, and optimal CVaR hedging.

By the definition of CVaR, we have

(19) \begin{equation} CVaR_\alpha(X)=RVaR_{\alpha,1-\alpha}(X)\end{equation}

that is, $\beta=1-\alpha,$ and hence, we get

\begin{equation*} I_{\{X\leq v_{\alpha+\beta}\}}=1;\:\;\:P-a.s\end{equation*}

According to (17), the optimal hedged loss function satisfies the form

(20) \begin{align} f^\ast(x)&=\big((x-d)^+-(x-u)^+\big)I_{\{x\leq v_{\alpha+\beta}\}}\nonumber\\[5pt] &=(x-d)^+-(x-u)^+\end{align}

Cong et al. (Reference Cong, Tan and Weng2014) considered the optimal CVaR hedging and showed the optimal hedged loss function $f^*_{CVaR}$ admitted the form

(21) \begin{equation} f^\ast_{CVaR}(x)=(x-d)^+-(x-u)^+\end{equation}

which is consistent with (20). Therefore, the optimal CVaR hedging can be seen as a special case of the RVaR-based hedging.

As for VaR, we have

\begin{equation*}VaR_{\alpha}(X)=RVaR_{\alpha,0}(X)\end{equation*}

that is, $\beta=0$ . In this case, (16) becomes

\begin{equation*} g_f(x)=\min\big\{(x-d)^+,\bar{u}\big\}I_{\{x\leq v_\alpha\}}\end{equation*}

On the one hand, we know that

(22) \begin{align} RVaR_{\alpha,0}\big(R_{f}(X)\big)&= VaR_{\alpha}\big(R_{f}(X)\big)\nonumber\\[5pt] &=VaR_{\alpha}(X)-f\big(VaR_{\alpha}(X)\big)\end{align}

On the other hand, we have

(23) \begin{align} RVaR_{\alpha,0}\big(R_{g_f}(X)\big)&= VaR_{\alpha}\big(R_{g_f}(X)\big)\nonumber\\[5pt] &=VaR_{\alpha}(X)-g_{f}\big(VaR_{\alpha}(X)\big)\nonumber\\[5pt] &=VaR_{\alpha}(X)-\min\big\{(VaR_\alpha(X)-d)^+,\bar{u}\big\}\nonumber\\[5pt] &=VaR_{\alpha}(X)-\min\big\{f\big(VaR_{\alpha}(X)\big),\bar{u}\big\}\end{align}

Since $\bar{u}$ is derived from the equation

\begin{equation*} RVaR_{\alpha,0}\big(R_{g_f}(X)\big)= RVaR_{\alpha,0}\big(R_{f}(X)\big)\end{equation*}

comparing (22) with (23), we have $\bar{u}\geq f\big(VaR_{\alpha}(X)\big)$ , while from the previous discussions, we know $\bar{u}\leq VaR_{\alpha+\beta}(X)+f\big(VaR_{\alpha}(X)\big)-VaR_{\alpha}(X)$ . The right hand side is $f\big(VaR_{\alpha}(X)\big)$ when $\beta=0$ , and hence, we arrive to $\bar{u}= f\big(VaR_{\alpha}(X)\big)$ .

Meanwhile, the following relationship

\begin{align*} \Big(x+f\big(VaR_\alpha(X)\big)-VaR_\alpha(X)\Big)^+I_{\{x\leq VaR_\alpha(X)\}}\leq f\big(VaR_\alpha(X)\big)=\bar{u}\end{align*}

implies that the optimal hedged loss function has the form

(24) \begin{align} f^*(x)&=\min\big\{(x-d)^+,\bar{u}\big\}I_{\{x\leq v_\alpha\}}\nonumber\\[5pt] &=(x-d)^+I_{\{x\leq v_{\alpha}\}}\end{align}

This result is consistent with the conclusion in Cong et al. (Reference Cong, Tan and Weng2013) that the optimal hedged loss function $f^*_{VaR}$ of the VaR-based hedging problem has the form

(25) \begin{equation} f^\ast_{VaR}(x)=(x-d)^+I_{\{x\leq v_\alpha\}}\end{equation}

and thereby, the optimal VaR hedging can be seen as a special case of the RVaR hedging.

4. Application in Equity-Linked Life Insurance Contracts

One important application of partial hedging is to deal with pricing and hedging of equity-linked life insurance contracts. Well-known papers are Melnikov & Skornyakova (Reference Melnikov and Skornyakova2005), Kirch & Melnikov (Reference Kirch and Melnikov2005), where authors discussed the implementation of quantile hedging and efficient hedging in this area. In this section, we focus on the RVaR-based optimal hedging of life insurance contracts and would compare RVaR, CVaR, and VaR hedging results.

Let $(\Omega,\mathcal{F},P)$ be a standard probability space. Consider a financial market with a terminal time $T \in (0,\infty)$ consisting two assets. One is the riskless asset

(26) \begin{equation} S_t^0 =e^{rt}, \:\:\:t\in[0,T]\end{equation}

where r is the constant risk-free interest rate.

Another one is the risky asset, $(S_t)_{t \in [0,T]}$ , described by a Mixed Fractional Brownian motion (MFBM) model:

(27) \begin{equation} S_t=S_0\exp(\mu t+\sigma_1B_t+\sigma_2B_t^H),\:\:\:S_0>0\end{equation}

where B is a Brownian motion (BM) independent of a Fraction Brownian motion $B^H$ (FBM). The constant $\mu$ is the drift, $\sigma_1>0$ is the volatility of the Brownian motion B, and $\sigma_2>0$ is the volatility of $B^H$ .

The reason why we assume the risky asset follows MFBM model is that, for $H>\frac{1}{2}$ , this model has the property of long-range dependence. Equity-linked life insurance contracts usually have long-term maturities, and hence, the factor of long-term dependence should be included in the list of key factors having a certain influence on pricing and hedging.

Let us denote the mixed process $M_t^\sigma=B_t+\sigma B_t^H$ , $t\in[0,T]$ , where $\sigma=\frac{\sigma_2}{\sigma_1}$ . It is well known that except for $H=\frac{1}{2}$ , $B^H$ is neither a Markov process nor a semimartingale, and thus, the mixed process is also not a semimartingale with respect to the filtration generated by the BM and the FBM. However, Cheridito (Reference Cheridito2001) has proved that in the case $H\in(\frac{3}{4},1)$ , the process $M^\sigma$ is a semimartingale with respect to its natural filtration: $\mathbb{F}^M=\{\mathcal{F}_t^M\:\:\:0<t<T\}$ , $\mathcal{F}_t^M=\sigma\{M_u^\sigma,\:\:\:0\leq u\leq t\}$ . Moreover, it is equivalent in measure to a Brownian motion. In this paper, we consider the filtration $\mathbb{F}^M$ and assume $H\in (\frac{3}{4}, 1)$ , $\mathcal{F}=\mathcal{F}_T^M,$ and hence, there is a Brownian motion $\{W_t\}_{t\in[0,T]}$ and a unique real-valued Volterra kernal $r_\sigma\in L_2([0,T]^2)$ such that the following relationship holds:

(28) \begin{equation} M_t^\sigma=W_t+\int_0^t\int_0^s r_\sigma(s,u)dW_uds, \:\:\:t\in[0,T]\end{equation}

where $r_{\sigma}$ is the unique solution of the equation:

\begin{equation*} \sigma^2H(2H-1)(t-s)^{2H-2}=r_{\sigma}(t,s)+\int_0^sr_{\sigma}(t,x)r_{\sigma}(s,x)dx,\:\:\: 0\leq s<t\leq T\end{equation*}

The MFBM market is complete, and the unique martingale measure $P^\ast$ is defined by the relation (see, Melnikov & Mishura Reference Melnikov and Mishura2011):

(29) \begin{align} Z^*_t&=\frac{dP^\ast}{dP}|_{\mathcal{F}^M_t}=\exp\bigg\{-\int_0^t\Big(\frac{\mu-r}{\sigma_1}+\frac{\sigma_1}{2}+\int_0^s r_\sigma(s,u)dW_u\Big)dW_s\\[5pt] \nonumber &\quad -\frac{1}{2}\int_0^t\Big(\frac{\mu-r}{\sigma_1}+\frac{\sigma_1}{2}+\int_0^s r_\sigma(s,u)dW_u\Big)^2ds\bigg\}\end{align}

Note that, in this complete market, the utility indifference price of a claim is equivalent to its fair price, that $\Pi(X)=e^{-rT}E^*(X)$ , where $E^*()$ is the expectation under the martingale measure $P^*$ .

According to the Girsanov theorem, the process

(30) \begin{equation} \tilde{W}_t=W_t+(\frac{\mu-r}{\sigma_1}+\frac{\sigma_1}{2})t+\int_0^t\int_0^s r_\sigma(s,u)dW_uds\end{equation}

is a Wiener process under the measure $P^\ast$ .

With the help of (28) and (30), the stock price process can be rewritten as

(31) \begin{align} S_t&=S_0\exp\Big\{\mu t+\sigma_1\big(W_t+\int_0^t\int_0^s r_\sigma(s,u)dW_uds\big)\Big\} \\[5pt] &=S_0\exp\big\{\sigma_1 \tilde{W}_t+(r-\frac{\sigma_1^2}{2})t\big\} \nonumber \end{align}

For equity-linked life insurance contracts, there are two sources of risk: market risk associated with the underlying asset price and insurance risk reflected by the insureds’ mortality, and hence, besides characteristics of the financial market, we also need to describe mortality properties of insureds. Following actuarial traditions, let a random variable T(x) on an “actuarial” probability space $(\Omega, \tilde{\mathcal {F}},\tilde{P}$ ) denote the remaining life time of a person of current age x and ${}_{T}p_{x}=\tilde{P}(T(x)>T)$ be the survival probability for the next T years of the insured. Since usually the insurance risk and the financial market risk have no effect on each other, we would take a natural assumption that $(\Omega,\mathcal F,P)$ and $(\Omega, \tilde{\mathcal {F}},\tilde{P}$ ) can be treated as independent.

Let us consider a pure endowment life insurance contract with the payoff $X=\max(S_T,K)$ provided that an insured is alive at T, where K is a constant guarantee amount. Since the mortality risk is essentially independent of the financial market, according to Brennan & Schwartz (Reference Brennan and Schwartz1976), the premium of such a contract is defined as

(32) \begin{equation} {}_TU_x=E^\ast(e^{-rT}X) E^{\tilde{P}}(I_{\{T(x)>T\}})={}_{T}p_{x}V_0\end{equation}

where x is the insured’s age, T is the maturity time of the contract, and $V_0$ is perfect hedging costs of the claim X. According to the most recently published United States 2015 Life Table (National Vital Statistics Reports volume 67, Number 7), the survival probability ${}_{T}p_{x}$ of a given insured can be found. Obviously, ${}_{T}p_{x}V_0<V_0,$ and thus, a perfect hedging of the option is impossible and a partial hedging strategy should be constructed. With the help of methodologies in section 3, we derive the minimal value of RVaR that can be achieved with the initial wealth $\tilde{v}_0={}_{T}p_{x}V_0$ .

Proposition 5. The fair price of the claim $X=\max\{S_T, K\}$ is

(33) \begin{align} V_0&=e^{-rT}E^*(X)\nonumber\\[5pt] &=S_0\Phi\big(\Lambda_+(K)\big)+Ke^{-rT}\Phi\big(-\Lambda_-(K)\big) \end{align}

where $\Phi$ is the distribution function of a standard normal random variable and

\begin{equation*} \Lambda_{\pm}(x)=\frac{\ln\frac{S_0}{x}+(r\pm\frac{\sigma_1^2}{2})T}{\sigma_1\sqrt{T}}\end{equation*}

Proof. See Appendix A.

In addition, applying Theorems 1 and 4 to a claim $X=\max\{S_T,K\}$ , we arrive to the following results.

Here, $z_{1-s}$ is the $(1-s)$ quantile of a standard normal random variable. $\Phi^{(2)}(z_1,z_2,\rho)$ denotes the cumulative distribution function of two jointly normally distributed random variables $(Z_1,Z_2)$ with zero means, unit variances, and the correlation $\rho$ .

Proof. See Appendix B.

We illustrate our results with the following numerical example.

Example: Assume a MFBM market with following parameters:

\begin{equation*}\mu=0.1,\:\:\sigma_1=0.15,\:\:\sigma_2=0.1,\:\:S_0=100,\:\: H=0.8,\:\: r=0.05\end{equation*}

An insured at age 40 buys a 10 years contract $X=\max{\{S_T, K\}}$ with the guarantee amount $K=110$ . By the United States 2015 Life Table, the survival probability of the insured is ${}_{10}p_{40}=0.97462,$ and hence according to equations (32) and (33), the premium that the insurance company can receive is $ {}_TU_x=0.97462*104.16=101.521$ . For a fixed $\alpha=0.95$ , Table 1 shows optimal hedged loss functions and corresponding minimal values of RVaR for different levels of $\beta$ .

Table 1.

Optimal hedged loss and minimal RVaR for different levels of $\beta$ .

Meanwhile, subject to the same initial capital constraint, we provide optimal hedged loss functions of VaR-based hedging and CVaR-based hedging correspondingly.

For VaR-based hedging at the level 95%, with the help of the method in Cong et al. (Reference Cong, Tan and Weng2013), we derive the optimal hedged loss function which is

(35) \begin{equation} f^*_{VaR}(X)=(X-4.3584)^+I_{\{X\leq 1641.8\}}\end{equation}

For CVaR-based hedging at the level 95%, according to the method in Cong et al. (Reference Cong, Tan and Weng2014), we arrive to the conclusion that the optimal hedged loss function is

(36) \begin{equation} f^*_{CVaR}(X)=(X-4.3588)^+\end{equation}

Comparing (35) and (36) with results in Table 1, it is not hard to find that, as $\beta\rightarrow{0}$ , the optimal hedged loss function of RVaR hedging coincides with $f^*_{VaR}(X)$ , while in another extreme case $\beta=1-\alpha$ , it coincides with $f^*_{CVaR}(X)$ . Such results are consistent with conclusions in section 3.2. Moreover, for a small value of $\beta$ , the optimal RVaR hedging is close to the optimal VaR-based hedging while for a big value of $\beta$ , it is closer to the optimal CVaR hedging, which indicates that the measure RVaR is a bridge between VaR and CVaR.