Option Pricing with Given Risk Constraints and Its Application to Life Insurance Contracts

Abstract

This paper presents a method for hedging in markets of two-factor diffusion and jump diffusion models under the restriction of a specified probability of success. In addition, a method for hedging with a given shortfall amount is developed. A maximal perfect hedging set is constructed for options involving the exchange of one asset for another. The developed method is applied to the pricing of equity-linked life insurance contracts, such as “pure endowments with a guarantee”. Traditional pricing approaches for hedging options often yield minimal returns for investors. By accepting a predefined level of risk, investors can achieve higher returns. In light of this, this paper proposes risk management strategies applicable to these hybrid financial and insurance products.

1. Introduction

The financial industry is a dynamic world filled with uncertainties. Risk management is the main concern for individuals, corporations, and institutions. Therefore, the use of hedging for pricing is prevalent in financial mathematics to protect against movements in future outcomes. The standard approach of hedging is to construct a portfolio of financial instruments that replicates the cash flow corresponding to the hedged item. The primary objective of hedging through the replication of the cash flow of a contingent claim is to protect against potential losses. Therefore, a contingent claim priced though this approach does not allow for profit.

The theory of quantile hedging, developed by Follmer and Leukert (1999) [1] (see also [2]), provides an approach that maximizes the probability of a successful hedge with reduced initial capital. Alternatively, one can define a bound $ϵ$ for the probability of unsuccessful hedging and minimize the initial capital required such that the probability of successful hedging is at least $1-ϵ$. The quantile hedging concept does not allow for the selection of the probability of shortfall. Novikov (1999) [3] proposed the approach of hedging with a given probability in a complete market with one risky and one risk-free asset. This paper will extend the approach of hedging with a given probability to the two-dimensional case and in terms of both diffusion and jump diffusion models. Furthermore, in recent years, the problem of partial hedging has been considered and studied in terms of different risk measures. For instance, the problem has been studied with the help of a class of two-parametric risk measures, the Range of VaR, which includes the risk measures of VaR and CVaR.

Equity-linked life insurance contracts combine life insurance protection with investment options tied to the performance of specific equity or investment funds. They pay stochastic benefits linked to the movement in a financial market while providing guaranteed benefits. The benefits are paid only if conditions are met, such as the death or survival of the policyholders. Therefore, equity-linked insurance contract pricing is much more complicated.

As mortality cannot be traded in any market, it is not possible to hedge mortality risk. Hence, the insurance market is incomplete and perfect hedging is impossible. Applying quantile hedging to equity-linked insurance contracts allows for the separation of financial risk and mortality risk. Brennan and Schwartz (1976) [4] and Boyle and Schwartz (1977) [5] were the first to discuss the hedging of equity-linked insurance contracts. The approach of reducing equity-linked insurance contracts to a call or applying perfect hedging is widely used today. This paper adopts the approach developed by Brennan and Schwartz to integrate hedging with a predefined probability to hedge and price an equity-linked insurance contract.

This paper focuses on an equity-linked pure endowment contract, where the benefit payout is linked to the equity performance and is received only if the policyholder survives to the contract’s maturity date. The analysis assumes a financial market comprising one risk-free asset and two risky assets. In this paper, two models are considered: a diffusion model that incorporates two Wiener processes and their correlation coefficient $\rho$, as in Margrabe (1978) [6] and Fischer (1978) [7], and a jump diffusion model with a single Wiener process generating both risky assets’ prices.

In summary, this paper introduces an approach to partial hedging with a predefined probability for both two-factor diffusion and jump diffusion models, as well as introducing an efficient hedging method. Furthermore, these developed hedging strategies are applied to price equity-linked insurance contracts. Thus, we provide a practical framework for the management of combined financial and actuarial risks in hybrid insurance products.

The layout of this paper is as follows. Section 2 introduces the financial setting. This is followed by Section 3, illustrating Novikov’s approach of hedging with a given probability in two dimensions for the diffusion model. In Section 4, the hedging technique is applied in the setting of the jump diffusion model. In Section 5, we extend the approach to efficient hedging with a given shortfall. Section 6 is devoted to the application of hedging with a given probability to equity-linked insurance contracts. Following Section 6, Section 7 illustrates our results with real-world data. The paper is concluded with Section 8, which discusses the practical application of the developed method, possible future research areas, and its limitations and shortcomings.

2. Financial Background Setting

Consider an arbitrage-free two-factor diffusion model within a complete market defined on the stochastic basis $(\Omega ,\mathcal{F},{\mathcal{F}}_t,P)$, for $t\ge 0$. Assume that $P\in \mathrm{𝒫}$, where $\mathrm{𝒫}$ is a family of probability distributions and the processes are adapted to the filtration ${\mathcal{F}}_t$, generated by the independent Wiener process W and Poisson process $\Pi$. The investor’s wealth at time t is represented by the following equation

$$X_t^{\pi }={\beta }_t{B}_t+{\gamma }_t^1{S}_t^1+{\gamma }_t^2{S}_t^2$$

(1)

where $B_t$ is the process of the risk-free asset’s price with $B_0=1$, and $S_t^i,i=1,2,$ are the processes of the risky assets. For simplicity, here, we assume $B_t$ = 1, implying a 0 interest rate ($r_t=0$). Additionally, let $\pi ={\left({\pi }_t\right)}_{t\ge 0}={({\beta }_t,{\gamma }_t^1,{\gamma }_t^2)}_{t\ge 0}$ represent a predictable trading strategy (hedge). By definition, in an arbitrage-free complete market, there exists a unique local martingale measure $P^{\ast }$ under which the processes ${\left(S_t^i\right)}_{t\ge 0}$ are local martingales with respect to measure $P^{\ast }$. Likewise, the process ${\left(X_t^{\pi }\right)}_{t\ge 0}$ is a local martingale with respect to measure $P^{\ast }$ for self-financing strategies. A trading strategy is considered self-financing $\pi \in SF$ (self-financing) if the investor’s capital or portfolio value $X_t^{\pi }$ is realized without any external inflow or outflow of cash or assets. This condition is expressed as follows:

$$X_t^{\pi }=X_0^{\pi }+{\int }_0^t{\gamma }_u^{1}d{S}_u^1+{\int }_0^t{\gamma }_u^{2}d{S}_u^2.$$

(2)

Let T represent the maturity date, and let H denote the pay-off of a contingent claim. The traditional method of pricing a European option involves identifying a hedge $\pi \in SF$ that satisfies

$$P(X_T^{\pi }\ge H)=1.$$

(3)

The calculated risk-neutral price of the contingent claim is defined as the minimal initial investor capital ($X_0^{\pi }$) that is sufficient for (3) to hold.

Definition 1.

Let $\Pi (x,H)$ represent a set of self-financing strategies where $X_0^{\pi }=x$ and Equation (3) holds. The risk-neutral price of the contingent claim H is defined as

$$\mathbb{C}\left(H\right)=inf\{x:\Pi (x,H)\ne \varnothing \}.$$

(4)

There exists a hedge or strategy ${\pi }^{\ast }\in SF$ such that $\mathbb{C}\left(H\right)=X_0^{{\pi }^{\ast }}=E^{\ast }\left[H\right]$. The hedge ${\pi }^{\ast }$ is called a perfect hedge.

The traditional method of option pricing often results in minimal returns for investors. Specifically, for European options, the gain is practically zero due to the construction of the perfect hedge. To address this limitation, we can relax condition (3), allowing investors to hedge the contingent claim while accepting a certain level of risk. In other words, instead of achieving a perfect hedge with a probability equal to one, as required by condition (3), investors can hedge the contingent claim with a probability of less than one. This adjustment can be expressed mathematically as follows:

$$P^{\ast }(X_T^{\pi }\ge H)=1-\alpha$$

(5)

where $\alpha$ is the given significance level (risk level). This method was previously studied in a market consisting of one risk-free and one risky asset by Novikov (1999) [3]. In this paper, we extend this approach and apply it to the context of life insurance contracts.

3. Methodologies of Partial Hedging with Shortfall Risk Constraints

The second fundamental theorem of asset pricing states that, in a complete and arbitrage-free market, there exists a unique martingale probability measure $P^{\ast }$. Building on this concept, we introduce and define a new class of hedges, denoted as $SF\left(A\right)$. This represents a set of self-financing hedging strategies subject to a shortfall constraint determined by a specified constant A.

$$\mathrm{SF}\left(A\right)=\left\{\pi \in SF:X_T^{\pi }\ge H-A\mathrm{a}.\mathrm{s}.\right\}$$

(6)

where A is a non-negative constant, and H is the pay-off of the contingent claim. As outlined in the previous section, achieving a perfectly hedged position requires the portfolio value at maturity, $X_T^{\pi }$, to be at least equal to the pay-off H. In the class $SF\left(A\right)$, this requirement is relaxed by adding a non-negative constant A. This expands the set of admissible self-financing hedges by reducing the pay-off at maturity from H to $H-A$. In this paper, we focus exclusively on hedging strategies within the $SF\left(A\right)$ class. Using this framework and class $SF\left(A\right)$, we derive a lower bound for the initial capital required by the investor. This lower bound is established in the following lemma.

Lemma 1.

If $\pi \in$ SF$\left(A\right)$, then

$$E^{\ast }\left[H-AI\{X_T^{\pi }<H\}\right]\le X_0^{\pi }$$

(7)

The lower bound for the initial investment is equal to the expected value of the pay-off minus A if the hedge is lower than the pay-off.

Proof.  

By assumption of the class $SF\left(A\right)$, we have $X_T^{\pi }-H+A\ge 0$, a.s. We can apply the Chebyshev inequality and find that

$$A{P}^{\ast }(X_T^{\pi }\ge H)=A{P}^{\ast }(X_T^{\pi }-H+A\ge A)\le E^{\ast }[X_T^{\pi }-H+A]\le X_0^{\pi }-E^{\ast }\left[H\right]+A$$

$$E^{\ast }\left[H\right]+A{P}^{\ast }(X_T^{\pi }\ge H)-A\le X_0^{\pi }$$

$$E^{\ast }\left[H\right]-A(1-P^{\ast }(X_T^{\pi }\ge H))\le X_0^{\pi }$$

$$E^{\ast }\left[H\right]-A{P}^{\ast }(X_T^{\pi }\le H)\le X_0^{\pi }$$

$$E^{\ast }[H-AI\{X_T^{\pi }\le H\}]\le X_0^{\pi }.$$

□

We further refine the set of hedges under consideration by introducing additional constraints. This subset is constructed from the class $SF\left(A\right)$ and is defined in relation to the initial capital x, the pay-off H, the significance level $\alpha$, and the non-negative constant A. These parameters collectively establish the set of admissible self-financing hedging strategies, as described below.

$$\Pi (x,H,\alpha ,A)=\{\pi \in \mathrm{SF}\left(A\right):P^{\ast }(X_T^{\pi }\ge H)=1-\alpha \}$$

(8)

The risk-neutral or fair price of a European option, based on the newly defined set of hedges, is defined as follows:

$$\mathbb{C}(H,\alpha ,A)=inf\{x:\Pi (x,H,\alpha ,A)\ne \varnothing \}$$

(9)

$C(H,\alpha ,A)$ is the fair price of a European option, which is the minimal initial investor capital considering the parameters introduced previously. Under general assumptions, it can be observed that a hedge ${\pi }^{\alpha }\in \Pi (x,H,\alpha ,A)$ exists, characterized by the initial capital and given as

$$x=X_0^{{\pi }^{\alpha }}=E^{\ast }[X_T^{\pi }-AI\{X_T^{\pi }<H\}]=\mathbb{C}\left(H\right)-\alpha A.$$

(10)

The second equality is derived from the definition of the class of hedges $SF\left(A\right)$. Furthermore, the third equality is obtained using general statistic properties. From the equality in Equation (10), we can derive the following observation. Lemma 1 establishes that $E^{\ast }[X_T^{\pi }-AI\{X_T^{\pi }<H\}]$ serves as the lower bound for the initial capital x. At the same time, the fair price is defined as the infimum of all possible initial capital values. Consequently, by Lemma 1, the fair price can be expressed as

$$\mathbb{C}(H,\alpha ,A)=\mathbb{C}\left(H\right)-\alpha A.$$

(11)

The hedge ${\pi }^{\alpha }$ defined above is derived using statistical techniques based on the Neyman–Pearson Lemma; see Lehmann (1986) [8]. This lemma provides a solution to the problem of constructing a test and a decision rule that maximizes the power of the test subject to a given level of significance. Statistical power is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true. The given significance level controls the probability of type I error, usually denoted as $\alpha$.

Assume that there exists an event $E\in \mathcal{F}$ such that

$$\underset{P\in \mathrm{𝒫}}{min}P\left(E\right)=P^{\ast }\left(E\right)=1-\alpha .$$

(12)

The worst-case measure is the measure $P\in \mathrm{𝒫}$ that minimizes the probability of successful hedging. In other words, it is the measure that makes it the most difficult to meet the constraint $1-\alpha$. The worst-case measure in this case is the risk-neutral measure $P^{\ast }$. In mathematical terms, the comparison of the likelihood ratios under the two hypotheses is expressed as follows. Define

$$L_t=L_t\left(Q\right)={\displaystyle \frac{dQ}{d{P}^{\ast }}}{|}_{{\mathcal{F}}_t},$$

(13)

which represents a Radon–Nikodym derivative. This derivative serves as the likelihood ratio associated with some measure $Q\in \mathrm{𝒫},Q\ne P^{\ast }$. The perfect hedging set then can be considered as the event.

$$E=\{L_T\ge \lambda \left(\alpha \right)\}$$

(14)

Consider a test with hypotheses $H_0$: $P^{\ast }$ and $H_1$: Q, and let the rejection region be $R=\{L_T<\lambda \left(\alpha \right)\}$. The Neyman–Pearson Lemma states that the most powerful test satisfies the following: for some $\lambda \left(\alpha \right)\ge 0$,

• the observation (sample) belongs to R;
• the observation (sample) belongs to $R^c$;
• $P^{\ast }$ (the observation (sample) belongs to R) = $\alpha$, where $\alpha$ is a pre-fixed significance level.

The event E ensures that the hedging strategy satisfies the condition that the probability of covering the pay-off is at least $1-\alpha$. It identifies the scenarios where the likelihood ratio equals a critical level $\lambda \left(\alpha \right)$. In this context, the null hypothesis implies that achieving a perfect hedge is not possible, while the alternative hypothesis suggests that it is feasible to hedge the pay-off H.

Let ${\pi }^{\ast }=({\gamma }_t^{1^{\ast }},{\gamma }_t^{2^{\ast }})$ be the perfect hedge or risk-neutral hedge in the classical pricing problem (when $\alpha =0$). Let $M_t=E^{\ast }\left[H\right|{\mathcal{F}}_t]$, which is a martingale under measure $P^{\ast }$, and

$$M_T=H\mathrm{a}.\mathrm{s}.$$

(15)

We express $M_t$ using the martingale representation theorem. Given the assumption of a complete market, there exists a predictable process ${\gamma }_t^{\ast }$ such that

$$\begin{array}{cc}\hfill M_t& =M_0+{\int }_0^t{\gamma }_t^{1^{\ast }}d{S}_t^1+{\int }_0^t{\gamma }_t^{1^{\ast }}d{S}_t^2\hfill \\ \hfill & =Y_0+{\int }_0^t{\gamma }_t^{1^{\ast }}d{S}_t^1+{\int }_0^t{\gamma }_t^{1^{\ast }}d{S}_t^2=C\left(H\right)+{\int }_0^t{\gamma }_t^{1^{\ast }}d{S}_t^1+{\int }_0^t{\gamma }_t^{1^{\ast }}d{S}_t^2\hfill \end{array}$$

Having established all the necessary constructions, we proceed to present the final result in the form of a theorem.

Theorem 1.

Let hedge ${\pi }^{\alpha }=({\gamma }_{\alpha ,t}^1,{\gamma }_{\alpha ,t}^2)$ be defined by the following equations:

$$X_t^{{\pi }^{\alpha }}=X_0^{{\pi }^{\alpha }}+{\int }_0^t{\gamma }_{\alpha ,t}^{1}d{S}_t^1+{\int }_0^t{\gamma }_{\alpha ,t}^{2}d{S}_t^2$$

$$X_0^{{\pi }^{\alpha }}=C\left(H\right)-\alpha A,{\gamma }_{\alpha ,t}^i={\gamma }_t^{i^{\ast }}-{\phi }_{t}A,i=1,2$$

$$m_t=P^{\ast }(L_T<\lambda \left(\alpha \right)|{\mathcal{F}}_t)=\alpha +{\int }_0^t{\phi }_t^{1}d{S}_t^1+{\int }_0^t{\phi }_t^{2}d{S}_t^2(\mathit{Martingale} \mathit{representation})$$

Then,

$$\alpha =P^{\ast }(X_T^{{\pi }^{\alpha }}<H),{\pi }^{\alpha }\in \Pi (X_0^{{\pi }^{\alpha }},H,\alpha ,A),$$

and

$$C(H,\alpha ,A)=X_0^{{\pi }^{\alpha }}=C\left(H\right)-\alpha A$$

Proof.  

According to the definition of the hedge ${\pi }^{\alpha }$, we have

$$\begin{array}{cc}\hfill X_t^{{\pi }^{\alpha }}& =X_0^{{\pi }^{\alpha }}+{\int }_0^t{\gamma }_t^{1^{\alpha }}d{S}_t^1+{\int }_0^t{\gamma }_t^{1^{\alpha }}d{S}_t^2=X_0^{{\pi }^{\alpha }}+{\int }_0^t({\gamma }_t^{1^{\ast }}-{\phi }_{t}A)d{S}_t^1+{\int }_0^t({\gamma }_t^{2^{\ast }}-{\phi }_{t}A)d{S}_t^2\hfill \\ \hfill & =X_0^{{\pi }^{\alpha }}+{\int }_0^t{\gamma }_t^{1^{\ast }}d{S}_t^1+{\int }_0^t{\gamma }_t^{1^{\ast }}d{S}_t^1-A\left({\int }_0^t{\phi }_t^{1}d{S}_t^1+{\int }_0^t{\phi }_t^{1}d{S}_t^1\right)\hfill \\ \hfill & =X_0^{{\pi }^{\alpha }}+M_t-C\left(H\right)-A(m_t-\alpha )=C\left(H\right)-\alpha A+M_t-C\left(H\right)-A{m}_t+A\alpha \hfill \\ \hfill & =M_t-A·m_t\hfill \\ \hfill X_T^{{\pi }^{\alpha }}& =M_T-A·m_T=H-A·P^{\ast }(L_T<\lambda \left(\alpha \right)|\mathcal{F})=H-A·I\{L_T<\lambda \left(\alpha \right)\}\hfill \end{array}$$

Therefore,

$$P^{\ast }(X_T^{{\pi }^{\alpha }}<H)=P^{\ast }(L_T<\lambda \left(\alpha \right))=\alpha$$

Since $M_T=H\ge 0$ and $m_t\le 1$,

$$X_T^{{\pi }^{\alpha }}=M_T-A{m}_T\ge H-A{m}_T\ge H-A\Rightarrow {\pi }^{\alpha }\in \mathrm{SF}\left(A\right)$$

Under the martingale measure $P^{\ast }$, the probability of under-hedging, which is failing to meet the pay-off H, is exactly $\alpha$. Thus,

$$P^{\ast }(L_T\left(P\right)\ge \lambda \left(\alpha \right))=P^{\ast }(X_{\alpha }^{\pi }\ge H)=1-\alpha ,$$

and ${\pi }^{\alpha }\in \Pi (x,H,\alpha ,A)$, defined by (8). By the results in Lemma 1 and the construction of ${\pi }^{\alpha }$, the initial capital $X_0^{{\pi }^{\alpha }}$ is minimized and satisfies (11). □

By construction, $m_t$ is equivalent to the changes in the significance level with respect to the movements in asset prices ($S_t^1$ and $S_t^2$) in addition to $\alpha$. The changes in the probability $m_t$, represented by ${\phi }_t^1$ and ${\phi }_t^2$, can be considered as the correction terms to the hedges ${\gamma }_{\alpha ,t}^i$ for $i=1,2$. In the theorem, ${\gamma }_t^{i^{\ast }}$ represents the risk-neutral hedges obtained in the classical approach of pricing. A is considered to be the shortfall amount that investors are willing to compromise. The partial hedges ${\gamma }_{\alpha ,t}^1$ and ${\gamma }_{\alpha ,t}^2$ are defined as the risk-neutral hedges adjusted for shortfall amount A with the correction terms.

4. Partial Hedging of Spread Options

4.1. Partial Hedging of Spread Options in Two-Factor Diffusion Model

As mentioned earlier, this approach will be applied to pricing and hedging exchange options. An exchange option is a type of financial instrument that allows the holder to forfeit ownership of the initial asset in return for acquiring the underlying asset, typically under pre-specified conditions or terms outlined in the contract. An exchange option can be used to price and hedge equity-linked insurance contracts. The pay-off function of such a contingent claim is given as follows:

$$H=max\{S_T^1,S_T^2\},\mathrm{which}\mathrm{can}\mathrm{be}\mathrm{reduced}\mathrm{to}H=max\{S_T^1-S_T^2,0\}.$$

where $S_T^1$ and $S_T^2$ are two correlated risky assets following the geometric Brownian motion.

$$d{S}_t^i=S_t^i({\mu }_{i}dt+{\sigma }_{i}d{W}_t^i)i=1,2,$$

(16)

$W_t^1$ and $W_t^2$ are correlated Wiener processes with correlation $cov(W_t^1,W_t^2)=\rho t$. The martingale probability density or Radon–Nikodym derivative of this model is given by

$$Z_T^{\ast }=exp\left({\phi }_1{W}_T^1+{\phi }_2{W}_T^2-{\displaystyle \frac{1}{2}}{\sigma }_{\phi }^{2}T\right)$$

(17)

where

$${\phi }_1={\displaystyle \frac{r({\sigma }_2-{\sigma }_1\rho )+\rho {\mu }_2{\sigma }_1-{\mu }_1{\sigma }_2}{{\sigma }_1{\sigma }_2(1-{\rho }^2)}};{\phi }_2={\displaystyle \frac{r({\sigma }_1-{\sigma }_2\rho )+\rho {\mu }_1{\sigma }_2-{\mu }_2{\sigma }_1}{{\sigma }_1{\sigma }_2(1-{\rho }^2)}}$$

(18)

$$\mathrm{and}{\sigma }_{\phi }^2={\phi }_1^2+{\phi }_2^2+2\rho {\phi }_1{\phi }_2$$

Additional assumptions for this particular diffusion model include $B_t=e^{rt}$; $S_t^i=S_0^{i}exp\left({\sigma }_i{W}_t^i+{\mu }_{i}t-{\displaystyle \frac{1}{2}}{\sigma }_i^{2}t\right)\forall i=1,2$. $B_t$ represents the bank account B at time t (as mentioned in Section 1, $B_0$ = 1). Following the framework established in Novikov’s paper, the Neyman–Pearson Lemma is utilized to construct the maximal hedging set. In this context, $\lambda \left(\alpha \right)$ represents a random variable used in the construction process. It is formulated as an expression analogous to the likelihood ratio, modified to include a parameter ${\theta }_{\alpha }$, which is a function of the significance level $\alpha$,

$$\lambda \left(\alpha \right)=exp\left({\phi }_1{\theta }_{\alpha }+{\phi }_2{\theta }_{\alpha }-{\displaystyle \frac{1}{2}}{\sigma }_{\phi }^{2}T\right)$$

(19)

where T is the maturity date. We then impose the condition that the likelihood ratio is greater than or equal to $\lambda \left(\alpha \right)$.

$$\begin{array}{cc}\hfill \{L_T\left(P\right)\ge \lambda \left(\alpha \right)\}& =\left\{exp\left({\phi }_1{W}_T^1+{\phi }_2{W}_T^2-{\displaystyle \frac{1}{2}}{\sigma }_{\phi }^{2}T\right)\ge exp\left({\phi }_1{\theta }_{\alpha }+{\phi }_2{\theta }_{\alpha }-{\displaystyle \frac{1}{2}}{\sigma }_{\phi }^{2}T\right)\right\}\hfill \\ \hfill & =\{{\phi }_1{W}_T^1+{\phi }_2{W}_T^2-{\displaystyle \frac{1}{2}}{\sigma }_{\phi }^{2}T\ge {\phi }_1{\theta }_{\alpha }+{\phi }_2{\theta }_{\alpha }-{\displaystyle \frac{1}{2}}{\sigma }_{\phi }^{2}T\}\hfill \\ \hfill & =\{{\phi }_1{W}_T^1+{\phi }_2{W}_T^2\ge ({\phi }_1+{\phi }_2){\theta }_{\alpha }\}\hfill \end{array}$$

The two Wiener processes are assumed to follow a normal distribution, with $W_T^i\sim N(0,T)$ for $i=1,2$, and their covariance is given by $cov(W_T^1,W_T^2)=\rho T$. Consider a stochastic differential equation of the form $d{X}_t=\beta dt+{\sigma }_{1}d{W}_t^1+{\sigma }_{2}d{W}_t^2$. By Levy’s characterization of Brownian motion, it is possible to represent this using $\sigma W_t$, where $\sigma$ is a fixed constant and $W_t$ is a different Wiener process. This is illustrated by the stochastic differential equation $d{X}_t=\beta dt+\sqrt{{\sigma }_1^2+2{\sigma }_1{\sigma }_2\rho +{\sigma }_2^2}d{W}_t$. In particular, we have

$${\phi }_1{W}_T^1+{\phi }_2{W}_T^2\sim N(0,{\sigma }^2)\mathrm{where}{\sigma }^2=T({\phi }_1^2+{\phi }_2^2+2\rho {\phi }_1{\phi }_2)$$

Let $W_T^{\ast }={\phi }_1{W}_T^1+{\phi }_2{W}_T^2$. We can determine ${\theta }_{\alpha }$ with the following condition:

$$P(W_T^{\ast }\ge ({\phi }_1+{\phi }_2){\theta }_{\alpha })=1-\alpha$$

$$P(L_T\ge \lambda \left(\alpha \right))=P\left({\displaystyle \frac{W_T^{\ast }}{\sigma }}\ge {\displaystyle \frac{({\phi }_1+{\phi }_2){\theta }_{\alpha }}{\sigma }}\right)=1-\alpha$$

(20)

Therefore,

$${\displaystyle \frac{({\phi }_1+{\phi }_2){\theta }_{\alpha }}{\sigma }}=Z_{\alpha }\Rightarrow {\theta }_{\alpha }={\displaystyle \frac{\sigma Z_{\alpha }}{{\phi }_1+{\phi }_2}}\mathrm{where}{Z}_{1-\alpha }={\Phi }^{-1}(1-\alpha )$$

As established in the previously mentioned theorem, $\mathbb{C}(H,\alpha ,A)=\mathbb{C}\left(H\right)-\alpha A$, where $\mathbb{C}\left(H\right)$ is determined using Margrabe’s formula, $\mathbb{C}\left(H\right)=S_0^{1}N\left(d_1\right)-S_0^{2}N\left(d_2\right)$.

$$\mathrm{where}{d}_1=ln\left({\displaystyle \frac{S_0^1}{S_0^2}}\right)+{\displaystyle \frac{{\sigma }^2}{2}}T;d_2=d_1-\sigma \sqrt{T};\mathrm{and}\sigma =\sqrt{{\sigma }_1^2+{\sigma }_2^2-2\rho {\sigma }_1{\sigma }_2}$$

We have

$$\begin{array}{cc}\hfill P(L_T\left(P\right)<\lambda \left(\alpha \right)|{\mathcal{F}}_t)& =P(W_T^{\ast }-W_t^{\ast }<({\phi }_1+{\phi }_2){\theta }_{\alpha }-W_t^{\ast })\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & =\Phi \left({\displaystyle \frac{({\phi }_1+{\phi }_2){\theta }_{\alpha }-W_t^{\ast }}{{\sigma }_{\psi }}}\right)\hfill \end{array}$$

(22)

Applying the two-dimensional Ito formula, we calculate the correction term (${\phi }_t^i$) for the hedge ${\gamma }_t^{i^{\ast }}$ as follows:

$${\phi }_t^i=-{\displaystyle \frac{B_t}{S_t^i{\sigma }_i}}\times {\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{t}exp\left(-{\displaystyle \frac{{[({\phi }_1^2+{\phi }_2^2){\theta }_{\alpha }-W_s^{\ast }]}^2}{{\sigma }_{\psi }^2}}\right)ds$$

(23)

4.2. Partial Hedging on Spread Options Using Jump Diffusion Model

We can extend this approach in terms of the jump diffusion model by adding a jump component. In this section, the approach of the partial hedging of exchange options is developed based on a two-factor jump diffusion model. Consider a financial market with two risky assets $S^1$ and $S^2$, described by a two-factor jump diffusion model

$$d{S}_t^i=S_t^i({\mu }_{i}dt+{\sigma }_{i}d{W}_t-{\nu }_{i}d{\Pi }_t),i=1,2$$

(24)

where ${\nu }_i$ is the size of the jump, and ${\Pi }_t$ is the number of jumps. ${\Pi }_t$ follows a Poisson process with intensity (arrival rate of jumps) equal to $\lambda$. In this particular jump diffusion model, it is assumed that the jump size of every jump is constant at ${\nu }_i$, for $i=1,2$. Following the approach for the two-factor diffusion model, we utilize the Radon–Nikodym derivative of such a jump diffusion model, which is given by (26); see [9,10,11,12,13].

$$Z_t^{\ast }={\displaystyle \frac{d{P}^{\ast }}{dP}}{|}_{{\mathcal{F}}_t}=exp\left[{\alpha }^{\ast }{W}_t-{\displaystyle \frac{{\alpha }^{{\ast }^2}}{2}}t+(\lambda -{\lambda }^{\ast })t+(ln{\lambda }^{\ast }-ln\lambda ){\Pi }_t\right]$$

(25)

where

$${\alpha }^{\ast }={\displaystyle \frac{{\mu }_2{v}_1-{\mu }_1{v}_2}{{\sigma }_2{v}_1-{\sigma }_1{v}_2}}\mathrm{and}{\lambda }^{\ast }={\displaystyle \frac{{\mu }_1{\sigma }_2-{\mu }_2{\sigma }_1}{{\sigma }_2{v}_1-{\sigma }_1{v}_2}}$$

The processes $W_t^{\ast }=W_t-{\alpha }^{\ast }t$ and ${\Pi }_t$ are independent Wiener and Poisson processes (with intensity ${\lambda }^{\ast }>0$). Next, we construct an event or a set such that the probability of such an event is equivalent to the predetermined significance level $1-\alpha$. Following the approach of Novikov, this set is constructed based on the Radon–Nikodym derivative.

$$P(L_T\left(P\right)\ge \lambda \left(\alpha \right))$$

where $L_T\left(P\right)$ is equal to $Z_T^{\ast }$ and $\lambda \left(\alpha \right)$ is of similar form to $Z_T^{\ast }$, while $\lambda \left(\alpha \right)$ is defined with respect to the significance level $\alpha$.

$$\lambda \left(\alpha \right)=exp\left({\alpha }^{\ast }{\theta }_{\alpha }-{\displaystyle \frac{{\alpha }^{{\ast }^2}}{2}}T+(\lambda -{\lambda }^{\ast })T+(ln{\lambda }^{\ast }-ln\lambda ){\Pi }_{\alpha }\right)$$

(26)

The difference between $\lambda \left(\alpha \right)$ and $Z_T^{\ast }$ is that the Wiener process $W_T$ in $Z_T^{\ast }$ is replaced with variable ${\theta }_{\alpha }$, and the Poison process ${\Pi }_T$ is replaced with variable ${\Pi }_{\alpha }$. The variables are determined based on the significance level $\alpha$. With some calculations, the set can be reduced to the following set given below.

$$\begin{array}{cc}\hfill \{L_T\left(P\right)\ge \lambda \left(\alpha \right)\}& =\left\{exp\left({\alpha }^{\ast }{W}_T-{\displaystyle \frac{{\alpha }^{{\ast }^2}}{2}}T+(\lambda -{\lambda }^{\ast })T+ln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}{\Pi }_T\right)\ge exp\left({\alpha }^{\ast }{\theta }_{\alpha }-{\displaystyle \frac{{\alpha }^{{\ast }^2}}{2}}T+(\lambda -{\lambda }^{\ast })T+ln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}{\Pi }_{\alpha }\right)\right\}\hfill \\ \hfill & =\left\{{\alpha }^{\ast }{W}_T-{\displaystyle \frac{{\alpha }^{{\ast }^2}}{2}}T+(\lambda -{\lambda }^{\ast })T+ln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}{\Pi }_T\ge {\alpha }^{\ast }{\theta }_{\alpha }-{\displaystyle \frac{{\alpha }^{{\ast }^2}}{2}}T+(\lambda -{\lambda }^{\ast })T+ln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}{\Pi }_{\alpha }\right\}\hfill \\ \hfill & =\left\{{\alpha }^{\ast }{W}_T+ln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}{\Pi }_T\ge {\alpha }^{\ast }{\theta }_{\alpha }+ln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}{\Pi }_{\alpha }\right\}\hfill \end{array}$$

Notice that the Poisson process ${\Pi }_t$ in the above set cannot be cancelled. To determine the probability of the set $\{L_T\left(P\right)\ge \lambda \left(\alpha \right)\}$, we will apply a conditional probability on the Poisson process. We define a new variable ${\theta }_{\alpha }^{\prime }$, and let it represent the right side of the above inequality. The equations below illustrate this concept.

$$\begin{array}{cc}\hfill P(L_T\left(P\right)\ge \lambda \left(\alpha \right))& =P\left({\alpha }^{\ast }{W}_T+ln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}{\Pi }_T\ge {\alpha }^{\ast }{\theta }_{\alpha }+ln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}{\Pi }_{\alpha }\right)=P\left({\alpha }^{\ast }{W}_T+ln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}{\Pi }_T\ge {\theta }_{\alpha }^{\prime }\right)\hfill \\ \hfill & =P\left(P\left({\alpha }^{\ast }{W}_T+ln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}{\Pi }_T\ge {\theta }_{\alpha }^{\prime }\right)|{\Pi }_T\right)\hfill \\ \hfill & =\sum _{n=0}^{\infty }P\left({\alpha }^{\ast }{W}_T+ln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}{\Pi }_T\ge {\theta }_{\alpha }^{\prime }|{\Pi }_T=n\right)p_{n,T}^{\ast }\hfill \end{array}$$

We have broken down the set into smaller subsets such that the probability of the set is the sum of the probabilities conditioned on the number of jumps, where $p_{n,T}^{\ast }$ is the Poisson probability with intensity ${\lambda }^{\ast }T$.

$$p_{n,T}^{\ast }={\displaystyle \frac{e^{-{\lambda }^{\ast }T}{\left({\lambda }^{\ast }T\right)}^n}{n!}}$$

(27)

Note that, with $W_T$ being a Wiener process and normally distributed, ${\alpha }^{\ast }{W}_T$ is normally distributed with the parameters $N(0,{\alpha }^{{\ast }^2}T)$. Rearranging the above probability, we find that $P(L_T\left(P\right)\ge \lambda \left(\alpha \right))$ is equivalent to the sum of normally distributed variables multiplied by Poisson probabilities.

$$P(L_T\left(P\right)\ge \lambda \left(\alpha \right))=\sum _{n=0}^{\infty }P\left({\alpha }^{\ast }{W}_T\ge {\theta }_{\alpha ,n}^{\prime }-nln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}\right)p_{n,T}^{\ast }=\sum _{n=0}^{\infty }\Phi \left({\displaystyle \frac{{\theta }_{\alpha ,n}^{\prime }-nln\frac{{\lambda }^{\ast }}{\lambda }}{{\alpha }^{\ast }}}\right)p_{n,T}^{\ast }$$

(28)

The probability $P(L_T\left(P\right)\ge \lambda \left(\alpha \right))$ is now a linear equation. As in the Novikov approach, we set this probability equal to significance level $1-\alpha$ and solve for ${\theta }_{\alpha ,n}^{\prime }$ for each n. Hence, ${\theta }_{\alpha ,n}^{\prime }$ can be considered as the $1-\alpha$th quantile for the case with n jumps. Let $\Phi \left({\displaystyle \frac{{\theta }_{\alpha ,n}^{\prime }-nln\frac{{\lambda }^{\ast }}{\lambda }}{{\alpha }^{\ast }}}\right)={\beta }_n$ for $n=0,1,2,\dots$, which can be considered as the coefficients of the linear equation. To solve for ${\theta }_{\alpha ,n}^{\prime }$, we need to approximate the coefficients using numerical methods. The process is formulated below.

$$\begin{array}{cc}\hfill 1-\alpha & =\sum _{n=0}^{\infty }\Phi \left({\displaystyle \frac{{\theta }_{\alpha ,n}^{\prime }-nln\frac{{\lambda }^{\ast }}{\lambda }}{{\alpha }^{\ast }}}\right)p_{n,T}^{\ast }=\sum _{n=0}^{\infty }{\beta }_n{p}_{n,T}^{\ast }={\beta }_0{p}_{0,T}^{\ast }+{\beta }_1{p}_{1,T}^{\ast }+{\beta }_2{p}_{2,T}^{\ast }+\cdots \hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & ={\beta }_0{e}^{-{\lambda }^{\ast }T}+{\beta }_1{e}^{-{\lambda }^{\ast }T}\left({\lambda }^{\ast }T\right)+{\beta }_2{\displaystyle \frac{e^{-{\lambda }^{\ast }T}{\left({\lambda }^{\ast }T\right)}^2}{2!}}+\cdots \hfill \end{array}$$

(30)

Note that, as n approaches infinity, the Poisson probability tends to 0, and the coefficients become insignificant after a certain number of terms. Thus, we can consider the above equation as a finite sum series and only solve for ${\theta }_{\alpha ,n}^{\prime }$ for significant coefficients. Given the construction of the hedging set, we can determine the “correction term” of the hedge in a similar way. Suppose that the probability $P(L_T\left(P\right)<\lambda \left(\alpha \right))$ is conditioned on the filtration ${\mathcal{F}}_t$ up to time t.

$$\begin{array}{cc}\hfill P(L_T\left(P\right)<\lambda \left(\alpha \right)|{\mathcal{F}}_t)& =P\left({\alpha }^{\ast }{W}_T-{\alpha }^{\ast }{W}_t+ln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}{\Pi }_T<{\theta }_{\alpha }^{\prime }-{\alpha }^{\ast }{W}_t|{\mathcal{F}}_T\right)\hfill \\ \hfill & =P\left(P\left({\alpha }^{\ast }{W}_T-{\alpha }^{\ast }{W}_t+ln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}{\Pi }_T<{\theta }_{\alpha }^{\prime }-{\alpha }^{\ast }{W}_t|{\mathcal{F}}_T\right)|{\Pi }_T\right)\hfill \\ \hfill & =\sum _{n=0}^{\infty }P\left({\alpha }^{\ast }{W}_T-{\alpha }^{\ast }{W}_t<{\theta }_{\alpha ,n}^{\prime }-nln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}-{\alpha }^{\ast }{W}_t|{\mathcal{F}}_T\right)p_{n,T-t}^{\ast }\hfill \end{array}$$

By subtracting the process $w_t$ from both sides of the inequality, we create an increment in the Poisson process. Since the probability is conditional on filtration ${\mathcal{F}}_t$, the increments in the Wiener process are independent, stationary, and normally distributed. Therefore, we can apply the same technique as before to arrive at the following equation.

$$P(L_T\left(P\right)<\lambda \left(\alpha \right)|{\mathcal{F}}_t)=\Phi \left({\displaystyle \frac{{\theta }_{\alpha ,n}^{\prime }-nln\frac{{\lambda }^{\ast }}{\lambda }-{\alpha }^{\ast }{W}_t}{{\alpha }^{\ast }\sqrt{T-t}}}\right)p_{n,T-t}^{\ast }$$

(31)

By construction, taking the derivative with respect to $S_t^1$ and $S_t^2$ yields the correction terms ${\phi }_t^1$ and ${\phi }_t^2$ for the hedges ${\gamma }_t^1$ and ${\gamma }_t^2$ at time t. Applying Ito’s lemma, the correction term is calculated as

$${\phi }_t^i=\sum _{n=0}^{\infty }{\displaystyle \frac{1}{S_t^i{\alpha }^{\ast }\sqrt{2\pi (T-t)}}}exp\left({\displaystyle \frac{1}{2(T-t)}}({\theta }_{\alpha ,n}^{\prime }-nln{\displaystyle \frac{{\lambda }^{\ast }}{\lambda }}-{\alpha }^{\ast }{W}_t)\right)p_{n,T-t}^{\ast },i=1,2.$$

(32)

Therefore, the hedges are represented as follows:

$${\gamma }_{\alpha ,t}^i={\gamma }_t^{i^{\ast }}-{\phi }_t^{i}A,i=1,2,$$

(33)

where ${\gamma }_t^{i^{\ast }}$ is the hedge determined using the classical approach of pricing and hedging. According to Theorem 1, the probability of the constructed set $\{L_T\left(P\right)<\lambda \left(\alpha \right)\}$ coincides with the probability that the investor capital at maturity is smaller than the pay-off, which is the probability of an unsuccessful hedge. Furthermore, by Theorem 1, $\mathbb{C}(H,\alpha ,A)=\mathbb{C}\left(H\right)-\alpha A$, where $\mathbb{C}\left(H\right)$ is calculated by the following generalization of Margrabe’s formula to the jump diffusion model (see Wang (2016) [14]):

$$\begin{array}{cc}\hfill E^{\ast }\left[{(S_t^1-S_t^2)}^{+}\right]& =\sum _{n=0}^{\infty }{\mathbb{C}}^{Mar}(S_0^1{v}_{n,T}^1,S_0^2{v}_{n,T}^2,T)p_{n,T}^{\ast }\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\mathbb{C}}^{Mar}({\tilde{S}}_{0,n}^1,{\tilde{S}}_{0,n}^2,T)& ={\tilde{S}}_0^1\Phi \left(b_{+}({\tilde{S}}_{0,n}^1,{\tilde{S}}_{0,n}^2,T)\right)-{\tilde{S}}_0^2\Phi \left(b_{-}({\tilde{S}}_{0,n}^1,{\tilde{S}}_{0,n}^2,T)\right)\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\tilde{S}}_{0,n}^1& =S_0^i{v}_{n,T}^i=S_0^i{(1-v_i)}^n{e}^{v_i{\lambda }^{\ast }T}\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill b_{\pm }({\tilde{S}}_{0,n}^1,{\tilde{S}}_{0,n}^2,T)& ={\displaystyle \frac{ln\frac{{\tilde{S}}_{0,n}^1}{{\tilde{S}}_{0,n}^2}\pm {({\sigma }_1-{\sigma }_2)}^2\frac{T}{2}}{({\sigma }_1-{\sigma }_2)\sqrt{T}}}\hfill \end{array}$$

(37)

where $p_{n,T}^{\ast }=e^{-{\lambda }^{\ast }T}{\displaystyle \frac{{\left({\lambda }^{\ast }T\right)}^n}{n!}}$ are components of the Poisson distribution with intensity ${\lambda }^{\ast }T$.

5. Efficient Hedging with a Given Expected Loss

Partial hedging with a given probability provides the investor with a higher profit by allowing some probability of potential loss. However, the amount of loss cannot be known beforehand or controlled. Considering this problem, we can develop an approach that allows for hedging with a given expected loss based on Novikov’s approach. The goal of this problem is to find an optimal strategy in which the shortfall amount is given and equal to c, illustrated by the equation below.

$$E\left[l{(H-X_T^{{\pi }^{\ast }})}^{+}\right]=c$$

(38)

Assume that $l\left(x\right)$ is the loss function and equal to $x^p$, where $p=1$ (linear loss function), and the optimal strategy is represented by ${\pi }^{\ast }$. In this particular setup of the loss function, the investor is risk-indifferent, and the risk of losing capital is neither magnified nor unconcerned.

Follmer and Leukert [15] (see also [16]) developed an approach that minimizes the expected shortfall amount under the constraint of smaller initial capital. The minimization problem is equivalent to the optimization problem of a modified contingent claim. In other words, the efficient hedge ${\pi }^{\ast }$ coincides with the perfect hedge of a modified contingent claim $H_p$.

$$H_p=H·{\mathbb{I}}_{\{Z_t^{-1}>c_p\}}$$

(39)

where $Z_t$ is the likelihood ratio or the Radon–Nikodym derivative and $c_p$ is a variable that can be solved based on the expected shortfall amount. By combining the approaches of Foellmer and Leukert and Novikov, we can construct an efficient hedging price and an efficient hedging strategy. From the Novikov approach, we introduce a non-negative constant c, which would be the expected capital loss. Similarly, we restrict the possible efficient hedging strategies to the following self-financing hedges set with respect to c.

$$SF\left(c\right)=\{\pi \in SF:X_T^{{\pi }^{\ast }}\ge H_p-c\}$$

(40)

In the above set of hedging strategies, we utilize Foellmer and Leukert’s approach, replacing the pay-off H with the modified contingent claim pay-off $H_p$.

$$E\left[l{(H-X_T^{{\pi }^{\ast }})}^{+}\right]=E\left[{(H-X_T^{{\pi }^{\ast }})}^{+}\right]=E[H-H_p]=c$$

(41)

By Novikov’s approach, the efficient hedging price is equivalent to the sum of the expected shortfall amount and the expected value of the modified contingent claim.

$$E\left[H\right]-E\left[H_p\right]=c\Rightarrow \mathbb{C}\left(f\right)=c+E\left[H_p\right]$$

(42)

where

$$E\left[H_p\right]=E\left[S_T^1{\mathbb{I}}_{\{Z_t^{-1}>a_p\}}{\mathbb{I}}_{\{S_T^1\ge S_T^2\}}\right]+E\left[S_T^1{\mathbb{I}}_{\{Z_t^{-1}>a_p\}}{\mathbb{I}}_{\{S_T^2<S_T^2\}}\right]$$

(43)

and

$$\{Z_t^{-1}>c_p\}=\left\{exp\left({\phi }_1{W}_T^1+{\phi }_2{W}_T^2-{\displaystyle \frac{1}{2}}{\sigma }_{\phi }^{2}T\right)<{\displaystyle \frac{1}{c_p}}\right\}=\left\{{\phi }_1{W}_T^1+{\phi }_2{W}_T^2<{\displaystyle \frac{1}{2}}{\sigma }_{\phi }^{2}T-ln\left(c_p\right)\right\}$$

(44)

6. Partial Hedging with a Given Probability and Application to Life Insurance

We can extend partial hedging with a specified probability to risk management for insurance products, specifically in the case of equity-linked insurance contracts. Compared to traditional life or property insurance products, equity-linked insurance contracts are fairly new products. The premiums from equity-linked insurance contracts are invested partially or entirely in ordinary financial shares. The pay-off at maturity depends on the performance of the financial market, thus transferring the investment risk to the policyholder. We will use an equity-linked pure endowment insurance contract to construct the risk-neutral premium. In this type of insurance contract, the policyholder receives a payment if they survive to maturity; otherwise, the policyholder would not receive any benefit payment.

Let $T\left(x\right)$ represent the remaining lifetime of an individual whose current age is X at the beginning of the contract. $T\left(x\right)$ is defined on the stochastic basis $(\tilde{\Omega },\tilde{\mathcal{F}},\widetilde{\mathrm{𝒫}})$, where the two probability measures P and $\tilde{P}$ are independent. By the standard assumption, the investment risks and mortality risks are independent of each other.

By using the previous work on exchange options, we can determine the risk-neutral premium. At maturity, the pay-off for an equity-linked pure endowment insurance contract is expressed by the following equation.

$$max\{S_T^1-S_T^2,0\}{\mathbb{I}}_{\left\{T\right(x)>T\}}$$

(45)

where T is the maturity time of the contract. The chance of survival is illustrated by the addition of the indicator function. The indicator function is equal to 1 if the individual survives to or past the end of the contract and equal to 0 if the individual passes away before the contract ends. The risk-neutral premium of this type of insurance contract is given as

$$\begin{array}{cc}\hfill \mathbb{P}& =E^{\ast }\times \tilde{E}[max\{S_T^1-S_T^2,0\}{\mathbb{I}}_{\left\{T\right(x)>T\}}]\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill & =E^{\ast }\left[H\right]\times \tilde{P}(T\left(x\right)>T)\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill & =E^{\ast }{\left[H\right]}_T{p}_x.\hfill \end{array}$$

(48)

_T$p_x$ is the standard actuarial notation of the survival probability of T years for an individual currently at age x. Note that the insurance market is incomplete, as mortality is not traded in a market, where a portfolio can be constructed to replicate the underlying asset’s cash flows. Brennand and Schwartz (see also [17,18]) determined an equilibrium price for an equity-linked insurance contract. In the equilibrium value, the investor will neither lose nor gain money. Following the approaches of Brennan and Schwartz (1976) [4] and Melnikov and Nosrati (2017) [19], we have the following equality:

$${}_{T}p_x^{ }={\displaystyle \frac{E^{\ast }[H-A{\mathbb{I}}_{\{X_T^{{\pi }^{\alpha }}<H\}}]}{E^{\ast }\left[H\right]}}={\displaystyle \frac{\mathbb{C}\left(H\right)-\alpha A}{\mathbb{C}\left(H\right)}}.$$

(49)

The equality above provides a connection between the mortality risks from insurance and the investment risks from finance. The ratio of the partial hedging price and the risk-neutral price is the survival rate, as the partial hedging price is lower. From the survival rate with life tables, we can determine the age needed to attain the survival rate. This allows the insurance to tailor the contract to individuals at a specific age to reach the equilibrium between mortality risks and investment risks. In other words, the insurance company can balance out the investment risk and the mortality risk. The following sections include numerical examples to illustrate this concept.

7. Numerical Examples

7.1. Numerical Examples for Two-Factor Diffusion Model

The data used in this section, consisting of daily prices of stocks from the Waters Corporation and Berkshire Hathaway, Inc. between 2017 and 2019, were extracted from Bloomberg (see [20]). These data provide the basis for the estimation of the parameters required for the simulations and calculations presented below. Let ${\mu }_1$ and ${\mu }_2$ represent the average daily returns of the Waters Corporation and Berkshire Hathaway, Inc., respectively. The parameters are estimated to be the following:

$${\mu }_1=0.00082106,{\mu }_2=0.00048693,{\sigma }_1=0.0145,{\sigma }_2=0.0106,\rho =0.7572$$

The stock price is simulated following a geometric Brownian motion with the above parameters. Based on a Monte Carlo simulation and Margrabe’s formula, the price of a 3-year exchange option is calculated to be 77.7241.

Table 1. Exchange option prices for different combinations of significance levels and expected shortfall amounts.

$\mathit{\alpha }$	A
	5	10	20	30	40	50
0.01	77.6741	77.6241	77.5241	77.4241	77.3241	77.2241
0.02	77.6241	77.5241	77.3241	77.1241	76.9241	76.7241
0.05	77.4741	77.2241	76.7241	76.2241	75.7241	75.2241
0.10	77.2241	76.7241	75.7241	74.7241	73.7241	72.7241
0.20	76.7241	75.7241	73.7241	71.7241	69.7241	67.7241

below illustrates the corresponding prices for various significance levels $\alpha$ and non-negative constants A.

The price of an equity-linked pure endowment insurance contract with the above underlying assets is calculated as follows:

$$\mathbb{P}=E^{\ast }{\left[H\right]}_T{P}_x$$

(50)

Applying the Brennan and Schwartz equilibrium, the required survival probability for each combination of significance level $\alpha$ and non-negative A is given below in

Table 2. Survival probabilities for different combinations of significance levels and expected shortfall amounts.

$\mathit{\alpha }$	A
	5	10	20	30	40	50
0.01	0.9994	0.9987	0.9974	0.9961	0.9949	0.9936
0.02	0.9987	0.9974	0.9949	0.9923	0.9897	0.9871
0.05	0.9968	0.9936	0.9871	0.9807	0.9743	0.9678
0.10	0.9936	0.9871	0.9743	0.9614	0.9485	0.9357
0.20	0.9871	0.9743	0.9485	0.9228	0.8971	0.8713

.

As anticipated, the ratio between the partial hedging price and the risk-neutral price is low for the high-risk combination of a large shortfall probability and expected loss. With higher risk, the partial hedging price is reduced substantially. Therefore, we require a low survival rate to reach equilibrium. From here, we then determine the ages needed to attain the survival probabilities for T years. Based on the Statistic Canada 3-year life table for 2018 to 2020 (see [21]), for a 3-year contract,

Table 3. Minimum age of insured for different combinations of significance levels and expected shortfall amounts.

$\mathit{\alpha }$	A
	5	10	20	30	40	50
0.01	22	41	50	55	58	61
0.02	41	50	58	63	66	69
0.05	53	61	69	73	76	78
0.10	61	69	76	80	82	84
0.20	69	76	82	86	89	90

provide minimum ages that are calculated to match the survival rate.Furthermore, in addition to using life tables, stochastic mortality models can also be used to calculate the required minimum ages (see [22]).

Table 3 translates the survival probabilities into the required ages, guiding insurers in structuring policies that are appropriately priced for different age groups. For a fixed contract length of 3 years, to maintain a hedged position, the current age of potential policyholders can be selected according to the required survival probability. As mentioned before, the high-risk combination requires a low survival rate. In the table above, it is reflected that, with higher investment risks, the insurance company needs to select older-aged individuals to balance out the investment risks. This aligns with the notion that older individuals face higher mortality risks, thus balancing the insurer’s risk exposure with the client age. By specifying minimum ages based on the desired survival rate, insurers can design tiered insurance products that cater to different age demographics. For example, policies with lower $\alpha$ and A could target younger clients, while higher-risk products might be more suitable for older policyholders, helping insurers to manage their exposure and maintain profitability.

7.2. Numerical Examples for Two-Factor Jump Diffusion Model

In this section, we use the same set of data as in Section 7 to illustrate partial hedging with a given probability using a two-factor jump diffusion model. Besides the parameters ${\mu }_1,{\mu }_2,{\sigma }_1,{\sigma }_2,$ and $\rho$ estimated in Section 7, we estimate the following additional parameters for the jump component:

$${\nu }_1=0.0010,{\nu }_2=0.00049838,\lambda =0.5014$$

where ${\nu }_1$ and ${\nu }_2$ are the jump sizes and lambda is the intensity or arrival rate of jumps. Similarly, the stock prices are simulated following a geometric Brownian motion with the addition of a jump component. Based on the Monte Carlo simulation and the generalized version of Margrabe’s formula, the price of a 3-year exchange option is calculated to be 61.9041. As mentioned in Section 4, the partial hedge is determined by solving a linear equation of the following form:

$$1-\alpha =\sum _{n=0}^{\infty }{\beta }_n{p}_{n,T}^{\ast }={\beta }_0{p}_{0,T}^{\ast }+{\beta }_1{p}_{1,T}^{\ast }+{\beta }_2{p}_{2,T}^{\ast }+\cdots$$

(51)

We observe that, as the number of jumps (n) increases, the Poisson probability ($p_{i,T}^{\ast }$) decreases to 0. Hence, to determine the coefficients ${\beta }_i$s, we can use a finite number of terms and set the rest of the terms as error terms. The following

Table 4. Probability of successful hedge at different numbers of terms.

	Number of Terms Used
	25	50	75	100
Probability of Successful Hedge	0.0079	0.9488	0.9500	0.9500

demonstrates the number of terms necessary to ensure that the given significance level is satisfied. A significance level of $5\%$ is used.

From the above table, we can infer that the minimum number of terms needed to satisfy the significance level is between 50 and 75. To be exact, the number of terms needed for estimation for a 5% significance level is 52. This balance point indicates that increasing terms beyond this do not significantly improve the accuracy but do increase the computational complexity. This finding is useful for financial institutions aiming to optimize their resources. Limiting the terms reduces the computational load without sacrificing the hedge effectiveness. The next

Table 5. Minimum number of terms required at difference significance levels.

	Significance Level ($\mathit{\alpha }$)
	0.01	0.02	0.05	0.10	0.20
Minimum Number of Terms Required	58	58	52	48	45

presents the minimum number of terms necessary to satisfy different significance levels.

Generally, as the significance level increases, the number of terms required to satisfy it decreases. With this in mind, we move on to calculating the partial hedging price according to the significance level and given shortfall amount A. The following

Table 6. Exchange option prices for different combinations of significance levels and expected shortfall amounts.

$\mathit{\alpha }$	A
	5	10	20	30	40	50
0.01	61.8541	61.8041	61.9041	61.6041	61.5041	61.4041
0.02	61.8041	61.7041	61.7041	61.3041	61.1041	60.9041
0.05	61.6541	61.4041	60.9041	60.4041	59.9041	59.4041
0.10	61.4041	61.1041	59.9041	58.9041	57.9041	56.9041
0.20	60.9041	60.3041	57.9041	55.9041	53.9041	51.9041

illustrates the corresponding prices for various significance levels $\alpha$ and non-negative constants A.

The jump diffusion model introduces the jump risk, which is naturally reflected in lower option prices. By incorporating the jump risk, the model provides a more realistic picture of the market behavior, especially in volatile markets. Insurance companies might prefer this model when the market conditions are unstable, as it offers a hedge at a lower cost while accounting for additional market risks. Following the same procedure as in Section 7.1 and applying the Brennan and Schwartz equilibrium, the required survival probability for each combination of significance level $\alpha$ and non-negative A is given below.

As a result of the smaller fair price for the exchange option, the ratios of the partial hedging price to the fair price are slightly lower compared to the ratios calculated based on the two-factor diffusion model. From here, we then determine the ages needed to attain the survival probabilities for T years, presented in

Table 7. Survival probabilities for different combinations of significance levels and expected shortfall amounts.

$\mathit{\alpha }$	A
	5	10	20	30	40	50
0.01	0.9992	0.9984	0.9968	0.9952	0.9935	0.9919
0.02	0.9984	0.9968	0.9935	0.9903	0.9871	0.9838
0.05	0.9960	0.9919	0.9838	0.9758	0.9677	0.9596
0.10	0.9919	0.9838	0.9677	0.9515	0.9354	0.9192
0.20	0.9838	0.9677	0.9354	0.9031	0.8708	0.8385

. Based on the Statistic Canada 3-year life table for 2018 to 2020, for a 3-year contract, the minimum ages are calculated to match the survival rate, given in

Table 8. Minimum age of insured for different combinations of significance levels and expected shortfall amounts.

$\mathit{\alpha }$	A
	5	10	20	30	40	50
0.01	32	47	55	60	63	65
0.02	47	55	63	67	70	72
0.05	58	65	72	77	79	81
0.10	65	72	79	83	86	88
0.20	72	79	86	89	92	94

.

The age of the individual needed to reach equilibrium for the jump diffusion model is higher for all combinations compared to the obtained age for the diffusion model. This is a result of the smaller fair price, which leads to a decreased survival probability. Thus, insurers might prefer to offer these contracts to older clients, where higher mortality offsets the increased market risk.

8. Discussion

From the perspective of the prior study by Novikov, the present research advances the concept of controlled partial hedging by offering practical ways to manage the risk–reward balance. The application of partial hedging to life insurance, particularly equity-linked pure endowment contracts, demonstrates an approach to blending financial and mortality risks.

In the insurance industry, partial hedging introduces a way to integrate actuarial and financial risk management effectively. By linking survival probabilities with hedging costs, insurance companies can adjust their policyholder age requirements and policy terms to balance mortality risks with financial market exposure. This integration represents a shift from traditional actuarial approaches, where mortality and financial risks were often considered separately. The model presented here enables insurance companies to structure equity-linked contracts with tailored risk–reward profiles, making them more appealing to clients and financially sustainable for providers.

A possible future consideration is the application of partial hedging in multi-asset portfolios, which could involve more complex models that account for additional correlation structures and stochastic dependencies. Investigating such models may reveal further insights into the optimal hedging strategies across diverse asset classes, expanding the utility of these methods beyond equity-linked insurance products to other structured financial instruments. Furthermore, the age of selection is determined using Statistic Canada life tables. A future consideration would be to use stochastic mortalities (see [22]) to determine the selection age of the equity-linked insurance policy.

While this approach works in theory, its application in real-world financial markets faces limitations. The market conditions often deviate from the assumptions of completeness, constant parameters, and frictionless trading, leading to challenges in implementation. Additionally, the partial hedging framework may be sensitive to the accuracy of the input data, such as volatility or correlation estimates, which are subject to estimation errors. The practical applicability is further constrained by the need for advanced computational resources, which may limit its adoption in the current fast-paced financial environment.