Quadratic hedging of equity-linked life insurance contracts under the real-world measure in discrete time

Definition 3.1.

A self-financing strategy φ = ( γ , β ) with strictly positive capital X = ( X t ) t ⩾ 0 , X 0 = 1 is called a P-discounting portfolio if for any self-financing portfolio π = ( ξ , η ) a discounted value process V / X = ( V t / X t ) t ⩾ 0 is a P-martingale (or, equivalently, discounted price processes S / X = ( S t / X t ) t ⩾ 0 and B / X = ( B t / X t ) t ⩾ 0 are P-martingales).

Finding a P-discounting portfolio enables us to work exclusively with a real-world, “physical”, probability model, describing the true nature of the processes. Moreover, such a discounting portfolio is unique if it exists.

Lemma 3.1.

If a P-discounting portfolio φ = ( γ , β ) with initial capital X 0 = 1 exists, then it is unique.

Proof.

Let φ ′ = ( γ ′ , β ′ ) be another P-discounting portfolio with the capital X ′ = ( X t ′ ) t ⩾ 0 and X 0 ′ = 1 . Then by Definition 3.1, ( X t / X t ′ ) t ⩾ 0 and ( X t ′ / X t ) t ⩾ 0 are P-martingales. Therefore, if Y t : = X t / X t ′ , then E ( Y t ) = E ( 1 / Y t ) = 1 . Since φ ( y ) = 1 / y , y > 0 , is strictly convex downward function, then by Jensen’s inequality,2 Y t = E ( Y t ) = 1 . □

Let κ t = γ t S t / X t denote a proportion of risky capital in a P-discounting portfolio φ at time t. Then it follows from Eqs (3), (1) and (2) that Δ X t = γ t − 1 Δ S t + β t − 1 Δ B t = X t − 1 κ t − 1 ρ t + X t − 1 ( 1 − κ t − 1 ) r = X t − 1 ( r + κ t − 1 ( ρ t − r ) ) , and hence (4) X t = X t − 1 ( 1 + r + κ t − 1 ( ρ t − r ) ) .

Thus, the capital ( X t ) t ⩾ 0 (with X 0 = 1 ) of a P-discounting portfolio can be defined by a sequence ( κ t ) t ⩾ 0 , as the following lemma shows.

Lemma 3.2.

A P-discounting portfolio φ = ( γ , β ) with strictly positive capital X = ( X t ) t ⩾ 0 , X 0 = 1 , exists if and only if there exists an F -adapted sequence κ = ( κ t ) t ⩾ 0 = ( γ t S t / X t ) t ⩾ 0 such that (i)

1 + r + κ t ( ρ t + 1 − r t ) > 0 (P-a.s.);

(ii)

a process M = ( M t ) t ⩾ 0 , defined by (5) M 0 = 0 , M t = ∑ i = 1 t ρ i − r 1 + r + κ i − 1 ( ρ i − r ) , is a P-martingale.

Proof.

a) Necessity.

Let φ = ( γ , β ) be a P-discounting portfolio with strictly positive capital X = ( X t ) t ⩾ 0 , X 0 = 1 , and proportion of risky capital κ t = γ t S t / X t . Since X t > 0 , t ⩾ 0 , part (i) of the lemma follows from Eq. (4).

Let π = ( ξ , η ) be an arbitrary self-financing portfolio with capital V = ( V t ) t ⩾ 0 and proportion of risky capital α t = ξ t S t / V t . Using a well-known fact that self-financing portfolios remain self-financing under a change of numeraire (see Geman et al. [5]), we can write Δ V t X t = ξ t − 1 Δ S t X t + η t − 1 Δ B t X t = ξ t − 1 Δ S t X t + V t − 1 − ξ t − 1 S t − 1 B t − 1 Δ B t X t = V t − 1 X t − 1 [ X t − 1 B t − 1 Δ B t X t + α t − 1 ( X t − 1 S t − 1 S t X t − X t − 1 B t − 1 B t X t ) ] (6) = V t − 1 X t − 1 ( α t − 1 − κ t − 1 ) Δ M t , where Δ M t = X t − 1 S t − 1 S t X t − X t − 1 B t − 1 B t X t (7) = ρ t − r 1 + r + κ t − 1 ( ρ t − r ) .

The second equality in (7) follows from (4), (1) and (2). The processes ( S t / X t ) t ⩾ 0 and ( B t / X t ) t ⩾ 0 are martingales by definition of a P-discounting portfolio (Definition 3.1), hence ( Δ M t ) t ⩾ 1 is a martingale difference.

b) Sufficiency.

Let κ = ( κ t ) t ⩾ 0 = ( γ t S t / X t ) t ⩾ 0 be an F -adapted sequence such that conditions (i) and (ii) hold. Then X = ( X t ) t ⩾ 0 with X 0 = 1 and X t = X t − 1 ( 1 + r + κ t − 1 ( ρ t − r ) ) is strictly positive F -adapted sequence.

It follows from the definition of a self-financing portfolio (Definition 2.1) and representation Δ X t = X t − 1 κ t − 1 ρ t + X t − 1 r ( 1 − κ t − 1 ) that X is a capital of self-financing portfolio φ = ( γ , β ) with γ t = κ t X t / S t and β t = ( 1 − κ t ) X t / B t .

Finally, from condition (ii) and representation (6) it follows that for any self-financing portfolio π = ( ξ , η ) with capital V = ( V t ) t ⩾ 0 a sequence V / X = ( V t / X t ) t ⩾ 0 is a martingale. Hence, φ = ( γ , β ) is a P-discounting portfolio by Definition 3.1. □

Thus, the existence of the sequence ( κ t ) t ⩾ 0 determines the conditions for existence of a P-discounting portfolio. We further assume that a P-discounting portfolio with capital X = ( X t ) t ⩾ 0 , X 0 = 1 , exists.

3.1. Numerical example

To illustrate the concept, consider the two-step binomial market. The evolution of prices if given by Eqs (1) and (2) with B 0 = $ 1 , S 0 = $ 100 , with constant interest rate r = 0.12 , and with return per unit of stock ρ t taking two values: b = 0.25 or a = − 0.1 . Let probability of an up move, p, be 0.4, then probability of a down move is 0.6. Find a price of European call option with payoff f ( S 2 ) = max ( S 2 − K , 0 ) , where strike price K = $ 110 .

Figure 1 shows two-period binomial tree for the evolution of stock prices S. The price of a contingent claim at a terminal time is easily computed. At the next step, a risk-neutral probability is usually found as p ∗ = ( r − a ) / ( b − a ) = 0.6286 and the call price is then the discounted expectation with respect to this new probability: C = E ∗ [ f ( S 2 ) / ( 1 + r ) 2 ] = $ 15.50 . The price at time t = 1 is either $26.79 or $1.40. These prices are shown in Fig. 1 in parentheses.

OPEN IN VIEWER

Fig. 1.

Binomial tree for the stock price, price of a call option, and the capital of a P-discounting portfolio. Note: The upper numbers denote stock prices, the middle numbers in parentheses denote the price of a contingent claim, and the lower numbers in square brackets represent a capital of a P-discounting portfolio.

To solve the same problem without switching to a new probability, find the discounting portfolio X as described in Lemma 3.2. In this binomial model proportion of risky capital is constant ( κ t = κ for all t) and can be found by the formula: κ = ( 1 + r ) ( μ − r ) ( b − r ) ( r − a ) , with μ = E ( ρ t | F t − 1 ) , from where we have κ = − 3.13 . Further calculations give: X 0 = 1 ; X 1 = 0.71 in case of an upward move, or X 1 = 1.81 in case of a downward move; X 2 = 0.51 for the upper node, 1.29 for the middle mode, and 3.27 for the lower node (shown in square brackets in Fig. 1). Now, the price at time t = 0 is C = E [ f ( S 2 ) / X 2 ] = $ 15.50 , and at time t = 1 it is either $26.79 or $1.40.

As expected, both methods lead to the same results.

Definition 4.1.

Strategy π = ( ξ , η ) with investment process I = ( I t ) t ⩾ 0 is called P-mean-self-financing, or simply mean-self-financing, if E ( Δ I t X t | F t − 1 ) = 0 , i.e. ( Δ I t / X t ) t ⩾ 1 is a martingale difference.

It is clear that the class of mean-self-financing strategies includes the self-financing portfolios. We can also give a martingale characterization of mean-self-financing strategies.

Lemma 4.1.

Strategy π = ( ξ , η ) with capital V = ( V t ) t ⩾ 0 is mean-self-financing if and only if V / X = ( V t / X t ) t ⩾ 0 is a martingale.

Proof.

a) Necessity.

Let strategy π = ( ξ , η ) with capital V = ( V t ) t ⩾ 0 be mean-self-financing. Then, similarly to (6) we have: Δ V t X t = ξ t − 1 Δ S t X t + η t − 1 Δ B t X t + Δ I t X t (8) = V t − 1 X t − 1 ( α t − 1 − κ t − 1 ) Δ M t + Δ I t X t , where M t is defined in (5).

By Lemma 3.2 and Definition 4.1, a sequence ( Δ ( V t / X t ) ) t ⩾ 1 is a martingale difference.

b) Sufficiency.

Let V / X = ( V t / X t ) t ⩾ 0 be a martingale and find arbitrary sequences ξ = ( ξ t ) t ⩾ 0 and η = ( η t ) t ⩾ 0 = ( ( V t − ξ t S t ) / B t ) t ⩾ 0 , so that a strategy π = ( ξ , η ) had a capital V. Then by Eq. (8) π is mean-self-financing. □

Further, we will be interested in finding the “cheapest” replicating mean-self-financing strategy. Note the additional investments at each time, ( Δ I t ) t ⩾ 1 , are random and a risk-averse investor will want to minimize uncertainty over the time of a contract. These considerations lead to a concept of risk-minimization introduced by Föllmer and Sondermann [4], who suggested to determine the trading strategy by minimizing the variance of the future costs.

We consider a global risk-minimization problem. Among all strategies with V T = f T , where f T is the payout of a contingent claim, we seek a mean-self-financing trading strategy minimizing the variance of all the additional investments over the life of a contract: minimize (9) R = Var ( ∑ t = 1 T Δ I t X t ) , (10) subject to E ( Δ I t X t | F t − 1 ) = 0 .

The resulting strategy will be called risk-minimizing. Due to condition (10), the risk-minimization problem (9) can be simplified to a minimization of R = E ∑ t = 1 T ( Δ I t / X t ) 2 . Theorem 4.2.

A unique risk-minimizing mean-self-financing hedging strategy π ∗ for the contract with payout f T has capital (11) V t ∗ = X t E ( f T X T | F t ) , 0 ⩽ t ⩽ T , and structure (12) ξ t ∗ = γ t V t ∗ X t + X t S t E [ Δ V t + 1 ∗ X t + 1 Δ M t + 1 | F t ] E [ ( Δ M t + 1 ) 2 | F t ) ] , (13) η t ∗ = V t ∗ − ξ t ∗ S t B t .

Risk associated with a contingent claim f T is R π ∗ = ∑ t = 1 T [ E ( Δ V t ∗ X t ) 2 (14) − E E 2 ( Δ V t ∗ X t Δ M t | F t − 1 ) E [ ( Δ M t ) 2 | F t − 1 ] ] .

Proof.

Due to mean-self-financing requirement (10), V ∗ / X = ( V t ∗ / X t ) t ⩾ 0 is a martingale by Lemma 4.1, and we require V T ∗ = f T , hence Eq. (11) is true. The strategy clearly exists as the condition V T ∗ = f T can always be achieved, for example, by the appropriate choice of η T at terminal time T as it is adapted.

Similarly to Eq. (8) we have: Δ V t ∗ X t = S t − 1 X t − 1 ( ξ t − 1 ∗ − γ t − 1 V t − 1 ∗ X t − 1 ) Δ M t (15) + Δ I t ∗ X t .

It follows from Eqs (15) and (12) that (16) Δ I t ∗ X t = Δ V t ∗ X t − E [ Δ V t ∗ X t Δ M t | F t − 1 ] E [ ( Δ M t ) 2 | F t − 1 ] Δ M t .

From Eq. (16) we get the expression (14) for the risk of a contingent claim.

Now, let π = ( ξ , η ) be another mean-self-financing hedging strategy with V 0 = V 0 ∗ . Hence, (17) V t = X t E ( f T X T | F t ) = V t ∗ .

Similarly to (15) we have Δ V t X t = S t − 1 X t − 1 ( ξ t − 1 − γ t − 1 V t − 1 X t − 1 ) Δ M t (18) + Δ I t X t .

From (18), (17) and (15) we get (19) Δ I t X t = S t − 1 X t − 1 ( ξ t − 1 ∗ − ξ t − 1 ) Δ M t + Δ I t ∗ X t .

Using Eq. (19) and noting that [ ( Δ I t / X t ) Δ M t ] t ⩾ 1 is a martingale difference, as follows from (16), we arrive at: R π = E ∑ t = 1 T ( Δ I t X t ) 2 = ∑ t = 1 T E S t − 1 2 X t − 1 2 ( ξ t − 1 ∗ − ξ t − 1 ) 2 ( Δ M t ) 2 + R π ∗ .

Hence, any other mean-self-financing strategy with the same initial capital will involve a higher risk. □

5. Hedging equity-linked life insurance contracts

In equity-linked life insurance contracts benefit the policyholder receives at a terminal time T, provided he or she is still alive at this time, is directly linked to the development of the stocks or stock indices. For instance, the amount paid to the policyholder, f ( S T ) , can be: (20) f ( S T ) = S T or (21) f ( S T ) = max ( S T , K ) .

The contract (20) is called a pure equity-linked contract; in such contracts the financial risk associated with stock prices can be entirely charged to the policyholder. The contract (21) is known as equity-linked contract with guarantee (here K is the guarantee) and represents an example of the contract when the financial risk is shared between a policyholder and an insurance company.

We assume that n policyholders buy the same type of a contract at time 0 at the age of x and their remaining lifetimes T 1 , … , T n are independent and identically distributed. We also assume that the remaining life times are independent of the discounted stock price process S / X = ( S t / X t ) t ⩾ 0 (or, more generally, that financial market is independent of the insurance risk). The random variable Y t denotes the number of policyholders who survived to time t. The survival probability of the insured is denoted P ( T i > t ) = t p x .

The number of survivors, Y t , can be seen as the sum of independent Bernoulli random variables, each taking on value 1 with probability t p x , or 0 with probability 1 − t p x . Then the expected number of the survivors at terminal time T is E ( Y T ) = E ( ∑ i = 1 n I [ T i > T ] ) = ∑ i = 1 n E ( I [ T i > T ] ) = n T p x , and the variance is Var ( Y T ) = Var ( ∑ i = 1 n I [ T i > T ] ) = ∑ i = 1 n Var ( I [ T i > T ] ) = n T p x ( 1 − T p x ) .

Thus, the insurer’s liability, f T , depends on the number of survivors, Y T , and the evolution of the stock prices f ( S T ) : f T = Y T f ( S T ) .

Formally, to construct a model combining insurance risk (uncertainty regarding number of policyholders who will survive to terminal time) and financial risk (uncertainty regarding the development of the stock prices), we need to start with two separate filtered probability spaces. Let ( Ω 1 , F , F = ( F t ) t ⩾ 0 , P 1 ) be a space carrying the financial risk, with filtration F = ( F t ) t ⩾ 0 containing information about the evolution of the financial market, i.e. F t = σ ( S 0 , … , S t ) . Let ( Ω 2 , H , H = ( H t ) t ⩾ 0 , P 2 ) be a space incorporating the insurance risk, with filtration H = ( H t ) t ⩾ 0 containing information about policyholders, i.e. H t = σ ( Y 0 , … , Y t ) . These two models are then embedded into a product space ( Ω , G , G = ( G t ) t ⩾ 0 , P ) , where the filtration G = ( G t ) t ⩾ 0 , defined by G t = σ ( F t ∪ H t ) , contains all the available information up to time t. Thus, it is assumed that the insurance company at any time t has access to the current information about both stock performance and the number of survived policyholders. Filtrations ( F t ) t ⩾ 0 and ( H t ) t ⩾ 0 are independent under P.

Market defined on a described product space will always be incomplete (even if the model of the financial market is complete), as contingent claim is allowed to depend on an additional source of risk independent of the financial risk. Hence, we can view the equity-linked life insurance contract as a contingent claim in an incomplete market, and therefore a risk-minimizing trading strategy for such a contract is defined similarly to Section 4.

Let π f = ( ξ f , η f ) denote a risk-minimizing hedging strategy for the financial contract with payout f ( S T ) . The capital of π f is given by V t f = X t E ( f ( S T ) / X T | F t ) , where f ( S T ) = V T f . Let π ∗ = ( ξ ∗ , η ∗ ) be a risk-minimizing trading strategy for the equity-linked life insurance contract with liability f T = V T ∗ = Y T V T f . Due to independence of financial and insurance risks, risk-minimizing mean-self-financing strategy π ∗ has the value process: V t ∗ = X t E ( Y T f ( S T ) X T | G t ) = X t E ( Y T | H t ) E ( f ( S T ) X T | F t ) (22) = Y t T − t p x + t V t f , and structure: ξ t ∗ = γ t V t ∗ X t + X t S t E [ Δ V t + 1 ∗ X t + 1 Δ M t + 1 | G t ] E [ ( Δ M t + 1 ) 2 | G t ] = Y t T − t p x + t ( γ t V t f X t + X t S t E [ Δ V t + 1 f X t + 1 Δ M t + 1 | F t ] E [ ( Δ M t + 1 ) 2 | F t ] ) = Y t T − t p x + t ξ t f , η t ∗ = V t ∗ − ξ t ∗ S t B t = Y t T − t p x + t η t f , t ⩾ 0 .

In Eq. (22), E ( Y T | H t ) = Y t T − t p x + t , as Y T | H t ∼ Binomial ( Y t , T − t p x + t ) , where Y t is actual number of survivors at time t, and T − t p x + t is a conditional probability of survival to time T given that the insured is alive at time t.

The risk associated with a contingent claim f T = Y T f ( S T ) is R π ∗ = n T p x ∑ t = 1 T [ T − t p x + t E ( V t f X t ) 2 − T − ( t − 1 ) p x + ( t − 1 ) × ( E ( V t − 1 f X t − 1 ) 2 + E E 2 [ Δ M t Δ V t f X t | F t − 1 ] E [ ( Δ M ) t 2 | F t − 1 ] ) ] + n ( n − 1 ) T p x 2 × ∑ t = 1 T [ E ( Δ V t f X t ) 2 (23) − E E 2 [ Δ M t Δ V t f X t | F t − 1 ] E [ ( Δ M ) t 2 | F t − 1 ] ] .

In cases when incomplete market arises from a complete financial market by allowing contingent claims to depend on an insurance risk that is independent of the financial risk, the second sum in Eq. (23) becomes 0.