2.5 The derived price process
The insurance contract forms the basis for introduction of two price processes, F and V:
Ft = the price at time t of the contractual payments to the insurance company over [0, T] , i.e. premiums less benefits, Vt = the price at time t of the contractual payments from the
insurance company over (t, T] , i.e. benefits less premiums.
We make some preliminary comments on these processes as a preparation and mo- tivation for a detailed study.
By the price at timetof contractual payments we mean the amount against which the payments stipulated by a contract are taken over by one agent from another.
Thus, buying and selling means ’taking over’ and ’handing over’, respectively, the contractual payments over some specified period of time. This consideration of contractual payments as dynamically marketed objects is called ’securitization’ of insurance contracts and plays an important role in the adaptation and application of financial theory to insurance problems. Important contributions are Delbaen and Haezendonck [17] and Sondermann [63].
By the securitization of contractual payments, we have implicitly taken as given the existence of a market, on which these contractual payments are allowed to be traded and, furthermore, that these contractual payments actually are bought and sold by the agents on the market. We shall assume thatZ constitutes such a market.
In many countries government regulations appear to prohibit a securitization of insurance contracts. One of the reasons may be that the supervisory authorities are not at all prepared for a free exchange of the kind of financial interests appearing on the insurance market. On the other hand, traditional reinsurance contracts actually represent one allowable way of forwarding risk to a third party. The agents on the insurance market, i.e. the customers, the direct insurance companies, and the reinsurance companies are, of course, the primary investors, but also other parties may consider the contractual payments of insurance contracts as possible investment objects. This statement is substantiated by the fact that F can be interpreted as the surplus of the company stemming from the insurance contract. This surplus is reflected in the equity, which is definitely a relevant investment object for all investors.
We have introduced two price processes, one covering all contractual payments and one covering future payments only. Even though we may be interested in the price of the future payments only, we shall work with the processF since this process
forms the asset, in a financial sense, arising from the securitization of the insurance contract. One could consider the introduction of the process F as a preliminary step leading to the definition of and derivation of formulas for the process V. In practice, one should have trading and marketing of V in mind. This also explains why we have not taken the investment strategy (to be introduced below) chosen by the insurance company as an integrated part of the insurance contract. As we shall see, this strategy affects the price of past payments but not the price of future payments.
In actuarial terminology, the outstanding liabilities are called the reserve, and these liabilities can be calculated under various assumptions. Since F and V are price processes arising on a market, it seems natural to callVt the market reserve at time t. We will, however, suppress the word ’market’, and simply speak ofVtas the reserve at timet. One should carefully note that, whereas the reserve is traditionally defined as the expected present value of future payments, we take the reserve to be the market price of future payments.
Our approach to the price processF is the following: Assuming that the market Z is arbitrage free, we require that also the market (Z, F) be arbitrage free. We use the essential equivalence between arbitrage free markets and existence of a so- called martingale measure, i.e. a measure under which discounted asset prices are martingales. If the no arbitrage condition is fulfilled for (Z, F), we shall speak ofB as an arbitrage free insurance contract and about V as the corresponding arbitrage free reserve.
Already at this stage we will argue for side conditions on the price process V. These side conditions are due to the no arbitrage condition on the market (Z, F) and the structure of the payment process B. Here it is important to state clearly the problems we actually want to solve: Given S, including Z, we wish to determine a payment process B such that no arbitrage possibilities arise from marketing the insurance contract. Afterwards, given the payment process B we wish to determine arbitrage free prices of the insurance contract.
When determining the payment process, this process is to be considered as a balancing tool and is as such comparable with the delivery price of a future or the price of an option. However, the payment process contains a continuum of balancing elements (premiums and benefits) and in practice all but one of these elements are predetermined by the customer and the last one acts as the balancing tool of the insurance company. Which elements are predetermined and which element is the balancing tool depends on the type of insurance contract (defined benefits, defined contributions etc.). Since the contract can be entered into at time 0 with no past payments, B should be balanced such that the equivalence relation
F0−=−V0−= 0 (2.4)
is fulfilled in order to prevent the obvious arbitrage possibility that arises if an agent can enter into the insurance contract and immediately sell the same contract on the market at a price different from 0. IfF0S is trivial such that B0 is deterministic, the
2.5. THE DERIVED PRICE PROCESS 33 equivalence relation (2.4) can also be written as
V0 =B0.
If e.g. B0 is fixed at 0, the remaining elements ofB are to be determined subject to V0 = 0. Hereby, the insurance contract is somewhat similar to a future contract.
The side condition at time T is also given by a no arbitrage argument. Since the price at time T of a payment of ∆BT at timeT in an arbitrage free market must be
∆BT, we have
VT− = ∆BT. (2.5)
So, the side conditions (2.4) and (2.5), imposed by the no arbitrage condition, should be included in the basis for balancing the payment processB. GivenB, this payment process is to be considered as an, indeed unusual, contingent claim and achieves as such at least one arbitrage free price at any time in an arbitrage free market. Here again, the insurance contract is somewhat similar to the future contracts which has a price, positive or negative, at any time during the term of the contract.
The insurance company receives payments in accordance with the insurance con- tract B, and we assume that these are currently deposited on (withdrawn from) an account which is invested in a portfolio with positive value process U, generated by a self-financing investment strategyθ ∈Rn+1, i.e.
Ut = θtãZt= Xn
i=0
θitZti >0, dUt = θtãdZt.
The strategy is furthermore assumed to comply with whatever institutional require- ments there may be. Throughout this chapter one can think of θ as the strategy corresponding to a constant relative portfolio, i.e. a strategy θ such that for a constant (n+ 1)-dimensional vector γ, θitZti− = γiUt−, i = 0, . . . , n. This strategy reflects an investment profile possibly restricted by the supervisory authorities, e.g.
such that θi is non-negative for all i if short-selling is not allowed. We emphasize that θ, in general, is not a strategy aiming at hedging some contingent claim.
Consequently, the present value at timet of the contractual payments over [0, T] becomes Ut
RT 0
1
UsdBs, where U is the value process corresponding to the chosen trading strategy θ. This present value is composed of an FtS-measurable part,
Lt=Ut
Z t 0
1 Us
dBs, and a part which is not in general FtS-measurable,
Ut Z T
t
1 Us
dBs.
If the price operator, denoted by πt, is assumed to be additive, pricing the contrac- tual payments over [0, T] amounts to replacingUt
RT t
1
UsdBs by some Ft-measurable
process, the price process −Vt. Thus, Ft=πt
Ut
Z T 0
1 Us
dBs
=Lt−Vt.
We restrict ourselves to prices allowingVtto be written in the formV (t, St). This restriction seems reasonable sinceS is Markov and since the payments byB and the intensities of N depend only on time and the current value ofS, but it is actually a restrictive assumption on the formation of prices in the market. It corresponds to a restrictive It corresponds to the restrictive structure of the measure transformation in Section 2.6.
If X jumps to state j at timet,S will jump to St−+βãtj−, and thus Vt jumps to Vtj−, where Vtj ≡V t, St+βãtj
. Each Vtj− is FtS-predictable, and we can introduce the J-dimensionalFtS-predictable row vector
VtJ− =
Vt1−, . . . , VtJ− .
Assume that the partial derivatives ∂tV (t, s), ∂sV (t, s), and ∂ssV (t, s) exist and are continuous, abbreviate
∂sVt=∂sV (t, St) = ∂sV (t, s)|s=St, and denote 12tr σTt∂ssVtσt
by ψt. Then Ito’s lemma applied to the processV gives the differential form,
dVt =
∂tVt+ (∂sVt)T αt+ψt
dt+ VtJ−−Vt1−×J
dNt+ (∂sVt)T σtdWt. Ito’s lemma also gives the differential form of the process L,
dLt = bctdt−bdt−dNt+Lt−
Ut−dUt
= bctdt−bdt−dNt+Ltrtdt−Ltrtdt+ Lt−
Ut−
dUt
= bctdt−bdt−dNt+Ltrtdt+ Lt−Zt0 Ut− d
Ut
Zt0
. It should be noted that we can also write dLt = dBt+ θUtLt−
t− dZt and consider the processLas a value process corresponding to a trading strategy given byυt= θUtLt−t−. Because of the payment process B, this strategy is not self-financing, though.
Now, collecting terms gives the differential form of the process F, dFt = dLt−dVt
= rtFtdt+
bct +rtVt−∂tVt−(∂sVt)T αt−ψt− bdt +VtJ −Vt1×J àt
dt
−(∂sVt)T σtdWt− bdt−+VtJ−−Vt1−×J
dMt+ Lt−Zt0 Ut−
d Ut
Zt0
.