60 T. Siegl, R.F. Tichy Insurance: Mathematics and Economics 26 2000 59–73
regular coupon. The bond is re-emitted after the event. The proposed model may help in choosing the optimal terms of such a risk exchange.
In the following we will present the model for arbitrary claim size distributions F and prove existence and uniqueness for the general case, while an analytic solution is found for the special case F y
= 1 − exp−y. Furthermore, we compare the performance of hybrid and pseudo-Monte Carlo sequences for this case. We will
show that a significant improvement by a factor of 2–3 is possible as far as the required sample size for a given error is concerned.
2. Model with upper barrier
We assume that premiums are paid at a constant rate of ˜c = ˆc + c c 0 per unit time, that the claim
number follows a Poisson process N
λ
t with parameter λ 0, and that the individual i.i.d. claims Y
i
≥ 0 have distribution F y with expected value µ
∞. Furthermore, the devaluations are rare events, and therefore the number of devaluations also follows a Poisson process N
γ
t with parameter γ 0. Upon devaluation, the reserve
is multiplied by the factor 0 α 1. In this case, the embedded option pays an amount of K 0x → αx + K.
It is financed by the continuous payment of ˆc. At the time t n of the nth devaluation, the free reserve x changes to
αx + K yielding the stochastic process of the free reserve R
t
at time t: R
t
= R + ct −
Nλt
X
n =1
Y
n
−
N
γ
t
X
n =1
1 − αR
t n
− K. If the reserve R
t
drops below 0, we say ruin has occurred. The infinite time survival probability is defined for the initial free reserve x
= R by
U x
= P inf
t ≥0
R
t
≥ 0 .
However, for 0 α 1 we can always find a capital level x such that the expected capital gain per unit time which is approximately
c − λµ − γ 1 − αx − K −C,
for C ∈ R
+
is less than any arbitrary constant. Thus, without further restrictions, the infinite time ruin probability 9x =
1 − Ux is 1. Therefore, we introduce an absorbing horizontal barrier at x
max
. If the process reaches the barrier, it is absorbed and the company has survived. This corresponds to the situation where the company will then decide
to pursue other forms of investment or securitization. By including a barrier, the process stops in finite time with probability 1. However, the restriction by the upper
barrier also forces us to restrict our choice of K for the solution method we are going to present now. We will formulate the model equation as an integro-differential equation with deviating arguments. When we allow αx
+ K to exceed x
max
we would obtain a retarded integro-differential equation. The difficulties inherent in solving this type of equation can be shown for the special case where λ
= 0. Then the model equation becomes through the standard use of the law of total probability a retarded differential equation:
cU
′
x − γ Ux + γ Uαx + K = 0,
∀x ≥ x
max
: U x
= 1. While the solution is readily seen for this boundary type the solution is U x
≡ 1 finding a solution for other boundary conditions becomes quite a challenge. As an example we consider a boundary on [x
max
, ∞ defined by
U x
max
= 1 and Ux = 0|
x ∈x
max
, ∞
. As we do not want to go too much into the details of this type of problem,
T. Siegl, R.F. Tichy Insurance: Mathematics and Economics 26 2000 59–73 61
Fig. 1. Risk model.
we will just show the solution for α = 2 and K = 0 which corresponds to the case presented later where we allow
α 1. We find for x
max
2 ≤ x x
max
: U x
= exp γ
c x
− x
max
, and for x
max
4 ≤ x x
max
2: U x
= exp γ
c 2x
− x
max
+ 1
+ exp −
γ x
max
2c exp
γ c
x −
x
max
2 ,
which can be continued ad infinitum with increasingly complex piecewise defined functions. This solution does not even fit our model as λ
= 0 and should just serve as a demonstration of the difficulties encountered when we leave out the condition that after devaluation the capital may not exceed x
max
. Thus, we restrict ourselves to the natural condition αx
+ K ≤ x
max
for x ∈ [0, x
max
]. This is equivalent to K ≤ 1 − αx
max
. This means that the securitization payout must not exceed the maximal loss by devaluation. Thus, there is no speculative element in the
securitization, but only the hedging element. The situation is depicted in Fig. 1. For further reference concerning models with absorbing horizontal barrier see Dickson 1991, and Dickson and
Gray 1984, and for general linear barriers Gerber 1981, and Siegl and Tichy 1996. The known solutions of the problems with general linear barriers are given by infinite series in terms associated with the case without barrier.
We expect the solution of our problem with a general linear barrier to be substantially more difficult, as the simpler case of a horizontal barrier leads to an infinite series already.
The law of total probability in an infinitesimal time interval τ yields the model equation for the probability of survival U x depending on the initial surplus x
U x = 1 − λτ − γ τ Ux + cτ + λτ
Z
x
U x − y dF y + γ τ Uαx + K + Oτ
2
, with the boundary condition
U x
max
= 1, 1
at the horizontal absorbing upper barrier x
max
. Note that we have a one-point boundary in contrast to the example presented before, where we had α 1. We can transform the model equation by a Taylor expansion and some
simplifications to the integro-differential equation with deviating argument: cU
′
x − λ + γ Ux + λ
Z
x
U x − y dF y + γ Uαx + K = 0.
2
62 T. Siegl, R.F. Tichy Insurance: Mathematics and Economics 26 2000 59–73
3. Analytic solution
