178 A.R. Bacinello Insurance: Mathematics and Economics 27 2000 177–188
the scheme which grants the GOB value, this is just an option to receive the GOB GOB option henceforth instead of the initially specified benefit.
The object of this paper is to value, on an individual basis, such an option. It can also be regarded, along the lines of Margrabe 1978, as an option to exchange one asset, asset A, the defined-benefit contribution pension,
for another, asset B, the defined-contribution benefit pension.
1
To this end we assume an economic framework consistent with the two-factor model proposed by Moriconi 1994 to describe the stochastic evolution of real
and nominal interest rates. In this framework we consider two additional state-variables: the individual salary, and the value of an index, representing the unit price of a reference portfolio where contributions are deemed
to be invested. The return on this particular portfolio determines the interest rate credited to the accumulated contributions.
The individual under consideration is always entitled to receive benefits of the same type of those included in asset A type A until the moment in which heshe exercises the GOB option. Hence it is obviously crucial, in
the valuation problem, to specify: a whether this option is of European or of American style; b when it comes to maturity. If the GOB option is of European style, i.e., if it can be exercised only at maturity, all benefits due
for contingencies occurred before this maturity are still of type A. Then the assets to be exchanged include only benefits due for contingencies occurred after maturity. In particular, if this maturity coincides with the retirement
date, the assets to be exchanged consist only of the retirement annuity usually reversible to survivors. On the contrary, if the GOB option, although being of European style, matures before the retirement date, the assets to be
exchanged include also disability, withdrawal and death benefits due in case these events occur between maturity and the retirement date besides retirement benefits. Even more so, if the option is of American style, i.e., if it can
be exercised in any moment on or before its maturity, benefits for all kinds of events, no matter when they happen, are relevant in the valuation problem.
In the paper we assume neutrality with respect to all risks affecting life contingencies mortality, disability, withdrawal, survivors, etc., and independence of all these risks from those affecting financial markets. In this
setting, by exploiting the martingale approach,
2
we first obtain a pricing formula for a GOB option of European style maturing at the retirement date. Some comparative statics properties of this formula are then analysed by
means of Monte Carlo methods. We consider the case in which the initial pension scheme which delivers ben- efits of type A is of the defined-benefit type as well as the one in which it is of the defined-contribution type
instead.
The paper is structured as follows. In Section 2 we introduce our notation and assumptions concerning the economic framework. Section 3 is devoted to the description of our valuation model for a European-style GOB
option with maturity at retirement, and Section 4 presents our simulation results. Section 5 concludes the paper.
2. The economic framework
This section is initially devoted to introduce our notation and assumptions concerning the economic framework. Then, we make some remarks about these assumptions, and finally we present a pricing formula for traded securities
which constitutes the building block of our valuation model. We delay until the next section definitions and notation concerning life contingencies.
2.1. Notation and assumptions As is usual in the financial literature, we assume a perfectly competitive and frictionless market, populated by
rational and non-satiated agents, all sharing the same information revealed by a filtration. In this very theoretical
1
See also Gerber and Shiu 1996, p. 202.
2
For details on this theory we refer to the fundamental papers of Harrison and Kreps 1979, Harrison and Pliska 1981, 1983, Artzner and Delbaen 1989, or to that of Duffie 1996 that summarises the basic results.
A.R. Bacinello Insurance: Mathematics and Economics 27 2000 177–188 179
framework we consider the following state-variables: rt
nominal instantaneous interest rate, pt
retail price index, gt
unit value of a portfolio in which contributions are deemed to be invested, st
individual salary. We assume that, under a risk-neutral measure Q, the random behaviour of these variables is described by the
following stochastic differential equations: drt = a
r
[θ
r
− rt ] dt + σ
r
p rt dZ
r
t , a
r
, θ
r
, σ
r
0, 2.1.1
dpt pt
= µ
p
dt + σ
p
dZ
p
t , σ
p
0, 2.1.2
dgt gt
= rt dt + σ
g
dZ
g
t , σ
g
≥ 0,
2.1.3 dst
st =
µ
s
dt + σ
s
dZ
s
t , σ
s
σ
p
, 2.1.4
where a
r
, θ
r
, σ
r
, µ
p
, σ
p
, σ
g
, µ
s
, σ
s
are real constants and Z
r
, Z
p
, Z
g
, Z
s
are correlated standard Brownian motions determining the sources of uncertainty in the economy. More precisely, we assume that
dZ
r
t dZ
p
t = ρ
rp
dt, −
1 ≤ ρ
rp
≤ 1,
2.1.5 dZ
r
t dZ
g
t = ρ
rg
dt, −
1 ≤ ρ
rg
≤ 1,
2.1.6 dZ
p
t dZ
g
t = ρ
pg
dt, −
1 ≤ ρ
pg
≤ 1,
2.1.7 dZ
p
t dZ
s
t = σ
p
σ
s
dt, 2.1.8
while dZ
r
t dZ
s
t = dZ
g
t dZ
s
t = 0. 2.1.9
2.2. Remarks Observe that relations 2.1.1 and 2.1.2 are compatible with those proposed by Moriconi 1994 under the
physical measure and nay, they have the same form but risk-adjusted parameters. In particular, the nominal interest rate follows the mean-reverting square root diffusion process proposed by Cox et al. 1979. The latter has
the pleasant property of producing almost surely strictly positive interest rates, at least when 2a
r
θ
r
≥ σ
r 2
see Feller, 1951.
As for the instantaneous inflation rate, dptpt dt, it can take also negative values, and its expected value, µ
p
, is assumed to be deterministic and constant in time. An alternative specification for µ
p
suggested by Moriconi 1994 is a deterministic function of time, whereas in the multifactor model of Cox et al. 1985 it is an additional
state-variable, following a mean-reverting diffusion process.
3
However, the simplified version of the model with deterministic expected inflation rate has the appealing property of producing real interest rates that, consistent with
the empirical evidence, are allowed to be negative, within a lower barrier equal to − µ
p
. In fact, according to the celebrated Fisher equation, the real interest rate, which we denote by et, is given by
et = rt − µ
p
, 2.2.1
3
See also Moriconi 1995.
180 A.R. Bacinello Insurance: Mathematics and Economics 27 2000 177–188
so that, by exploiting relation 2.1.1, det = drt = a
r
[θ
e
− et ] dt + σ
r
q et + µ
p
dZ
r
t , 2.2.2
with θ
e
= θ
r
− µ
p
. Finally, among the numerous models for inflation proposed in literature, we recall Chan 1998, who models the behaviour of the inflation rate directly by means of an Ornstein–Uhlenbeck diffusion process.
As far as the portfolio index gt is concerned, it follows a diffusion process when σ
g
0 see relation 2.1.3. In the very special case when it is risk-free i.e., when σ
g
= 0, gt = g0 exp{
R
t
ru du} is usually referred to as the deposit account or money market account.
As for the modelling of the salary or labour income, there are many debates in the actuarial literature e.g., Carriere and Shand 1998 propose a deterministic trend for it. Undoubtedly, stochastic models are the most suitable,
but the question is which ones. Salaries are generally characterised by a discontinuous behaviour since their paths are stepwise functions of time, so that jump processes seem to be realistic enough. However, especially in the financial
literature, they are often modelled by geometric Brownian motions see, e.g., Duffie and Zariphopoulou, 1993; Koo, 1998. Hence they are characterised by continuous paths. Here, as an approximation, we also make the geometric
Brownian motion assumption 2.1.4, derived by the following considerations. Assume, for simplicity, that the expected variations in the individual income are due to seniority, according to a constant rate a
s
, and to inflation. Assume also that the unexpected ones due, for instance, to promotion, demotion, sickness, etc. are generated by
a standard Brownian motion Z uncorrelated with Z
r
, Z
g
and, in particular, with Z
p
. The behaviour of st is then described by the stochastic differential equation
dst st
= a
s
dt + dpt
pt +
σ dZt , σ 0,
2.2.3 where a
s
and σ are real constants. Exploiting relation 2.1.2 we have also: dst
st =
a
s
+ µ
p
dt + σ
p
dZ
p
t + σ dZt , 2.2.4
with dZ
p
t dZt = 0. Then relation 2.1.4 is simply obtained by letting µ
s
= a
s
+ µ
p
, σ
s
= q
σ
2
+ σ
p 2
, dZ
s
t = σ
p
σ
s
dZ
p
t + s
1 − σ
p
σ
s 2
dZt . 2.3. The martingale representation for the price of traded securities
To conclude Section 2, we recall that, in our frictionless market framework, all traded securities can be priced according to a martingale representation. More precisely, consider a finite-variance contingent-claim that delivers, at
a future date T, a random payoff X
T
which is functionally dependent on the values up to time T of the state-variables introduced in Section 2.1. Then its price at time t, which we denote by 5
t
X
T
, is given by the following relation: 5
t
X
T
= E
t
X
T
exp −
Z
T t
ru du ,
2.3.1 where E
t
denotes expectation taken with respect to the risk-neutral measure Q and conditional on information up to time t.
This fundamental equation, expressible in closed-form only for very particular and simple expressions of the payoff X
T
, can be readily approached via Monte Carlo methods and will be the building-block in all our valuation framework.
We observe, however, that using this formula in the actual world can be problematic. Indeed, a very interesting topic of research is the study of valuation models that take into account the presence of friction in the market. Moreover,
A.R. Bacinello Insurance: Mathematics and Economics 27 2000 177–188 181
in an incomplete market, there can be several risk-neutral measures and therefore the price of a non-redundant security can be not uniquely determined.
3. The valuation model for a European-style GOB option with maturity at retirement
