Long-Term Risk Management of Nuclear Waste: A Real Options
Approach∗
Henri Loubergé†, Stéphane Villeneuve‡
and Marc Chesney§
First version: August 1998
This version: April 2001
Keywords: Risk management, optimal stopping, real and American options.
Abstract
In this paper, we investigate the optimal timing for deep geological disposal of nuclear
waste. Our model is based on the real options approach to investment under uncertainty. In
this context, the problem is similar to the optimal exercise policy for a perpetual American
spread option. The potential usefulness of such a model for actual decision-making on a
sensitive issue is illustrated by some numerical simulations.
∗
Work on this topic was initiated when Henri Loubergé visited the Finance-Insurance Laboratory of CREST
in Paris. The support of this institution is gratefully acknowledged. The authors also thank participants in
the CEPN seminar on radiological risk management, the Bachelier seminar in Paris, the FUR IX meeting in
Marrakesh, the seminar of the European Group of Risk and Insurance Economists in Rome, and three referees of
this journal for their comments on previous versions. The usual caveat applies.
†
Department of Economics, University of Geneva, Uni Mail, CH-1211 Geneva 4, Switzerland. Email:
[email protected].
‡
Department of Mathematics, University of Evry, France. Email: [email protected].
§
Department of Finance, HEC, Jouy-en-Josas, France. Email: [email protected].
1
Introduction
The management of radioactive waste is a very sensitive socio-political issue involving technical,
scientific, economic and ethical aspects. Although nuclear energy has already been used for many
years in several industrialized countries, discussions are still going on about what to do with
the waste produced by the nuclear fuel cycle, and more particularly with the most active and
dangerous components of the waste1 . Two main options are considered for the latter category,
surface storage and deep geological disposal. However, neither option has gained widespread
social acceptance so far, as this waste will remain a threat for several thousand years, even
though the decay in radioactivity proceeds at a faster pace for some radioisotopes than for
others (e.g., cesium 137 has a half-life of 30 years, compared with half-lives of 24,000 years for
plutonium 239 and 2.1 million years for neptunium 237). The management of such long-lived
hazardous waste represents a challenge for our society. In economic terms, it is a cost derived
from past decisions to take advantage of nuclear energy. The minimization of this cost implies the
definition of an appropriate dynamic procedure taking into account, as far as possible, the social
and scientific uncertainties involved when consequences in the very long term are considered2 .
Deep geological disposal was long considered by experts in nuclear energy as the appropriate
solution for nuclear waste. As the argument went, this would relieve future generations of any
cost associated with the management of nuclear waste. However, it has been recognized recently
that this approach does not solve all the problems. Given the time frames considered, various
risk scenarios must be taken into account. For instance, it is certain that the containers we are
able to construct today will not withstand the radioactivity of highly active waste for several
thousand years. Leakages will occur, with hazardous consequences in the presence of water
infiltrations and/or tectonic movements. Moreover, human activity (e.g., boring) may lead to
accidents, especially if deep disposal has made the population less aware of the underlying danger
(see Hériard Dubreuil et al., 1999). Thus, deep geological disposal is associated with random
future costs which should not be overlooked in the present decision process.
The alternative is surface storage, or one of its variants: subsurface storage and reversible
deep storage3 . Surface storage implies permanent monitoring of the site by present and future
generations. As such, it is a precautionary measure. It precludes accidents with catastrophic
consequences due to surprise and unpreparedness. However, surface storage also has a cost
arising from the combination of different elements: the use of surface land to store nuclear
waste, the maintenance and protection of storage sites and the accidental radiation releases
1
Part of the low-level waste is released into the environment in gaseous or liquid form, with a limited radiological
impact. Low-level waste in solid form and medium-level waste with a short half-life are stored on surface or
subsurface. The concern remains with high-level waste and with medium-level waste having a long half-life.
2
In the management of nuclear waste, the mid-term is in the range 0 to 10,000 years, the long term in the
range 10,000 to 100,000 years, and the very long term beyond 100,000 years. (See Hériard Dubreuil et al., 1999).
3
As pointed out by one referee, reversible deep storage introduces an additional cost into the picture, the cost of
reversal. However, reversible deep storage may conceptually be considered as a particular case of surface storage
since it shares the main characteristics of the latter: the trading of the opportunity to make better decisions
in the future against the burden of current maintenance costs (which may extend far into the future). Like
surface storage, reversible deep storage means active management of the waste and this cannot last forever. It is
eventually substituted by irreversible deep disposal, as the result of a conscious decision process, or as the result
of involuntary abandonment of the site. This last possibility is excluded in the present paper. A more complete
study should take this possibility into account, as well as the cost trade-off represented by the choice of surface
storage vs. reversible deep storage.
2
which may be caused by negligence or sabotage4 . For this reason it is not generally considered a
feasible permanent solution. This was recognized, for instance, in the following recommendation
of the International Atomic Energy Agency, dated 1989: ”The burden on future generations
shall be limited by implementing, at an appropriate time, a safe disposal option which does not
rely on long-term institutional control” (our italics). (IAEA, 1989; see also OECD-NEA, 1995).
In this paper, our purpose is to set up a stylized model which determines whether it is
optimal to switch from surface storage to deep geological disposal, at each point in time. The
optimal solution minimizes the expected present value of costs due to nuclear waste, including
the random costs of future unanticipated accidents (in the case of deep disposal) and the random
costs of institutional control and hazard management (in the case of surface storage).
Our problem is an optimal stopping problem, comparable to the definition of the optimal
exercise policy for a perpetual American spread option. Thus, our model fits into the real
options approach to investment under uncertainty (see Dixit and Pyndick, 1994; McDonald and
Siegel, 1986). An investment project is an option which may be exercised, or not. Even though
the project may have positive present value, given current information on future cash flows, it
may be optimal to postpone it, in order to keep alive the option to make better decisions in the
future. In our case, the current cost of radiation releases is a decreasing function of time, due
to the tendency of radioactivity to decay. It will be possible to reduce this cost in the future if
technological progress opens up the prospect of more efficient pre-disposal processing of nuclear
waste (e.g., transmutation of radioactive elements) and more resistant containers. This cost will
fall further if technological progress leads to valuable uses of some nuclear waste components in
the future. Thus, in line with the real options approach, expectations of future technological
progress may prompt the postponement of deep geological disposal. In this case, nuclear waste
is stored on the surface until deep disposal has become optimal, which arises almost surely in
our context5 . A random cost at some date in the future is traded against a random cost in
continuous time6 .
In financial theory, such a problem may be solved using the theory of dynamic optimization
under uncertainty, or risk-neutral valuation. In the case of investments in the oil industry,
for example, risk neutral valuation is used. This approach is justified by the possibility of
replicating the underlying random variables using options and futures on oil or shares in oil
companies. However, in our case, this approach is not applicable because of the lack of replicating
portfolios tracking the path of the underlying random variables. Dynamic optimization with
an exogenously specified discount rate7 should be used. It is not even possible to relate the
4
Note that this cost is random, even if large-scale accidents are precluded by continuous monitoring of the site.
In a different, but more speculative context, one could imagine that nuclear waste may become useful in the
distant future, e.g., as an input to an industrial process or a medical treatment. Then, it could be optimal to
postpone deep disposal forever.
6
Our model could be compared to previous models in the real options literature where the firm has the
flexibility to switch to less costly inputs to produce a given output (see, e.g., Kulatilaka and Trigeorgis, 1994).
However, in these models, the flexibility to switch exists on several occasions and may be analyzed as a sequence
of European put options on the minimum of two assets. In our case, the decision to switch is irreversible and the
date is unknown. The real option is unique, perpetual and American.
7
The problem of defining an appropriate discount rate for long-term public investments in safety is one of the
most vexing problems in economics today. It is often argued that uncertainty of future wealth commands a lower
discount rate and that the discount rate should tend towards zero as the horizon lengthens (see Gollier 1998,
Weitzman 1998). But Gollier (1999) showed that it is efficient to discount the future at a decreasing rate only if
relative risk aversion is decreasing and if there is no risk of recession in the future. If a recession is allowed, the
conditions on preferences of the representative agent become much more stringent. It has also been argued that
5
3
discount rate to the riskless rate and to the market price of risk, using, e.g., the CAPM. Indeed,
financial markets do not span the stochastic fluctuations of the underlying variable and there are
no financial instruments with durations corresponding to our context of long-term radioactive
decay. The adjustment for risk must take place in the financial evaluation of future consequences,
the usual procedure in the economics of uncertainty. Our approach is therefore comparable to
other approaches in the real options literature where, even with shorter horizons, the BlackScholes valuation method cannot be applied. This is the case, for example, in recent literature
on valuations of health care technologies (see Palmer and Smith, 2000).
We set up our model in section 1, and derive our main results in section 2 by drawing on
recent literature on the valuation of American spread options. Two cases are considered: the
case of zero processing costs, and the case of positive processing costs. The results of numerical
simulations illustrating possible uses of the model are presented in section 3. Section 4 shows
that our model may also be applied to choose the optimal location for deep disposal. Section 5
concludes by pointing out some directions for future applied research along the lines traced in
this paper.
1
The model
Our model is based on a general cost function to be minimized. Different costs must be considered. First, the cost of organizing deep geological disposal8 . For reasons of intergenerational
ethics, this cost is incurred initially (at time t = 0), whether deep disposal occurs at t = 0 or
later. The funds required are set aside if deep disposal is postponed. They earn a return, which
is assumed riskless. To avoid unnecessary complications, we also assume that this safe return
compensates exactly for cost increases over time. Thus, this cost is constant in real terms and
it does not affect our optimal stopping problem. Secondly, the cost of nuclear waste processing,
including (if feasible) the transmutation of radio-elements. This cost will be represented by C.
It is incurred at the deep disposal date9 . Thirdly, we have to consider the monetary cost of
an accidental radiation release once deep disposal has occurred. This cost is represented by a
random process, (St )t≥0 , assumed to follow a geometric Brownian motion:
dSt
= −µ1 dt + σ1 dWt1
St
(1)
where µ1 represents the positive drift in technical efficiency, σ1 its volatility, and both parameters
are assumed constant.
Assuming a positive drift in technical efficiency means that the efficiency of nuclear waste
processing is expected to increase in the long run. Therefore, the severity of accidental radiation
releases will decline in the long run due to more efficient pre-disposal processing. Assuming a
distant monetary and non-monetary (health) costs and benefits should be discounted differently if the value of
health grows over time (see Jones-Lee and Loomes, 1995). But these arguments are not supported by surveys of
public preferences (see Cropper et al., 1994). The present paper does not consider these issues and assumes an
exogenously specified discount rate.
8
This includes the social costs due to reluctance of the population living close to the disposal area.
9
Note, however, that this cost may be incurred once or twice. It is incurred once if deep geological disposal is
adopted initially, at time t = 0. It is also incurred once if nuclear waste processing occurs only at the future deep
disposal date. It may be incurred twice if deep disposal occurs at t > 0 but the institution in charge of waste
management decides to process the waste first for surface storage. In this case, the initial cost of surface storage,
Q0 , includes C.
4
geometric Brownian motion for St implies that we accept the possibility of random regressions in
technical know-how, even though the trend is positive. The assumption may also be justified by
invoking random changes in the monetary valuation of radiation releases. Indeed, the monetary
valuation of radiation releases incorporates direct and indirect effects. The direct effects concern
health only. They are represented by radiation-induced cancers. Their valuation is based on
the ”monetary value of the man-sievert”10 . This concept has been widely used to allocate funds
for investment in safety in nuclear power plants (see Lefaure, 1995). Its implementation is
based on an economic model which incorporates a risk aversion parameter, and uses either the
willingness-to-pay approach or the human capital approach to estimate the value of one lifeyear (see Lochard et al., 1996; Schneider et al., 1997). More recently, this model was adapted
to evaluate public exposure, taking into account lower levels of exposure and the absence of
a compensation system for radiation-induced cancers in the public (in contrast to workers).
Eeckhoudt et al. (1999) found that, depending on the relative risk aversion coefficient and on
the degree of compensation for workers’ cancers, the willingness-to-pay of workers for a given
reduction of their probability of contracting cancer should be multiplied by a factor which varies
between 2 and 6 when the model is applied to the general public11 . The indirect effects of an
accidental radiation release were evaluated in the framework of the ExternE project funded by
the European Commission (see Markandya, 1995; Markandya et al., 1998). They are mainly
represented by the decrease in or interruption of most economic activity in the affected zone,
and the induced effects in other zones. Here again, it is possible to calculate a multiplying factor
to be applied to the external cost of an accident in order to take risk perceptions into account
(see Eeckhoudt et al., 2000). Clearly, the monetary value of both direct and indirect effects is
bound to vary over time, due to economic, social and cultural considerations (including changes
in risk aversion), hence the additional justification for the random component of the process St .
Once the nuclear waste has been disposed of at time τ , we assume that the process (St )t≥τ
follows a new geometric Brownian motion:
dSt
= −αdt + Σd(Wt1 − Wτ1 )
St
(2)
In this expression, α represents the natural rate of decay in radiation activity, Σ represents the
volatility of random changes in the monetary valuation of radiation releases after disposal and
(St )t≥τ represents the monetary valuation at time t of a radiation release after processing at
time τ . It is assumed that µ1 > α which means that technological progress is expected. For
analytical convenience, we assume that the same Brownian motion Wt1 drives the stochastic
components of equations (1) and (2).
Once disposal has occurred, at time τ , an accident happens at a random time τ + T . We
assume that an accident happens only once and that the probability distribution of T is given
10
The sievert (Sv) is a unit used to quantify the health impact of radiations. For occupational context (workers
in nuclear power plants), the average exposure is 5 mSv (millisieverts) per year. For the general public, the
average dose is 0.1 mSv/year. Using the dose-effect relationship calculated by the International Commission on
Radiological Protection (ICRP) and assuming a lifetime of 75 years for the public and a working life of 45 years
for exposed workers, the risk of death due to a radiation-induced cancer (under normal conditions) is estimated
at 1% for workers and at 0.04 % for the public (see Eeckhoudt et al., 1999).
11
This is an application of the concept of multiplicative risk premium developed by Pratt (1964). According
to Eeckhoudt et al. (1999), the monetary value of the man-sievert in the case of public exposure may thus be
estimated at 75,000 to 200,000 euros.
5
by the exponential law of parameter λ:
IP (T < t) =
Z
t
λe−λs ds = 1 − e−λt .
0
The parameter λ may be adjusted to reflect the likelihood of the accident occurring before a
given future date. A lower λ means that an accident is less probable in the next few centuries.
A fourth cost to be considered is the instantaneous cost of surface storage. It is represented
by a random variable, Qt , assumed to follow a geometric Brownian motion with drift µ2 and a
volatility parameter σ2 as follows:
dQt
= µ2 dt + σ2 dWt2
Qt
(3)
The random component in the definition of Qt reflects the ”ongoing” variation in costs, including
the monetary costs of accidental radiation releases which may be caused by negligence or sabotage (see footnote 4). As before, this cost may be evaluated taking risk aversion into account.
Let us assume that (W 1 , W 2 ) is a standard Brownian motion12 and let us denote by (Ft )t≥0
the filtration generated by this Brownian motion. Moreover, T is assumed to be independent of
(F∞ ).
Solving (1), (2) and (3) with y = S0 and x = Q0 yields
1
Sty = y exp(−µ1 t + σ1 Wt1 − σ12 t) t ≤ τ
2
1
St = Sτ exp(−α(t − τ ) + Σ(Wt1 − Wτ1 ) − Σ2 (t − τ )) t ≥ τ
2
and
1
Qxt = x exp(µ2 t + σ2 Wt2 − σ22 t).
2
From the third equation it can be seen that the net marginal convenience yield from raw material
storage often met in the real options literature becomes, in this new context, the net marginal
”inconvenience yield” from the surface storage of nuclear waste. Indeed, as soon as deep disposal
has occurred, a flow of negative dividends corresponding to the cost of surface storage is avoided.
The rate of these negative dividends is the ”inconvenience yield”.
Using the exponential law and denoting the discount rate by ρ, we compute the expected
present cost of radiation release if deep disposal occurs at time τ and the accident occurs at
random time τ + T . We thus obtain
h
i
h
i
1 2
1
1
IE e−ρ(τ +T ) Sτ +T
= IE e−ρτ Sτ e−(ρ+α)T +Σ(Wτ +T −Wτ )− 2 Σ T )
h
i
1 2
1
1
= IE e−ρτ Sτ IE[e−(ρ+α)T +Σ(Wτ +T −Wτ )− 2 Σ T |Fτ ]
Z ∞
−ρτ
−(ρ+α)s+Σ(Wτ1+s −Wτ1 )− 21 Σ2 s
−λs
= IE e Sτ
IE[e
|Fτ ]λe
ds
0
Z ∞
= IE e−ρτ Sτ
e−(ρ+α)s λe−λs ds
0
λ
−ρτ
= IE e Sτ
.
λ+α+ρ
12
We could assume that W 1 and W 2 are correlated. We assume independence of Brownian motions for the
sake of convenience. The methodology developed in this paper applies to the same kind of model with correlated
Brownian motions.
6
where the third equality follows from the independence between T and the Brownian motion.
Our objective is to minimize over all stopping times τ the mean cumulative discounted cost
of deep disposal at time τ , M CDC(τ ), given by:
Z τ
λ
−ρs
−ρτ
M CDC(τ ) = IE
e Qs ds + e
C+
Sτ
λ+α+ρ
0
where the first term in the brackets represents the cost of surface storage from 0 to τ , while C
represents the processing cost when deep disposal is decided.
λ
We set Γ = λ+α+ρ
and g(y) = C + Γy in the rest of this paper. This represents the expected
cost of immediate deep disposal, with a current monetary valuation of radiation releases equal
to y and the processing cost amounting to C today. Noting that (Qt , St ) is a time-homogeneous
Markov process, our problem is reduced to the valuation of an auxiliary cost function, denoted
from here onwards by G and defined as:
Z τ
−ρs x
−ρτ
y
G(x, y) = inf IE
e Qs ds + e g(Sτ )
(4)
τ ∈[0,+∞]
0
Remark 1.1 We easily deduce from (4) the following results:
1. For every (x, y) ∈]0, +∞[2 , G(x, y) ≤ g(y).
2. Immediate deep disposal is optimal if G(x, y) = g(y).
Using the analogy with the optimal exercise of American options, we introduce the following
set, which defines the optimal deep disposal region:
DDR = (x, y) ∈]0, +∞[2 ; G(x, y) = g(y)
(5)
According to optimal stopping theory (see, for instance, Villeneuve, 1999), the optimal date for
deep disposal is the entrance time of (Qt , St ) in DDR.
We close this section with a proposition which asserts that immediate deep disposal is not
optimal if x is small enough.
Proposition 1.1 If x < ρC + (ρ + µ1 )Γy then G(x, y) < g(y).
Proof of Proposition 1.1: For every t ≥ 0,
Z t
G(x, y) ≤ IE
e−ρs Qxs ds + e−ρt g(Sty )
0
Z t
= x
e(µ2 −ρ)s ds + e−ρt (C + ΓIE(Sty ))
0
Z t
= x
e(µ2 −ρ)s ds + e−ρt (C + Γye−µ1 t )
0
= g(y) + (x − (ρC + Γ(ρ + µ1 )y) t + o(t) as t tends to zero.
Since x < ρC + Γ(ρ + µ1 )y, we make t tend to zero to conclude. 2
7
2
Characterization of the deep disposal region
An American spread option gives its holder the right, but not the obligation, to exchange a
financial asset against another financial asset (see Margrabe, 1978). Let us denote by V the
value of an American spread option with strike price C.
V (x, y) = sup IE0 e−ρτ (Qxτ − Sτy − C)+ .
(6)
τ ∈[0,+∞]
The next proposition highlights the link between G and V .
Proposition 2.1 Assume ρ > µ2 . Then, we have
x
G(x, y) =
−V
ρ − µ2
x
, Γy .
ρ − µ2
Proof of Proposition 2.1: See appendix below. Proposition 2.1. asserts that the minimum cost function G depends on the value of an
option to exchange, at a cost C, a current random surface storage cost for the prospect of a
x
future accidental radiation release. Indeed, ρ−µ
represents the total expected discounted cost
2
of surface storage if the possibility of switching to deep disposal is not available. By definition,
G is the total expected discounted cost, with the flexibility to switch optimally to deep disposal.
The difference between these costs is therefore the capital amount saved thanks to the flexibility
of switching. It is an option value. Alternatively, it may be shown that our minimization
problem may be transformed into a maximization problem similar to the problem of valuing
spread options. The function to be maximized is the expected present value of the difference
between two stochastic variables: the saving in surface storage costs (once deep disposal occurs)
minus the cost of deep disposal including the processing cost C.
Using proposition 2.1, we observe that
x
x
2
, Γy) =
− g(y)
(7)
DDR = (x, y) ∈]0, +∞[ ; V (
ρ − µ2
ρ − µ2
Our initial problem is now reduced to computing the value V of a perpetual American spread
option, if possible, or determining DDR as the exercise region if no closed form solution exists
for the value of V .
We consider two cases: zero processing cost, namely C = 0, and strictly positive processing
cost, C > 0.
2.1
Zero processing cost
Assume C = 0. In this case, V coincides with the value of a perpetual American exchange
option:
(8)
V (x, y) = sup IE e−ρτ (Qxτ − Sτy )+ .
τ ∈[0,∞]
8
The value of such an option may be computed explicitly13 . We get
V (x, y) = (x − y)1{x≥
θ̂
y}
θ̂−1
+
θ̂
x
θ̂
θ̂ − 1
y
!θ̂−1
1{x<
θ̂
y}
θ̂−1
with θ̂ = w + ∆ where
2
− µ1ν+µ
,
2
q
1)
∆ = w2 + 2(ρ+µ
,
ν2
w=
1
2
ν 2 = σ12 + σ22 .
Notice that the dependance of the exercise bound θ̂ on the volatilities is restricted to their sum
σ12 + σ22 . This fact will be crucial in the next subsection. Thus,
(
)
θ̂Γ
2
DDR = (x, y) ∈]0, +∞[ ; x ≥
(ρ − µ2 )y
θ̂ − 1
θ̂Γ
(ρ − µ2 ) which is the boundary of DDR. A straightforward comθ̂ − 1
putation (see the appendix) shows that
2
ν
λ
lim β̂(ρ) =
+ µ1 + µ2
.
ρ→µ2
2
λ + α + µ2
Let us define β̂(ρ) =
Hence it turns out that, with zero processing cost, the analogy of our problem with the valuation of American spread options leads to an explicit solution. For given parameter values, the
expected cost of waiting, G(x, y), may be computed explicitly and compared with the expected
cost of immediate deep disposal, g(y). For G(x, y) < g(y) the future date of deep disposal
cannot be predicted. However, the probability distribution of the optimal stopping time may be
determined. Let us denote
τopt = inf{t ≥ 0 | (Qxt , Sty ) ∈ DDR}
= inf{t ≥ 0 | Qxt ≥ β̂(ρ)Sty }
Set
• νBt = σ2 Wt2 − σ1 Wt1 ,
µ1 + µ2 − 21 (σ22 − σ12 )
ν
1
y
• b = log β̂(ρ) ,
ν
x
• µ=
13
See, for instance, Broadie and Detemple (1996) or Gerber and Shiu (1996).
9
we obtain immediately
τopt = inf{t ≥ 0 | Bt + µt ≥ b}.
Using the continuity of Brownian motions and assuming x < β̂(ρ)y, this reduces to
τopt = inf{t ≥ 0 | Bt + µt = b}.
From there, using the density of drifted Brownian motion (see, for instance, Karatzas and
Shreve, 1987, p.197), we can compute the probability of nuclear waste being optimally disposed
of within, say, the next century,
Z 100
(b − µt)2
b
√
] dt.
exp[−
IP (τopt ≤ 100) =
2t
2πt3
0
We compute this probability in section 3 for selected parameter values.
2.2
Positive processing cost
In this subsection, we assume C > 0. The value V of the perpetual American spread option is
given by equation (6). We shall characterize the deep disposal region given by (7) although we
are not able to explicitly compute the value function V . To investigate the properties of DDR,
we introduce the following set
E = (x, y) ∈]0, +∞[2 ; V (x, y) = x − y − C .
Thus, (x, y) ∈ DDR is equivalent to (
We proceed in two steps:
x
, Γy) ∈ E.
ρ − µ2
- in the first step, we make the processing cost stochastic in a formal probability space,
which amounts to considering an auxiliary optimal stopping problem;
- in the second step, we use the first step and the Girsanov theorem to treat the case of a
fixed positive processing cost.
2.2.1
An auxiliary stopping time problem
Consider Bt = (Bt1 , Bt2 , Bt3 ) a standard (Ft )-Brownian
motion of dimension three under some
probability measure IP1 . Define U (x, y, z) = sup IE e−ρτ (Qxτ − Sτy − Cτz )+ , with
τ ∈[0,∞]
dSt = St [−µ1 dt + σ̂1 dBt1 + σ̂3 dBt3 ]
dQt = Qt [µ2 dt + σ̂2 dBt2 + σ̂3 dBt3 ]
dCt = Ct σ̂3 dBt3
Note that the first two stochastic processes in the above equations differ from the processes
defined by equations (1) and (3).
According to Theorem 4.7 of Hu and Oksendal (1998), we have the following identity:
Ê = (x, y, z) ∈]0, ∞[3 ; U (x, y, z) = x − y − z
= (x, y, z) ∈]0, ∞[3 ; x ≥ K21 y + K23 z ,
10
where K21 and K23 are the exercise boundaries of perpetual American exchange options, the
values of which are given by
sup IE e−ρτ (Qxτ − Sτy )+ ,
τ ∈[0,∞]
sup IE e−ρτ (Qxτ − Cτz )+ .
τ ∈[0,∞]
More precisely, equations (3.6)-(3.11) in Gerber and Shiu (1996, page 306) give us
θ̂21
θ̂23
K21 =
and K23 =
with
θ̂21 − 1
θ̂23 − 1
s
1 µ1 + µ2 2
2 + 2(ρ + µ1 )
θ̂21 = w21 + ∆21 with w21 = −
, ν21 = σ̂12 + σ̂22 and ∆21 = w21
2
2
ν21
ν21
and
θ̂23 = w23 + ∆23 with w23
1
µ2 2
= −
, ν = σ̂22 and ∆23 =
2 ν23 23
s
2 +
w23
2ρ
2
ν23
The next proposition, based on a simple application of the Girsanov theorem, shows the
equivalence between our problem and the formal one when the third state variable is equal to
C.
Proposition 2.2 We have U (x, y, C) = V (x, y).
Proof of Proposition 2.2: Let
(µ2 − 12 (σ̂22 +σ̂32 ))τ +σ̂2 Bτ2 +σ̂3 Bτ3
−ρτ
(−µ1 − 21 (σ̂12 +σ̂32 ))τ +σ̂1 Bτ1 +σ̂3 Bτ3
− 12 σ̂32 τ +σ̂3 Bτ3
A = IEIP1 e
xe
− ye
− Ce
+
1
1
1 2
3
2
2
2
1
= IEIP1 e−ρτ e− 2 σ̂3 τ +σ̂3 Bτ xe(µ2 − 2 σ̂2 )τ +σ̂2 Bτ − ye(−µ1 − 2 σ̂1 )τ +σ̂1 Bτ − C
+
Let us define IP by
1 2
dIP
3
= e− 2 σ̂3 t+σ̂3 Bt .
dIP1 |Ft
According to the Girsanov theorem, (Wt1 , Wt2 , Wt3 ) = (Bt1 , Bt2 , Bt3 − σ̂3 t) is a IP -Brownian motion
with respect to Ft . Set σ1 = σ̂1 and σ2 = σ̂2 to obtain
−ρτ
(µ2 − 21 σ22 )τ +σ2 Wτ2
(−µ1 − 12 σ12 )τ +σ1 Wτ1
A = IE e
xe
− ye
−C
.
+
Taking the supremum over all stopping times allows us to conclude. Therefore, remembering the definition of E, we get
E = (x, y) ∈]0, ∞[2 ; V (x, y) = x − y − C
= Ê ∩ {z = C}
= (x, y) ∈]0, ∞[2 ; x ≥ K21 y + K23 C .
11
From there, we obtain
DDR =
x
(x, y) ∈]0, ∞[ ; (
, Γy) ∈ E
ρ − µ2
2
(x, y) ∈]0, ∞[2 ; x ≥ (ρ − µ2 )(K21 Γy + K23 C)
= (x, y) ∈]0, ∞[2 ; x ≥ β1∗ (ρ)y + β2∗ (ρ)C
=
where β1∗ (ρ) = (ρ − µ2 )K21 Γ and β2∗ (ρ) = (ρ − µ2 )K23 .
2.2.2
Fixed processing cost
For fixed processing costs, we start from an artificial model as in section 2.2.1, such that σ1 = σ̂1
and σ2 = σ̂2 . It should be pointed out that the coefficients K21 and K23 defining the exercise
boundaries are independent of this artificial extension. Thus,
x
2
DDR =
(x, y) ∈]0, ∞[ ; (
, Γy) ∈ E
ρ − µ2
= (x, y) ∈]0, ∞[2 ; x ≥ (ρ − µ2 )(K21 Γy + K23 C)
= (x, y) ∈]0, ∞[2 ; x ≥ β0∗ (ρ) + β1∗ (ρ)y ,
where,
β0∗ (ρ) = (ρ − µ2 )K23 C and β1∗ (ρ) = (ρ − µ2 )K21 Γ.
As in Section 2.1, we determine the DDR when ρ tends to µ2 . We have
2
σ2
∗
+ µ2 C,
lim β (ρ) =
ρ→µ2 0
2
and
lim β ∗ (ρ)
ρ→µ2 1
=
σ12
+ µ1 + µ2
2
λ
.
λ + α + µ2
Thus, when the processing cost C is positive, the minimum expected cost G(x, y) cannot
be made explicit. However, the deep disposal region may be determined. For given values
of the parameters, our model allows us to say whether it is optimal to dispose of the nuclear
waste immediately, or whether it is preferable to wait. Note that, in this case, the probability
distribution of optimal stopping times may not be determined explicitly, although it is still
possible to compute the probability of being in the deep disposal region at any future date.
3
Numerical results
The objective of this section is to provide an illustration of our model with some numerical
simulations. We set the following parameters: µ2 = 0.001; σ1 = σ2 = 0.1; α = 0.00005, λ = 0.5
and we observe how the deep disposal region and the probability of the nuclear waste being
disposed of in a given time span vary depending on the more relevant parameters ρ, µ1 , x and
y. We consider two cases: zero processing cost (C = 0) and fixed processing cost (C = 100)14 .
14
We emphasize that these simulations are for illustration only. Future research along these lines will have to
proceed from estimations of actual parameter values.
12
3.1
Zero processing cost: C = 0.
Figure 1 shows how the minimum cost function G(x, y) varies with the initial cost of surface
storage Q0 = x and the initial monetary valuation of an accidental radiation release S0 = y. x
and y are both taken to vary between 0 and 100. We choose ρ = µ1 = 0.01. For the unrealistic
case y = 0, G(x, 0) = g(0) = 0, immediate deep disposal is obviously optimal.
’value.data’
G(Q(0),S(0))
100
90
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
Q(0)
70
80
90
100 0
10
20
30
40
50
60
70
80
S(0)
Figure 1: Minimum cost function G when C = 0
Figure 2 shows how the probability of the waste being disposed of within the next century
varies with ρ when we fix y = 10x and µ1 = 0.01. It appears that this probability is first unity
and is then a decreasing function of ρ. This may be surprising given that a higher discount rate
reduces the influence of future accidental costs. However, a higher discount rate also reduces the
influence of the cumulative discounted surface storage costs. Moreover, given the high value of
λ, the accident may occur early. When ρ, which can be viewed as the rate of preference for the
present, increases, the decision may be postponed because present continuous costs associated
with surface storage could be lower than short-term discrete costs due to waste processing and a
random accident. The role of λ is confirmed by an additional simulation (not displayed): if λ is
very low (λ = 5 × 10−5 ), the probability of the waste being disposed of within the next century
13
90
100
is unity for any value of ρ.
probability
1
0.95
0.9
0
0.05
0.1
0.15
0.2
discount factor
Figure 2: Probability of the waste being disposed of within the next century as a function of
the discount rate ρ when C = 0.
14
3.2
Fixed processing cost, C = 100
Figure 3 may be compared to Figure 1. The minimum cost function G is a non-decreasing and
concave function of x. The graph illustrates Proposition 1.1: when x is small enough G(x, y)
tends to zero and it is optimal to wait.
’value2.data’
G(Q(0),S(0))
200
180
160
140
120
100
80
60
40
20
0
0
10
20
30
40
50
60
Q(0)
70
80
90
100 0
Figure 3: Minimum cost function G when C = 100
15
10
20
30
40
50
60
70
80
90
S(0)
100
Figure 4 is comparable to Figure 2. The comparison illustrates the impact of the processing
cost. For convenience we reported on the same figure the probability of the waste being disposed
of within the next century as a function of ρ when C = 0 (upper graph) and when C = 100 (lower
graph). In both cases, y = 10x and µ1 = 0.01. It appears that the probability declines more
rapidly when C is positive. For a discount rate of 10 per cent, the probability is approximately
0.85, instead of 0.95 when C = 0. This illustrates the fact that a non-random future cost tends
to be postponed when the discount rate increases.
1
zero processing cost
positive processing cost (C=100)
Probabilities
0.95
0.9
0.85
0
0.05
0.1
0.15
0.2
Discount factor
Figure 4: Probability of the waste being disposed of within the next century as a function of
the discount rate ρ when C = 0 and C = 100.
16
Figure 5 displays the probability of the waste being disposed of as a function of µ1 when
ρ = 0.01 and y = 10x. This probability is a non-decreasing function of µ1 . This illustrates the
fact that, with the given parameters, a 10 per cent rate of technological progress is needed to
be almost sure that the waste is disposed of optimally in the next century.
0.95
probability
0.94
0.93
0.92
0.91
0.9
0
0.05
0.1
0.15
0.2
expected rate of increase in technical efficiency
Figure 5: Probability of the waste being disposed of within the next century as a function of µ1
when ρ = 0.01 and C = 100.
17
4
Extension: The case of optimal location
Our model can be extended as follows. Consider the problem of finding, not only the optimal
time to dispose of the nuclear waste but also the optimal location to perform this task. Indeed,
let us assume that there are n possible locations (regions, countries), characterized by different
sets of parameters. For location i, i = 1, . . . , n, we have for instance an initial cost of surface
disposal xi , an initial monetary cost of accidental releases yi , processing costs Ci and an intensity
parameter λi for the exponential law. For any location i, we are able to compute the minimum
cost of delayed deep disposal Gi (xi , yi ), the cost of immediate deep disposal gi (yi ) and/or to
determine the deep disposal region associated with this location.
Let us define the set I = {i ∈ {0, . . . , n} , Gi (Qi0 , S0i ) = gi (S0i )} as the set of locations where
immediate deep disposal is optimal. Among the integers i which belong to I, it is optimal to
choose the one which minimizes the cost of immediate deep disposal gi (yi ). If ever the set I
is empty, we are able to compute the probability of the nuclear waste being disposed of in any
location i within, e.g., the next century. We may then focus on those for which the probability
exceeds some arbitrary level.
5
Conclusion
Long-term risk management of nuclear waste is a very important social issue. Decision-making
on what to do with the waste produced by the nuclear fuel cycle is made difficult by the uncertainties concerning, inter alia, the development of our societies, the pace of technological progress
and the social acceptance of radiation risk in the coming centuries. But right now, as stated
by Croüail et al. (2000): ”a responsible attitude implies using as best as we can all available
information on the possible consequences of our present actions, even if this information does
reflect the lack of knowledge in the assessment of consequences far in the future”. The model
proposed in this paper starts from this premise. It addresses the deep disposal of nuclear waste
and provides a decision-making tool to help determine not only ”when” deep disposal becomes
optimal, but also ”where and for how much” deep disposal will have to be performed. We have
used the analogy between this problem and the financial valuation of perpetual American spread
options. The American spread option value is obtained by maximizing the expected value of
the discounted difference between the prices of two assets minus the strike price for all exercise
dates. Similarly, the optimal decision to opt for immediate deep disposal of nuclear waste or
not is obtained, after some technical transformations, by maximizing the expected value of the
discounted difference between two stochastic variables: the saving in surface storage cost (by
switching to deep disposal), and the cost of deep disposal. The processing cost at the disposal
date plays the same role as a strike price.
From a methodological point of view, option valuation is often based on the equivalent martingale measure and the riskless interest rate. Our model uses risk-adjusted monetary equivalents
and an appropriate discount rate for the very long term. We show how the problem may be
solved and we provide some numerical illustrations based on hypothetical parameter values.
However, practical applications will have to feed the model with more realistic parameter values. For this reason, additional numerical simulations are an obvious extension of our research.
Extensions may also be considered at the theoretical level. For instance, our model assumes a
given stock of nuclear waste, to be managed optimally. It would be stimulating to set up a model
18
analyzing the optimal risk management of a flow of nuclear waste. In addition, the exponential
distribution assumed for the accident date implies that the probability of incurring an accident
in a time interval ∆T decreases with time. This feature may be considered unrealistic if oblivion
is a credible source of future accidents. Other probability distributions may be assumed.
Appendix: Mathematical proofs
We set φ+ = max(0, φ) and φ− = (−φ)+ . We first prove the following lemma.
2 → R a continuous function satisfying the following conditions:
Lemma 5.1 Let φ : R+
p
2
1) ∃ K0 , K1 > 0, ∀(x, y) ∈ R+
|φ(x, y)| ≤ K0 x2 + y 2 + K1 .
2 such that φ(x , y ) > 0.
2) ∃ (x0 , y0 ) ∈ R+
0 0
−ρt −
3) lim IE e φ (Qt , St ) = 0.
t→+∞
We have
sup IE e−ρτ φ+ (Qxτ , Sτy ) = sup IE e−ρτ φ(Qxτ , Sτy ) .
τ ∈[0,∞]
τ ∈[0,∞]
Remark 5.1 The function φ(x, y) = x − y − C satisfies the conditions of lemma 5.1. Indeed,
√ p
1) |φ(x, y)| ≤ 2 x2 + y 2 + C.
2) φ(1 + C, 0) = 1
3) φ− (x, y) ≤ C thus lim IE e−ρt φ− (Qt , St ) = 0.
t→+∞
Proof of lemma 5.1: Clearly,
sup IE e−ρτ φ+ (Qxτ , Sτy ) ≥ sup IE e−ρτ φ(Qxτ , Sτy ) .
τ ∈[0,∞]
τ ∈[0,∞]
To show the converse inequality, we set
v(t, x, y) = sup IE e−ρτ φ+ (Qxτ , Sτy ) .
τ ∈[0,t]
2.
Using condition (2), we have v(t, x, y) > 0 for every (t, x, y) ∈]0, +∞[×IR+
Let us introduce the following stopping time:
τt∗ = inf{s ∈ [0, t] ; v(t − s, Qs , Ss ) = φ+ (Qs , Ss )}.
According to optimal stopping theory, τt∗ is an optimal stopping time. Thus,
φ+ (Qτt∗ , Sτt∗ ) = v(t − τt∗ , Qτt∗ , Sτt∗ ) on {τt∗ < t}
= φ(Qτt∗ , Sτt∗ )
19
since v is a positive function.
Thus,
i
h
∗
v(t, x, y) = IE e−ρτt φ+ (Qxτt∗ , Sτy∗ )
t
h
i
h
i
∗
y
−ρτt∗
x
= IE e
φ(Qτt∗ , Sτ ∗ )1{τt∗ <t} + IE e−ρτt φ+ (Qxτt∗ , Sτy∗ )1{τt∗ =t} .
t
t
As φ+ = φ + φ− , we have,
h
i
h
i
∗
v(t, x, y) = IE e−ρτt φ(Qxτt∗ , Sτy∗ ) + IE e−ρt φ− (Qt , St )1{τt∗ =t}
t
−ρτ
≤
sup IE e φ(Qxτ , Sτy ) + IE e−ρt φ− (Qt , St ) .
τ ∈[0,∞]
Let t tend to infinity to conclude. 2
Proof of Proposition 2.1: Since ρ > µ2 , the process
Z t
−ρt x
Mt = e Qt + (ρ − µ2 )
e−ρs Qxs ds
0
is a positive martingale for 0 ≤ t < ∞ which admits an integrable last element M∞ . Indeed,
Z ∞
−ρs x
IE(M∞ ) = (ρ − µ2 )IE
e Qs ds = x.
0
Therefore, (Mt )0≤t≤∞ is a martingale.
The optional sampling theorem (see Karatzas and Shreve, 1987, Theorem 3.22, p.19) yields:
IE[Mτ ] = x for every stopping time τ ∈ [0, ∞].
Hence,
Z τ
x − IE [e−ρτ Qxτ ]
−ρs x
e Qs ds =
.
IE
ρ − µ2
0
Thus,
IE
Z
0
Noting that
τ
e−ρs Qxs ds + e−ρτ (C + ΓSτy ) =
Qxτ
x
− IE e−ρτ
− (C + ΓSτy ) .
ρ − µ2
ρ − µ2
Qxτ
2
= Qx/ρ−µ
and ΓSτy = SτΓy , we obtain
τ
ρ − µ2
h
i
x
Γy
2
G(x, y) =
− sup IE e−ρτ (Qx/ρ−µ
−
S
−
C)
.
τ
τ
ρ − µ2 τ ∈[0,∞]
Lemma 5.1 and Remark 5.1 allow us to conclude that
x
x
G(x, y) =
−V
, Γy . ρ − µ2
ρ − µ2
20
Computation of lim β̂(ρ).
ρ→µ2
Set ρ = µ2 + ε with ε > 0. As θ̂ = w + ∆, we get
s
1 µ1 + µ2 2
µ1 + µ2
ε
1 µ1 + µ2
−
+
2
+
2
−
+
θ̂ − 1 =
2
ν2
ν2
ν2
2
ν2
s
1 µ1 + µ2 2
ε
1 µ1 + µ2
=
+
+2 2 −
+
2
ν2
ν
2
ν2
Now, for every real number a, b,
if we put a =
1
2
+
µ1 +µ2
ν2
√
and b =
a2 + 2bε − a is equivalent to
1
,
ν2
b
ε
a2
as ε tends to zero. Therefore,
2
we obtain that θ̂ − 1 is equivalent to ν2 + µ1 + µ2 /ε
which ends the computation of lim β̂(ρ). ρ→µ2
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