EXPLICIT SOLUTION SIMULATION METHOD
FOR THE 3/2 MODEL
IRO RENE´ KOUARFATE, MICHAEL A. KOURITZIN, AND ANNE MACKAY
Abstract. An explicit weak solution for the 3/2 stochastic volatility model is
obtained and used to develop a simulation algorithm for option pricing purposes.
The 3/2 model is a non-affine stochastic volatility model whose variance process is
the inverse of a CIR process. This property is exploited here to obtain an explicit
weak solution, similarly to Kouritzin (2018). A simulation algorithm based on
this solution is proposed and tested via numerical examples. The performance of
the resulting pricing algorithm is comparable to that of other popular simulation
algorithms.
1. Introduction
Recent work by Kouritzin (2018) shows that it is possible to obtain an explicit
weak solution for the Heston model, and that this solution can be used to simulate
asset prices efficiently. Exploiting the form of the weak solution, which naturally
leads to importance sampling, Kouritzin and MacKay (2020) suggest the use of
sequential sampling algorithms to reduce the variance of the estimator, inspired by
the particle filtering literature. Herein, we show that the main results of Kouritzin
(2018) can easily be adapted to the 3/2 stochastic volatility model and thus be used
to develop an efficient simulation algorithm that can be used to price exotic options.
The 3/2 model is a non-affine stochastic volatility model whose analytical tractabil-
ity was studied in Heston (1997) and Lewis (2000). A similar process was used
in Ahn and Gao (1999) to model stochastic interest rates. Non-affine stochastic
volatility models have been shown to provide a good fit to empirical market data,
sometimes better than some affine volatility models; see Bakshi et al. (2006) and
the references provided in the literature review section of Zheng and Zeng (2016).
The 3/2 model in particular is preferred by Carr and Sun (2007) as it naturally
emerges from consistency requirements in their proposed framework, which models
the variance swap rate directly.
As a result of the empirical evidence in its favor, and because of its analytical
tractability, the 3/2 model has gained traction in the academic literature over the
past decade. In particular, Itkin and Carr (2010) prices volatility swaps and option
on swaps for a class of Levy models with stochastic time change and uses the 3/2
Key words and phrases. 3/2 model, explicit solutions, weak solutions, stochastic volatility,
Monte Carlo simulations, option pricing, non-affine volatility.
Partial funding in support of this work was provided by an FRQNT grant and by NSERC
discovery grants.
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2 I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY
model as a particular case. The 3/2 model also allows for analytical expressions
for price different volatility derivatives; see for example Drimus (2012), Goard and
Mazur (2013) and Yuen et al. (2015). Chan and Platen (2015) considers the 3/2
model for pricing long-dated variance swaps under the real world measure. Zheng
and Zeng (2016) obtains a closed-form partial transform of a relevant density and
uses it to price variance swaps and timer options. In Grasselli (2017), the 3/2 model
is combined with the Heston model to create the new 4/2 model.
For the 3/2 model’s growing popularity, there are very few papers that focus on its
simulation. One of them is Baldeaux (2012), who adapts the method of Broadie and
Kaya (2006) to the 3/2 model and suggests variance reduction techniques. The ca-
pacity to simulate price and volatility paths from a given market model is necessary
in many situations, from pricing exotic derivatives to developing hedging strategies
and assessing risk. The relatively small size of the literature about simulating the
3/2 model could be due to its similarity with the Heston model, which allows for
easy transfer of the methods developed for the Heston model to the 3/2 one. In-
deed, the 3/2 model is closely linked to the Heston model; the stochastic process
governing the variance of the asset price in the 3/2 is the inverse of a square-root
process, that is, the inverse of the variance process under Heston.
This link between the Heston and the 3/2 model motivates the present work;
Kouritzin (2018) mentions that his method cannot survive the spot volatility reach-
ing 0. Since the volatility in the 3/2 model is given by the inverse of a “Heston
volatility” (that is, the inverse of a square-root process), it is necessary to restrict
the volatility parameters in such a way that the Feller condition is met, in order
to keep the spot volatility from exploding. In other words, by definition of the
3/2 model, the variance process satisfies the Feller condition, which makes it per-
fectly suitable to the application of the explicit weak solution simulation methods
of Kouritzin (2018).
It is also worthwhile to note that Kouritzin and MacKay (2020) notice that the
resulting simulation algorithm performs better when the Heston parameters keep
the variance process further from 0. It is reasonable to expect that calibrating the
3/2 model to market data give such parameters, since they would keep the variance
process (i.e. the inverse of the Heston variance) from reaching very high values.
This insight further motivates our work, in which we adapt the method of Kouritzin
(2018) to the 3/2 model.
As stated above, many simulation methods for the Heston model can readily be
applied to the 3/2 model. Most of these methods can be divided into two categories;
the first type of simulation schemes relies on discretizing the spot variance and the
log-price process. Such methods are typically fast, but the discretization induces a
bias which needs to be addressed, see Lord et al. (2010) for a good overview. Broadie
and Kaya (2006) proposed an exact simulation scheme which relies on transition
density of the variance process and an inversion of the Fourier transform of the
integrated variance. While exact, this method is slow, and has thus prompted several
authors to propose approximations and modifications to the original algorithm to
EXPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 3
speed it up (see for example Andersen (2007)). Be´gin et al. (2015) offers a good
review of many existing simulation methods for the Heston model.
The simulation scheme proposed by Kouritzin (2018) for the Heston model relies
on an explicit weak solution for the stochastic differential equation (SDE) describing
the Heston model. This result leads to a simulation and option pricing algorithm
which is akin to importance sampling. Each path is simulated using an artificial
probability measure, called the reference measure, under which exact simulation is
possible and fast. The importance sampling price estimator is calculated under the
pricing measure by multiplying the appropriate payoff (a function of the simulated
asset price and volatility paths) by a likelihood, which weights each payoff pro-
portionally to the likelihood that the associated path generated from the reference
measure could have come from the pricing measure. The likelihood used as a weight
in the importance sampling estimator is a deterministic function of the simulated
variance process, and is thus easy to compute. The resulting pricing algorithm has
been shown to be fast and to avoid the problems resulting from discretization of the
variance process.
In this paper, we develop a similar method for the 3/2 model by first obtaining a
weak explicit solution for the two-dimensional SDE. We use this solution to develop
an option price importance estimator, as well as a simulation and option pricing
algorithm. Our numerical experiments show that our new algorithm performs at
least as well as other popular algorithms from the literature. We find that the
parametrization of the model impacts the performance of the algorithm.
The paper is organized as follows. Section 2 contains a detailed presentation of
the 3/2 model as well as our main result. Our pricing algorithm is introduced in
Section 3, in which we also outline existing simulation techniques, which we use in
our numerical experiments. The results of these experiments are given in Section 4,
and Section 5 concludes.
2. Setting and main results
We consider a probability space (Ω,F ,P), where P denotes a pre-determined risk-
neutral measure1 for the 3/2 model. The dynamics of the stock price under this
chosen risk-neutral measure are represented by a two-dimensional process (S, V ) =
{(St, Vt), t ≥ 0} satisfying{
dSt = rSt dt+
√
VtStρ dW
(1)
t +
√
VtSt
√
1− ρ2 dW (2)t
dVt = κ Vt(θ − Vt) dt+ εV 3/2t dW (1)t ,
(2.1)
with S0 = s0 > 0 and V0 = v0 > 0, and where W = {(W (1)t ,W (2)t ), t ≥ 0} is a two-
dimensional uncorrelated Brownian motion, r, κ, θ and ε are constants satisfying
κ > − ε2
2
, and ρ ∈ [−1, 1]. The drift parameter r represents the risk-free rate and ρ
represents the correlation between the stock price S and its volatility V .
1Since our goal in this work is to develop pricing algorithms, we only consider the risk-neutral
measure used for pricing.
4 I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY
The restriction κ > − ε2
2
imposed on the parameters keeps the variance process
from exploding. This property becomes clear when studying the process U = {Ut, t ≥ 0}
defined by Ut =
1
Vt
for t ≥ 0. Indeed, it follows from Itoˆ’s lemma that
dUt = κθ
(
κ+ ε2
κθ
Ut
)
dt− ε
√
Ut dW
(1)
t
= κ˜(θ˜ − Ut) dt+ ε˜
√
Ut dW
(1)
t
where κ˜ = κθ, θ˜ = κ+ε
2
κθ
and ε˜ = −ε. In other words, with the restriction κ > − ε2
2
, U
is a square-root process satisfying the Feller condition κ˜θ˜ > ε˜
2
2
, so that P(Ut > 0) = 1
for all t ≥ 0.
In order to use results obtained for the Heston model and adapt them to the 3/2
model, we express (2.1) in terms of the inverse of the variance process, U , as follows{
dSt = rSt dt+
√
U−1t Stρ dW
(1)
t +
√
U−1t St
√
1− ρ2 dW (2)t
dUt = κ˜(θ˜ − Ut) dt+ ε˜
√
Ut dW
(1)
t ,
(2.2)
with S0 = s0 and U0 = 1/v0.
Although U is a square-root process, (2.2) is of course not equivalent to the Heston
model. Indeed, in the Heston model, it is the diffusion term of S, rather than its
inverse, that follows a square-root process. However, the ideas of Kouritzin (2018)
can be exploited to obtain an explicit weak solution to (2.2), which will in turn be
used to simulate the process.
It is well-known (see for example Hanson (2010)) that if n := 4κ˜θ˜
ε˜2
is an integer,
the square-root process U is equal in distribution to the sum of n squared Ornstein-
Uhlenbeck processes. Proposition 1 below relies on this result.
Proposition 1. Suppose that n = 4κ˜θ˜
ε˜2
∈ N+ and let W (2), Z(1), . . . , Z(n) be indepen-
dent standard Brownian motions on some probability space (Ω,F ,P). For t ≥ 0,
define
St = s0 exp
{
ρ
ε˜
log
(
Ut
U0
)
+
(
r +
ρκ˜
ε˜
)
t
−
(
ρ
ε˜
(
κ˜θ˜ − ε˜2/2
)
+
1
2
)∫ t
0
U−1s ds+
√
1− ρ2
∫ t
0
√
U−1s dW
(2)
s
}
,
Ut =
n∑
i=1
(
Y
(i)
t
)2
where
Y
(i)
t =
ε˜
2
∫ t
0
e−
κ˜
2
(t−u) dZ(i)u + e
− κ˜
2
tY
(i)
0 , with Y0 =
√
U0/n
EXPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 5
and
W
(1)
t =
n∑
i=1
∫ t
0
Y
(i)
u√∑n
j=1(Y
(j)
u )2
dZ(i)u .
Let X = (S, U), W = (W (1),W (2)) and let {Ft}t≥0 be the augmented filtration
generated by (W (2), Z(1), . . . , Z(n)). Then,
• W (1) is a standard Brownian motion, and
• (X,W ), (Ω,F ,P), {Ft}t≥0 is a weak solution to (2.2).
Proof. We first observe that Y (i), i ∈ {1, . . . , n}, are independent Ornstein-Uhlenbeck
processes, and that by Le´vy’s characterization, W (1) is a Brownian motion. It follows
from an application of Itoˆ’s lemma that
dUt =
n∑
i=1
(
ε˜2
4
− κ˜(Y (i)t )2
)
dt+ ε˜Y
(i)
t dZ
(i)
t
=
(
nε˜2
4
− κ˜
n∑
i=1
(Y
(i)
t )
2
)
dt+ ε˜
n∑
i=1
Y
(i)
t dZ
(i)
t
=
(
nε˜2
4
− κ˜Ut
)
dt+ ε˜
√
Ut dW
(1)
t , (2.3)
where the equality is obtained by multiplying and dividing the second term on the
right-hand side by
√∑n
j=1(Y
(i)
t )
2. Here, since we work under the assumption that
n = 4κ˜θ˜
˜2
, (2.3) can be re-written as
dUt = κ˜
(
θ˜ − Ut
)
dt+ ε˜
√
Ut dW
(1)
t .
An application of Itoˆ’s lemma to St completes the proof.
An alternative, systematic way to verify the functional form for St that avoids
our Itoˆ-lemma-based guess and verify technique can be found in Kouritzin (2018).
It is likely that for a given market calibration of the 3/2 model, n = 4κ˜θ˜
˜2
is not
an integer. For this reason, a more general result is needed to develop a simulation
algorithm based on an explicit weak solution.
We generalize the definition of n and let n = max
(
b4κ˜θ˜
ε˜2
+ 1
2
c, 1
)
. We further
define θ˜n by
θ˜n =
nε˜2
4κ˜
.
It follows that κ˜θ˜n =
nε˜2
4
.
While U above cannot hit 0 under the Feller condition, it can get arbitrarily
close, causing U−1t to blow up. To go beyond the case
4κ˜θ˜
˜2
∈ N, we want to change
measures, which is facilitated by stopping U from approaching zero.
6 I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY
Given a filtered probability space (Ω,F , {Ft}t≥0, P̂) with independent Brownian
motions Z(1), . . . , Z(n) andW (2), and a fixed δ > 0, we can define (Ŝ, Û) = {(Ŝt, Ût)}t≥0
by
Ŝt = s0 exp
{
ρ
ε˜
log(Ût/U0) +
(
r +
ρκ˜
ε˜
)
t
−
(
ρ
ε˜
(
κ˜θ˜ − ε˜2/2
)
+
1
2
)∫ t
0
Û−1s ds+
√
1− ρ2
∫ t
0
Û−1/2s dW
(2)
s
}
,
(2.4)
Ût =
n∑
i=1
(Y
(i)
t )
2, (2.5)
and τδ = inf{t ≥ 0 : Ût ≤ δ}, where
Y
(i)
t =
ε˜
2
∫ t∧τδ
0
e−
κ˜
2
(t−u) dZ(i)u + e
− κ˜
2
tY
(i)
0 , with Y0 =
√
U0/n (2.6)
for i ∈ {1, . . . , n}.
Theorem 1, to follow immediately, shows that it is possible to construct a prob-
ability measure on (Ω,F) under which (Ŝ, Û) satisfies (2.2) until Û drops below a
pre-determined threshold δ.
Theorem 1. Let (Ω,F , {Ft}t≥0, P̂) be a filtered probability space on which Z(1), . . . , Z(n),W (2)
are independent Brownian motions. Let (Ŝ, Û) be defined as in (2.4) and (2.5) and
let τδ = inf{t ≥ 0 : Ût ≤ δ} for some δ ∈ (0, 1). Define
L̂
(δ)
t = exp
− κ˜(θ˜n − θ˜)ε˜
∫ t
0
Û−1/2v dŴ
(1)
v −
κ˜2
2
(
θ˜n − θ˜
ε˜
)2 ∫ t
0
Û−1v dv
 (2.7)
with
Ŵ
(1)
t =
n∑
i=1
∫ t
0
Y
(i)
u√∑n
j=1(Y
(j)
u )2
dZ(i)u (2.8)
and Pδ(A) = Ê[1AL̂(δ)T ] ∀A ∈ FT for T > 0.
Then, under the probability measure Pδ, (W (1),W (2)), where
W
(1)
t = Ŵ
(1)
t + κ˜
θ˜ − θ˜n
ε˜
∫ t∧τδ
0
Û−1/2s ds,
EXPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 7
are independent Brownian motions and (Ŝ, Û) satisfies
dŜt =
{
rŜt dt+ Û
−1/2
t Ŝtρ dW
(1)
t + Û
−1/2
t Ŝt
√
1− ρ2 dW (2)t , t ≤ τδ
rδŜt dt+ σδŜt dW
(2)
t , t > τδ,
dÛt =
{
κ˜(θ˜ − Ût) dt+ ε˜Û1/2t dW (1)t , t ≤ τδ
0, t > τδ
(2.9)
on [0, T ], with
rδ = r +
ρ
2ε˜δ
(
2κ˜δ − 2κ˜δ + ε˜2 − ρε˜2) ,
σδ =
√
1− ρ2
δ
.
Proof. Let D = S(R2), the rapidly decreasing functions. They separate points
and are closed under multiplication so they separate Borel probability measures
(see Blount and Kouritzin (2010)) and hence are a reasonable martingale problem
domain.
To show that (X̂,W ), (Ω,F ,Pδ), {F̂t}t≥0, with X̂ = (Ŝ, Û) andW = (W (1),W (2)),
is a solution to (2.9), we show that it solves the martingale problem associated with
the linear operator
Atf(s, u) =
(
rsfs(s, u) + κ˜(θ˜ − u)fu(s, u) + 1
2
s2u−1fss(s, u) + ρε˜sfsu(s, u)
+
1
2
ε2ufuu(s, u)
)
1[0,τδ](t) +
(
rδsfs(s, u) +
1− ρ2
2δ2
fsss
2
)
1[τδ,T ](t)
where fs =
∂f(s,u)
∂s
, fu =
∂f(s,u)
∂u
, fss =
∂2f(s,u)
∂s2
, fuu =
∂2f(s,u)
∂u2
and fsu =
∂2f(s,u)
∂s∂u
. That
is, we show that for any function f ∈ D, the process
Mt(f) = f(Ŝt, Ût)− f(Ŝ0, Û0)−
∫ t
0
(Asf)(Ŝv, Ûv) dv,
is a continuous, local martingale.
First, we note by (2.4), (2.5), (2.6) as well as Itoˆ’s lemma that (Ŝ, Û) satisfies a
two-dimensional SDE similar to the 3/2 model (2.2), but with parameters κ, θn, rδ
and r̂t = r− κ˜ρε˜ (θ˜− θ˜n)Û−1t . That is, ((Ŝ, Û), Ŵ ), (Ω,F , P̂), {F̂t}t≥0, where {F̂t}t≥0
is the augmented filtration generated by (Z1, . . . , Zn,W
(2)), is a solution to
dŜt =
{
r̂tŜt dt+ Û
−1/2
t Ŝtρ dŴ
(1)
t + Û
−1/2
t Ŝt
√
1− ρ2 dW (2)t , t ≤ τδ,
rδŜt dt+ σδŜt dW
(2)
t , t > τδ,
(2.10)
dÛt =
{
κ˜(θ˜n − Ût) dt+ ε˜Û1/2t dŴ (1)t , t ≤ τδ,
0, t > τδ.
(2.11)
8 I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY
with Ŝ0 = s0, Û0 = 1/v0 and Ŵ
(1) defined by (2.8). It follows that for any function
f ∈ D,
df(Ŝt, Ût) = Ltf(Ŝt, Ût)dt
+
(
ρŜtÛ
−1/2
t fs(Ŝt, Ût) + ε˜Û
1/2
t fu(Ŝt, Ût)
)
1[0,τδ](t) dŴ
(1)
t
+
(
Û
−1/2
t 1[0,τδ](t) + δ
−1/2
1[τδ,T ]
)√
1− ρ2Ŝtfs(Ŝt, Ût) dW (2)t ,
(2.12)
where the linear operator L is defined by
Ltf(s, u) =
(
r̂tsfs(s, u) + κ˜(θ˜n − u)fu(s, u) + 1
2
s2u−1fss(s, u) + ρε˜sfsu(s, u)
+
1
2
ε2ufuu(s, u)
)
1[0,τδ](t) +
(
rδsfs(s, u) +
1− ρ2
2δ2
fsss
2
)
1[τδ,T ](t)
(2.13)
We observe that L̂
(δ)
t satisfies the Novikov condition, since by definition of Ût,
|κ˜(θ˜n − θ˜)|2
ε˜2Ût
≤ |κ˜(θ˜n − θ˜)|
2
ε˜2δ
,
P̂-a.s. for all t ≥ 0. It follows that L̂(δ)t is a martingale and that Pδ is a probability
measure.
We also have from (2.7) and (2.12) that for f(s, u) ∈ C2([0,∞]2),[
L̂(δ), f(Ŝ, Û)
]
t
=
∫ t∧τδ
0
L̂(δ)v
(
(r − r̂v)Ŝvfs(Ŝv, Ûv) + κ˜(θ˜ − θ˜n)fu(Ŝv, Ûv)
)
dv.
(2.14)
Next we define the process M̂(f) for any f ∈ D by
M̂t(f) = L̂
(δ)
t f(Ŝt, Ût)− L̂(δ)0 f(Ŝ0, Û0)−
∫ t
0
L̂(δ)v Avf(Ŝv, Ûv) dv
= L̂
(δ)
t f(Ŝt, Ût)− L̂(δ)0 f(Ŝ0, Û0)−
[
L̂(δ), f(Ŝ, Û)
]
t
−
∫ t
0
L̂(δ)v L(δ)v f(Ŝv, Ûv) dv
(2.15)
Using integration by parts, we obtain
M̂t(f) =
∫ t
0
L̂(δ)v df(Ŝv, Ûv) dv +
∫ t
0
f(Ŝv, Ûv) dL̂
(δ)
v −
∫ t
0
L(δ)v L(δ)v f(Ŝv, Ûv) dv
=
∫ t
0
L̂(δ)v
[
κ˜(θ˜ − θ˜n)Û−1/2v f(Ŝv, Ûv) +
(
ρŜvÛ
−1/2
v fs(Ŝv, Ûv)
+ε˜Û1/2v fu(Ŝv, Ûv)
)
1[0,τδ](v)
]
dŴ (1)v
+
∫ t
0
L̂(δ)v
(
Û−1/2v 1[0,τδ](v) + δ
−1/2
1[τδ,T ](v)
)√
1− ρ2Ŝvfs(Ŝv, Ûv) dW (2)v
EXPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 9
so M̂t(f) is a local martingale. However, since f is rapidly decreasing, sfs(s, u),
ufu(s, u), sfsu(s, u) and ufuu(s, u) are all bounded. We also have that Ûv ≥ δ and
L̂
(δ)
v is integrable for all v. Hence, it follows by (2.13), (2.14), (2.15) and Tonelli that
M̂(f) is a martingale.
To finish the proof, it suffices to follow the remark on p.174 of Ethier and Kurtz
and show that
E
[(
f(Ŝtn+1 , Ûtn+1)− f(Ŝtn , Ûtn)−
∫ tn+1
tn
Avf(Ŝv, Ûv) dv
) n∏
k=1
hk(Ŝtk , Ûtk)
]
= 0,
(2.16)
for 0 ≤ t1 < t2 < . . . < tn+1, f ∈ D, h ∈ B(R2) (the bounded, measurable functions)
and where Ê[·] denotes the P̂-expectation. To do so, we re-write the left-hand side
of (2.16) as
Ê
[
L̂
(δ)
tn+1
(
f(Ŝtn+1 , Ûtn+1)− f(Ŝtn , Ûtn)−
∫ tn+1
tn
Avf(Ŝv, Ûv) dv
) n∏
k=1
hk(Ŝtk , Ûtk)
]
= Ê
[(
L̂
(δ)
tn+1f(Ŝtn+1 , Ûtn+1)− L̂(δ)tn f(Ŝtn , Ûtn)−
∫ tn+1
tn
L̂(δ)v Avf(Ŝv, Ûv) dv
) n∏
k=1
hk(Ŝtk , Ûtk)
]
= Ê
[(
M̂tn+1(f)− M̂tn(f)
) n∏
k=1
hk(Ŝtk , Ûtk)
]
,
which is equal to 0 since M̂(f) is a martingale. We can then conclude that (Ŝ, Û)
solves the martingale problem for A with respect to P̂.
Remark 1. In Theorem 1, we indicate the dependence of the process L̂(δ) on the
threshold δ via the superscript. Indeed, L̂(δ) depends on δ through Û . Going forward,
for notational convenience, we drop the superscript, keeping in mind the dependence
of the likelihood process on δ.
3. Pricing algorithm
In this section, we show how Theorem 1 can be exploited to price a financial
option in the 3/2 model. First, we justify that (Ŝ, Û) defined in (2.4) and (2.5) can
be used to price an option in the 3/2 model, even if they satisfy (2.2) up to τδ. We
also present an algorithm to simulate paths of (Ŝ, Û) under the 3/2 model as well
as the associated importance sampling estimator for the price of the option.
3.1. Importance sampling estimator of the option price. For the rest of this
paper, we consider an option with maturity T ∈ R whose payoff can depend on
the whole path of {(St, Vt)}t∈[0,T ], or equivalently, {(St, Ut)}t∈[0,T ]. Indeed, since
Vt = U
−1
t for all 0 ≤ t ≤ T and to simplify exposition, we will keep on working
in terms of the inverse of the variance process U going forward. We consider a
payoff function φT (S, U) with E[|φT (S, U)|] < ∞. We call pi0 = E[φT (S, U)] the
10 I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY
price of the option and the function φ, its discounted payoff. For example, a call
option, which pays out the difference between the stock price at maturity, ST , and a
pre-determined exercise price K if this difference is positive, has discounted payoff
function e−rT max(ST −K, 0) and price E[e−rT max(ST −K, 0)].
Remark 2. We work on a finite time horizon and the option payoff function φT
only depends on (S, U) up to T . We use the index T to indicate this restriction on
(S, U).
The next proposition shows that it is possible to use (Ŝ, Û), rather than (S, U),
to price an option in the 3/2 model.
Proposition 2. Suppose (S, U) is a solution to the 3/2 model (2.2) on probaiblity
space (Ω,F ,P) and τδ = inf{t ≥ 0 : Ut ≤ δ}. Define (Ŝ, Û), by (2.4) and (2.5),
set τ̂δ = inf{t ≥ 0 : Ût ≤ δ} for δ ∈ (0, 1) and let φT (S, U) be a payoff function
satisfying E[|φT (S, U)|] <∞. Then,
lim
n→∞
E1/n[φT (Ŝ, Û)1{τ1/n>T}] = E[φT (S, U)],
where Eδ[·] denotes the expectation under the measure Pδ defined in Theorem 1.
Proof. By Theorem 1, (Ŝ, Û) satisfies (2.2) on [0, τ1/n] under the measure P1/n. It
follows that
E1/n[φT (Ŝ, Û)1{τ̂1/n>T}] = E[φT (S, U)1{τ1/n>T}].
Because U satisfies the Feller condition, limn→∞ 1τ1/n≤T = 0, P-a.s. and
lim
n→∞
E1/n[φT (Ŝ, Û)1{τ̂1/n>T}] = limn→∞
E[φT (S, U)1{τ1/n>T}] = E[φT (S, U)]
by the dominated convergence theorem.

We interpret Proposition 2 in the following manner: by choosing δ small enough,
it is possible to approximate pi0 by pi
(δ)
0 := Eδ[φT (Ŝ, Û)1{τδ>T}], that is, using (Ŝ, Û)
rather than (S, U). The advantage of estimating the price of an option via (Ŝ, Û)
is that the trajectories can easily be simulated exactly under the reference mea-
sure P̂ defined in Theorem 1. In practice, we will show in Section 4 that for rea-
sonable 3/2 model calibrations, it is usually possible to find δ small enough that
Eδ[φT (Ŝ, Û)1{τδ>T}] is almost undistinguishable from pi0.
In the rest of this section, we explain how pi
(δ)
0 can be approximated with Monte
Carlo simulation. As mentioned above, paths of (Ŝ, Û) are easily simulated under
the reference measure P̂, not under Pδ. It is therefore necessary to express pi(δ)0 using
Theorem 1 in the following manner
pi
(δ)
0 = Eδ[φT (Ŝ, Û)1{τδ>T}] = Ê[L̂T φT (Ŝ, Û)1{τδ>T}]. (3.1)
EXPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 11
From (3.1) and the strong law of large numbers, we can define pi
(δ)
0 , an importance
estimator for pi
(δ)
0 , by
pi
(δ)
0 =
∑N
j=1 φT (Ŝ
(j), Û (j))L̂
(j)
T 1{τ (j)δ >T}∑N
j=1 L̂
(j)
T 1{τ (j)δ >T}
, (3.2)
where
{
Ŝ(j), Û (j), L̂(j)
}N
j=1
are N ∈ N simulated paths of (Ŝ, V̂ , L̂).
3.2. Simulating sample paths. In light of Proposition 2, we now focus on the
simulation of (Ŝt, Ût, L̂t)t≤τδ . Using (2.4) and (2.6), Ŝ and Y can easily be discretized
for simulation purposes. To simplify the simulation of the process L̂, we write (2.7)
as a deterministic function of Û in Proposition 3 below.
Proposition 3. Let L̂t be defined as in Theorem 1, with Û defined by (2.5). Then,
for t ≤ τδ, L̂t can be written as
L̂t = exp
{
−(κ˜θ˜n − κ˜θ˜)
ε˜2
[
log(Ût/Û0) + κ˜t+
κ˜θ˜ − 3κ˜θ˜n + ε˜2
2
∫ t
0
Û−1s ds
]}
. (3.3)
Proof. An application of Itoˆ’s lemma to log Ût for t ≤ τδ yields
log(Ût/Û0) = (κ˜θ˜n − ε˜2/2)
∫ t
0
Û−1s ds− κ˜t+ ε˜
∫ t
0
Û−1/2s dŴ
(1)
s . (3.4)
Isolating
∫ t
0
Û
−1/2
s dŴ
(1)
s in (3.4) and replacing the resulting expression in (2.7) gives
the result.

For t ∈ [0, T ) and h ∈ (0, T − t), for simulation purposes, we can re-write (2.4),
(2.6) and (3.3) in a recursive manner as
Ŝt+h = Ŝt exp
{
ρ
ε˜
log(Ût+h/Ût) + ah− b
∫ t+h
t
Û−1s ds
+
√
1− ρ2
∫ t+h
t
Û−1/2s dW
(2)
s ,
}
(3.5)
Y
(i)
t+h = Y
(i)
t e
− κ˜
2
h +
ε˜
2
∫ (t+h)∧τδ
t
e−
κ˜
2
(t+h−u) dZu, for i = 1, . . . , n, (3.6)
and
L̂(t+h)∧τδ = Lt exp
{
c
(
log(Û(t+h)∧τδ/Ût) + κ˜(h ∨ (τδ − t))
+d
∫ (t+h)∧τδ
t
Û−1s ds
)}
(3.7)
12 I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY
where
a = r +
ρκ˜
ε˜
b =
ρ
ε˜
(κ˜θ˜ − ε˜2/2) c = − κ˜θ˜n − κ˜θ˜
ε2
d =
κ˜θ˜ − 3κ˜θ˜n + ε˜2
2
.
We now discuss the simulation of (Ŝt+h, Ût+h, L̂t+h) given (Ŝt, Ût, L̂t), as well as
{Y (i)t }ni=1. Typically, h will be a small time interval, that is, we consider h T .
It is easy to see from the above that given Y
(i)
t , Y
(i)
t+h follows a Normal distribution
with mean Y
(i)
t e
− κ˜
2
h and variance ε˜
2
4κ˜
(1− e−hκ˜). The simulation of Y (i)t+h given Y (i)t is
thus straightforward. Ût+h can then be obtained by (2.5) as the sum of the squares
of each Y
(i)
t+h, for i = 1, . . . , n.
Given simulated values Ût+h and Ût, the term
∫ t+h
t
Û−1s ds, which appears in both
Ŝt+h and L̂t+h, can be approximated using the trapezoidal rule by letting∫ t+h
t
Û−1s ds ≈
(Ût + Ût+h)
2
h. (3.8)
More precise approximations to this integral can be obtained by simulating inter-
mediate values Û−1t+ih for i ∈ (0, 1) and using other quadrature rules. In Kouritzin
(2018) and Kouritzin and MacKay (2020), Simpson’s 1
3
rule was preferred. In this
section, we use a trapezoidal rule only to simplify the exposition of the simulation
algorithm.
Given that Ût+h > δ and once an approximation for the deterministic integral∫ t+h
t
Û−1s ds is calculated, L̂t+h can be simulated using (3.7). To generate a value
for Ŝt+h, it suffices to observe that conditionally on {Ûs}s∈[t,t+h],
∫ t+h
t
Û
−1/2
s dW
(2)
s
follows a Normal distribution with mean 0 and variance
∫ t+h
t
Û−1s ds.
The resulting algorithm produces N paths of (Ŝ, Û , L̂) and the stopping times τδ
associated with each path; it is presented in Algorithm 1, in the appendix. These
simulated values are then used in (3.2) to obtain an estimate for the price of an
option.
4. Numerical experiment
4.1. Methods and parameters. In this section, we assess the performance of
the pricing algorithm derived from Theorem 1. To do so, we use Monte Carlo
simulations to estimate the price of European call options. These Monte Carlo
estimates are compared with the exact price of the option, calculated with the
analytical expression available for vanilla options in the 3/2 model (see for example
Lewis (2000) and Carr and Sun (2007)). More precisely, we consider the payoff
function φT (S, U) = max(ST − K, 0) for K > 0 representing the exercise price of
the option and we compute the price estimate according to (3.2).
The precision of the simulation algorithm is assessed using either the mean square
error or the relative mean square error, as indicated. We define the mean square
EXPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 13
error by
MSE = E[(pi0 − pi(δ)0 )2]
and the relative mean square error by
RelMSE =
E[(pi0 − pi(δ)0 )2]
pi0
,
where pi0 is the exact price of the option and pi
(δ)
0 is the estimate calculated with
(3.2). The expectations above are approximated by calculating the estimates a large
number of times and taking the mean over all runs.
Throughout this section, we consider the five parameter sets presented in Table
1. Parameter set 1 (PS1) was used in Baldeaux (2012). Parameter set 2 (PS2)
was obtained by Drimus (2012) via the simultaneous fit of the 3/2 model to 3-
month and 6-month S&P500 implied volatilities on July 31, 2009. The three other
parameter sets are modifications of PS2: PS3 was chosen so that 4κ˜θ˜
ε˜2
∈ N, and PS4
and PS5 were selected to have a higher n. Recalling that n = max
(
b4κ˜θ˜
ε˜2
+ 1
2
c, 1
)
represents the number of Ornstein-Uhlenbeck processes necessary to simulate the
variance process, we have that n = 204 for PS1, n = 5 for PS2 and PS3 and n = 12
for PS4 and PS5.
Throughout the numerical experiments, the threshold we use is δ = 10−5. For
all parameter sets, the simulated process U never crossed below this threshold.
Therefore, any δ below 10−5 would have yielded the same results.
Table 1. Parameter sets
S0 V0 κ θ ε ρ r 4κ˜θ˜/ε˜
2
PS1 1 1 2 1.5 0.2 −0.5 0.05 204
PS2 100 0.06 22.84 0.218 8.56 −0.99 0.00 5.25
PS3 100 0.06 18.32 0.218 8.56 −0.99 0.00 5.00
PS4 100 0.06 19.76 0.218 3.20 −0.99 0.00 11.72
PS5 100 0.06 20.48 0.218 3.20 −0.99 0.00 12.00
4.2. Results. In this section, we present the results of our numerical experiments.
We first test the sensitivity of our simulation algorithm to n, the number of Ornstein-
Uhlenbeck processes to simulate. We then compare the performance of our algorithm
to other popular ones in the literature.
4.2.1. Sensitivity to n. We first test the impact of n on the precision of the algorithm.
Such an impact was observed in Kouritzin and MacKay (2020) in the context of the
Heston model. To verify whether this also holds for the 3/2 model, we consider the
first three parameter sets and price at-the-money (that is, K = S0) European call
options. For PS1, we follow Baldeaux (2012) and compute the price of a call option
with maturity T = 1. The exact price of this option is 0.4431. PS2 and PS3 are
14 I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY
used to obtain the price of at-the-money call options with T = 0.5, with respective
exact prices 7.3864 and 7.0422. In all three cases, the length of the time step used
for simulation is h = 0.02.
Here we assess the precision of the algorithm using the relative MSE in order to
compare all three parameter sets, which yield vastly different prices. The relative
quadratic error is approximated by computing the price estimators 20 times, for
N ∈ {5000, 10000, 50000} simulations. The integral with respect to time (see step
(3) of Algorithm 1) is approximated using M ∈ {2, 4} sub-intervals and Simpson’s
1
3
rule.
The results of Table 2 show that the precision of the simulation algorithm seem
to be affected by n. Indeed, as a percentage of the exact price, the MSE of the price
estimator is higher for PS1 than for the other parameter sets. This observation
becomes clearer as N increases.
We recall that for PS3, 4κ˜θ˜
ε˜2
is an integer, while this is not the case for PS2. It
follows that for this latter parameter set, the weights L̂
(j)
T are all different, while they
are all equal to 1 for PS3. One could expect the estimator using uneven weights
to show a worse performance due to the possible great variance of the weights.
However, in this case, both estimators show similar a performance; the algorithm
does not seem to be affected by the use of uneven weights.
Finally, Table 2 shows that increasing M may not significantly improve the pre-
cision of the price estimator. Such an observation is important, since adding subin-
tervals in the calculation of the time-integral slows down the algorithm. Keeping
the number of subintervals low reduces computational complexity of our algorithm,
making it more attractive.
Table 2. Relative MSE as a percentage of pi0.
N PS1 PS2 PS3
M = 2 M = 4 M = 2 M = 4 M = 2 M = 4
5000 0.271 0.316 0.183 0.225 0.239 0.214
10000 0.203 0.158 0.111 0.112 0.172 0.143
50000 0.158 0.135 0.085 0.083 0.067 0.070
4.2.2. Comparison to other algorithms. In this section, we compare the performance
of our new simulation algorithm for the 3/2 model to existing ones. The first bench-
mark algorithm we consider is based on a Milstein-type discretization of the log-price
and variance process. The second one is based on the quadratic exponential scheme
proposed by Andersen (2007) as a modification to the method of Broadie and Kaya
(2006), which we adapted to the 3/2 model. These algorithms are outlined in the
appendix.
To assess the relative performance of the algorithms, we price in-the-money (K/S0 =
0.95), at-the-money (K/S0 = 1) and out-of-the-money (K/S0 = 1.05) call options
with T = 1 year to maturity. The exact prices of the options, which are used to
calculate the MSE of the price estimates, are given in Table 3. We consider all
EXPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 15
parameter sets with the exception of PS1, since this parametrization requires the
simulation of 204 Ornstein-Uhlenbeck process, which makes our algorithm exces-
sively slow.
Table 3. Exact prices pi0 of European call options.
K/S0 PS2 PS3 PS4 PS5
0.95 10.364 10.055 11.657 11.724
1 7.386 7.042 8.926 8.999
1.05 4.938 4.586 6.636 6.710
Figures 1 and 2 present the relative MSE of the price estimator as a function of
the number of simulations. We note that the parametrizations considered in Figure
1 are such that 4κ˜θ˜
ε˜2
/∈ N, while the opposite is true for Figure 2.
Overall, the precision of our weighted simulation algorithm is similar to that of the
other two algorithms studied. However, certain parameter sets result in more precise
estimates. Figure 1 shows that the MSE is consistently larger with the weighted
simulation algorithms than with the benchmark ones for PS2. However, with PS4,
the weighted algorithm performs as well as the other two algorithms, or better. We
note that for PS2, n = 5 while for PS4, n = 12. It was observed in Kouritzin and
MacKay (2020) in the case of the Heston model that as n increases, the weighted
simulation algorithm seems to perform better relatively to other algorithms. This
observation also seems to hold in the case of the 3/2 model.
For parametrizations that satisfy 4κ˜θ˜
ε˜2
∈ N, such as in Figure 2, we observe that
the weighted simulation algorithm is at least as precise, and often more, than the
other algorithms. In this case, all the weights L̂T are even, which tends to decrease
the variance of the price estimator and thus, to decrease the relative MSE. It is also
interesting to note that in the case of Figure 2, since 4κ˜θ˜
ε˜2
∈ N, it is not necessary to
simulate τ
(j)
δ and the trajectories L̂T . Indeed, in this case, it is possible to simplify
the algorithm using Proposition 1, which greatly speeds it up. When possible,
parametrization that satisfy this condition should therefore be considered.
16 I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY
0.00025
0.00050
0.00075
2e+04 4e+04 6e+04 8e+04 1e+05
N
rm
se
_
itm
(A) PS2, K/S0 = 0.95
2e−04
4e−04
6e−04
8e−04
2e+04 4e+04 6e+04 8e+04 1e+05
N
rm
se
_
a
tm
Algorithms M QE WE WE2
(B) PS2, K/S0 = 1
2e−04
4e−04
6e−04
8e−04
2e+04 4e+04 6e+04 8e+04 1e+05
N
rm
se
_
o
tm
(C) PS2, K/S0 = 1.05
0.00025
0.00050
0.00075
0.00100
2e+04 4e+04 6e+04 8e+04 1e+05
N
rm
se
_
itm
(D) PS4, K/S0 = 0.95
0.00025
0.00050
0.00075
0.00100
0.00125
2e+04 4e+04 6e+04 8e+04 1e+05
N
rm
se
_
a
tm
(E) PS2, K/S0 = 1
0.00025
0.00050
0.00075
0.00100
2e+04 4e+04 6e+04 8e+04 1e+05
N
rm
se
_
o
tm
(F) PS4, K/S0 = 1.05
Figure 1. Relative MSE as a function of N , PS2 and PS4, algo-
rithms: Milstein (dot), quadratic exponential (triangle), weighted,
M = 2 (square), weighted, M = 4 (cross).
EXPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 17
2e−04
4e−04
6e−04
2e+04 4e+04 6e+04 8e+04 1e+05
N
rm
se
_
itm
(A) PS3, K/S0 = 0.95
2e−04
4e−04
6e−04
2e+04 4e+04 6e+04 8e+04 1e+05
N
rm
se
_
a
tm
Algorithms M QE WE WE2
(B) PS3, K/S0 = 1
2e−04
4e−04
6e−04
2e+04 4e+04 6e+04 8e+04 1e+05
N
rm
se
_
o
tm
(C) PS3, K/S0 = 1.05
5e−04
1e−03
2e+04 4e+04 6e+04 8e+04 1e+05
N
rm
se
_
itm
(D) PS5, K/S0 = 0.95
0.00025
0.00050
0.00075
2e+04 4e+04 6e+04 8e+04 1e+05
N
rm
se
_
a
tm
(E) PS5, K/S0 = 1
3e−04
6e−04
9e−04
2e+04 4e+04 6e+04 8e+04 1e+05
N
rm
se
_
o
tm
(F) PS5, K/S0 = 1.05
Figure 2. Relative MSE as a function of N , PS3 and PS5, algo-
rithms: Milstein (dot), quadratic exponential (triangle), weighted,
M = 2 (square), weighted, M = 4 (cross).
5. Conclusion
In this paper, we present a weak explicit solution to the 3/2 model, up until
the inverse of the variance process drops below a given threshold. We develop a
simulation algorithm based on this solution and show that it can be used to price
options in the 3/2 model, since in practice, the inverse variance process stays away
from 0. Numerical examples show that our simulation algorithm performs at least
as well as popular algorithms presented in the literature.
18 I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY
It is important to note that the method that we present in this paper could
be significantly sped up by the use of sequential resampling, as implemented in
Kouritzin and MacKay (2020) for the Heston model. Such improvements, left for
future work, could give a significant advantage to our weighted simulation algorithm
for the 3/2 model.
Appendix
This section presents the simulation algorithms used to produce the numerical
examples in Section 4. Algorithm 1 stems from the results we present Theorem 1.
Algorithm 2 is a Milstein-type algorithm applied to the 3/2 model. Algorithm 3 is
Andersen (2007)’s approximation to the algorithm proposed by Broadie and Kaya
(2006), modified for the 3/2 model, since the original algorithm was developed for
the Heston model. Algorithms 2 and 3 are considered for comparison purposes.
For all algorithms, we consider a partition {0, h, 2h, . . . ,mh}, with mh = T of the
time interval [0, T ], and outline the simulation of N paths of (Ŝ, Û , L̂), as well as
the associated stopping times τδ.
To simplify the exposition of Algorithm 1, we define the following constants:
αh = e
− κ˜
2
h, σh =
ε˜2
4κ˜
(
1− e−hκ˜) .
We also drop the hats to simplify the notation.
Algorithm 1 (Weighted explicit simulation).
I. Initialize:
Set the starting values for each simulated path:
{(S(j)0 , L(j)0 , τ (j)δ ) = (S0, 1, T + h)}Nj=1, {Y (l,j)0 =
√
U0/n}n,Nl,j=1
II. Loop on time: for i = 1, . . . ,m
Loop on particles: for j = 1, . . . , N , do
(1) For l = 1, . . . , n, generate Y
(l,j)
ih using Y
(l,j)
ih ∼ N
(
αhY
(l,j)
(i−1)h, σ
2
h
)
.
(2) Set U
(j)
ih =
∑n
l=1(Y
(l,j)
ih )
2.
(3) Let IntU
(j)
ih ≈
∫ ih
(i−1)h(U
(j)
s )−1 ds using (3.8) (or another quadrature rule).
(4) Generate S
(j)
ih from S
(j)
(i−1)h using (2.4), with
∫ (i−1)h
ih
(̂U
(j)
s )−1/2dW
(2)
s ∼ N(0, IntU (j)ih ).
(5) If ih ≤ τ (j)δ ,
(i) If U
(j)
ih > δ, generate L
(j)
ih from L
(j)
(i−1)h using (3.7).
(ii) Otherwise, set τ
(j)
δ = t.
Algorithm 2. I. Initialize:
EXPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 19
Set the starting values for each simulated path:
{(S(j)0 , U (j)0 ) = (S0, U0}Nj=1
II. Loop on time: for i = 1, . . . ,m
Loop on particles: for j = 1, . . . , N , do
(1) Correct for possible negative values: u¯(j) = max(U((i− 1)h), 0)
(2) Generate U
(j)
ih from U
(j)
(i−1)h:
U
(j)
ih = U
(j)
(i−1)h + κ˜(θ˜ − u¯(j))h
+ ˜
√
u¯(j)hZ
(j)
1 +
1
4
ε˜2((Z
(j)
1 )
2 − 1)h,
with Z
(j)
1 ∼ N(0, 1).
(3) Generate S
(j)
ih from S
(j)
(i−1)h:
Sih = S(i−1)h exp
{(
r − 1
2u¯
)
h+
√
h
u¯
Z
(j)
2
}
,
with Z
(j)
2 ∼ N(0, 1).
Algorithm 3. I. Initialize:
(1) Set the starting values for each simulated path:
{(S(j)0 , U (j)0 ) = (S0, U0)}Nj=1
(2) Fix the constant φc ∈ [1, 2].
II. Loop on time: for i = 1, . . . ,m
Loop on particles: for j = 1, . . . , N , do
(1) Set the variables mi,j and si,j:
mi,j = θ˜ + (U
(j)
(i−1)h − θ˜)e−κ˜h
si,j =
U(i−1)hε˜2e−κ˜h
κ˜
(1− e−κ˜h) + θ˜ε˜
2
2κ˜
(1− e−κ˜h)2
(2) Set φi,j =
s2i,j
m2i,j
.
(3) If φi,j < φc,
Generate U
(j)
ih from U
(j)
(i−1)h:
U
(j)
ih = ai,j(bi,j + Z
(j))2,
where Z(j) ∼ N(0, 1) and
b2i,j = 2φ
−1
i,j − 1 +
√
2φ−1i,j
√
2φ−1i,j − 1
ai,j =
mi,j
1 + b2i,j
20 I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY
(4) If φi,j ≥ φc,
Generate U
(j)
ih from U
(j)
(i−1)h:
U
(j)
ih =
1
β
log
(
1− pi,j
1−X(j)
)
,
where X(j) ∼ Uniform(0, 1) and
pi,j =
ψi,j − 1
ψi,j + 1
, βi,j =
1− pi,j
mi,j
(5) Let IntU
(j)
ih ≈
∫ ih
(i−1)h(U
(j)
s )−1 ds using (3.8).
(6) Generate S
(j)
ih from S
(j)
(i−1)h using (2.4), with
∫ (i−1)h
ih
(̂U
(j)
s )−1/2dW
(2)
s ∼ N(0, IntU (j)ih ).
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Department of Mathematics, Universite´ du Que´bec a` Montre´al, Montreal, QC,
H3C 3P8 Canada
E-mail address: kouarfate.iro [email protected]
Department of Mathematical and Statistical Sciences, University of Alberta,
Edmonton, AB, T6G 2G1 Canada
E-mail address: [email protected]
Department of Mathematics, Universite´ du Que´bec a` Montre´al, Montreal, QC,
H3C 3P8 Canada
E-mail address: [email protected]

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