Submitted to Operations Researchmanuscript (Please, provide the manuscript number!)
Simulation of tempered stable Levy bridges and itsapplications
Kyoung-Kuk KimDepartment of Industrial and Systems Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701,
South Korea, [email protected]
Sojung KimDepartment of Mathematical Sciences, Korea Advanced Institute of Science and Technology, [email protected]
We consider tempered stable Levy subordinators and develop a bridge sampling method. An approximate
conditional PDF given the terminal values is derived with stable index less than one, using the double
saddlepoint approximation. We then propose an acceptance-rejection algorithm based on the existing gamma
bridge and the inverse Gaussian bridge as proposal densities. Its performance is comparable to existing
sequential sampling methods such as Devroye (2009) and Hofert (2011) when generating a fixed number of
observations. As applications, we consider option pricing problems in Levy models. First, we demonstrate
the effectiveness of bridge sampling when combined with adaptive sampling under finite-variance CGMY
processes. Second, further efficiency gain is achieved in terms of variance reduction via stratified sampling.
Key words : bridge sampling, Levy process, saddlepoint approximation, tempered stable subordinator
History : This paper was first submitted on May 19, 2014 and has been with the authors for 12 months for
4 revision.
1. Introduction
When it comes to simulating sample paths of a stochastic process, one fundamental idea applied
to a Brownian motion is bridge sampling. This refers to the procedure such that we first generate a
skeleton of a Brownian motion and fill in the details as needs arise by utilizing the readily available
conditional distributions. It is contrasted with the typical sequential sampling, moving forward
in time. And the underlying process, Brownian bridge, has found numerous applications such as
statistical inference, large deviations, and option pricing just to name a few. From the simulation
point of view, bridge sampling provides us with the control over the coarseness of a simulated
1
Kim and Kim: Simulation of tempered stable Levy bridges2 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
process, the freedom to combine with sequential sampling, and other possibilities to make use
of variance reduction techniques and low-discrepancy methods (Glasserman 2003). Naturally, the
extension of such bridge sampling to more advanced stochastic processes has received a growing
amount of attention.
In this paper, we consider a Levy process conditioned on the terminal values, that is, (Xt)0≤t≤T
given X0 and XT . This generalizes the notion of Brownian bridge and thus is called Levy bridge.
Along with various useful applications of Brownian bridge, there have been extensive studies on
the probabilistic properties of such conditional distributions of Levy processes. Concentrating on
rather practical aspects, in this work, we are particularly interested in simulating sample paths of
Levy bridge processes, that is, bridge sampling for Levy processes.
Despite its potential usefulness, simulation methods for Levy bridges have not been developed
except for some special processes. A few well-known examples include gamma bridge, inverse Gaus-
sian (IG) bridge, and squared Bessel bridge. Those processes allow explicit or semi-explicit forms
of probability density functions (PDF). Bridge sampling methods for gamma and variance gamma
(VG) processes are developed in Ribeiro and Webber (2004), Avramidis, L’Ecuyer, and Tremblay
(2003) and Avramidis and L’Ecuyer (2006). For IG and normal inverse Gaussian (NIG) processes,
see Ribeiro and Webber (2003). The paper applies the MSH method in Michael et al. (1976) and
the authors utilize quite a unique transformation of an IG bridge, resulting in large efficiency gains.
For a squared Bessel bridge, its Laplace transform is presented in Pitman and Yor (1982) and a
series expansion is available as well.
On the other hand, simulation algorithms for diffusion bridges have received much attention over
the last fifteen years. Early attempts at bridge sampling for diffusions are based on the Metropolis-
Hastings algorithm and a discrete-time approximation of diffusion processes in Roberts and Stramer
(2001) and Durham and Gallant (2002); see also Lin et al. (2010) for an extension. Beskos et al.
(2006) introduce a remarkable exact sampling algorithm of general diffusion processes. Their pro-
posed method accepts a sampled skeleton of a standard Brownian motion with some probability
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 3
proportional to the Radon-Nikodym derivative between the laws of the Brownian motion and the
diffusion bridge. Lastly, there is a simple approximation scheme for diffusion bridges of Bladt and
Sørensen (2014), using two diffusions, one moving forward in time and the other backward, and
constructing an approximate bridge.
Except for those mentioned above, Levy bridge sampling methods are hardly known. One of the
difficulties comes from the fact that the distribution of a Levy process is mostly specified via its
characteristic function (CHF) or the associated Levy measure. Therefore, the PDF is only available
via numerical inversion or infinite series expansion, which renders some of the prior algorithms
impractical. In this respect, the (approximate) PDF or related probabilistic features are reported in
the literature. Some representative works include Ruschendorf and Woerner (2002) and Figueroa-
Lopez and Houdre (2009). In Figueroa-Lopez and Tankov (2014), the authors obtain small-time
asymptotic behaviors of the exit probabilities of a Levy process out of certain intervals, using
Levy bridges. Unfortunately, such a small-time density expansion is not easily applicable from the
simulation perspective because an approximating quantity fails to be a legitimate PDF in general.
We, however, note that they describe an interesting bridge sampling algorithm for mid-points if the
PDF of a Levy process is known and unimodal, with an application to stopping times. For more
general instances, there are some approximation schemes such as the forward-backward method
for Markov bridges in Asmussen and Hobolth (2012) and the beta approximation method for Levy
processes in Glasserman and Kim (2008).
We consider stable and tempered stable processes, which have infinite jump activity and are
widely used for heavy-tailed models. We propose a simulation algorithm for the Levy bridges of
stable and tempered stable subordinators along with efficient simulation schemes for the stochastic
processes based on or derived from them. In fact, their bridge processes have the same law thanks
to exponential tilting when the stable index is less than one. For this purpose, we obtain an
approximate conditional PDF given the end-points via saddlepoint techniques. The resulting closed
form conditional PDF enables us to consider an acceptance-rejection algorithm in which two known
proposal densities are utilized, namely the gamma bridge and the IG bridge.
Kim and Kim: Simulation of tempered stable Levy bridges4 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
The saddlepoint method obtains asymptotic expansions by approximating a contour integral in
the complex plane near a saddlepoint, via the steepest descent method. It is first introduced in
statistics by Daniels (1954) to provide an approximate PDF of the mean of i.i.d. random variables,
and later the tail probabilities of the sample mean are derived in Lugannani and Rice (1980). In
financial applications, many authors have applied saddlepoint methods to option pricing under
various models such as affine jump-diffusions or credit models. The reader is referred to Rogers
and Zane (1999), Carr and Madan (2009), Glasserman and Kim (2009),Yang et al. (2006), Dembo
et al. (2004), and Gordy (2002) etc.
The performance and advantages of the proposed bridge sampling scheme are demonstrated via
several numerical studies. We particularly consider option pricing problems under Levy models.
The main messages from our numerical tests can be summarized as follows:
• It is comparable in cost and accuracy to three existing sequential simulation methods if we
generate a fixed number of observations. The first method is an acceptance-rejection method
based on the representation of a stable random variable in Chambers et al. (1976), the sec-
ond is the method of Devroye (2009) with uniformly bounded complexity. The last under
consideration is the fast rejection method proposed by Hofert (2011).
• It allows adaptive sampling for pricing path-dependent options under finite variation tem-
pered stable processes (e.g., CGMY processes). Such an adaptive technique extended from
Becker (2010) results in remarkable savings in simulation costs.
• It makes it possible to use stratified sampling techniques for pricing path-dependent options
beyond VG and IG processes, which can lead to significant variance reduction.
• It is combined with sequential sampling utilizing the strengths of both, and such a hybrid
method is applied for least square Monte Carlo (LSMC) methods for American options valua-
tion under asset dynamics with subordinated Brownian motions. This avoids the massive data
storage requirements for LSMC pricing without sacrificing precision. (See the e-companion to
this paper.)
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 5
In addition to the pricing of path-dependent options, it is worth noting that there are prospec-
tive applications of our approach to statistical inference and the information-based asset pricing
framework. Inference based on incomplete data, the so-called missing data problem, is a fundamen-
tal issue in statistics. By treating the data observed at a discrete set of points as a missing data
problem and generating intermediate values conditioned on endpoints, one can exploit inference
tools designed for continuous paths. On the other hand, the essential idea of the information-based
asset pricing is to model the flow of information in markets as stochastic processes. For a market
factor ST revealed at time T , the corresponding information process (ξtT )0≤t≤T conditioned on
ξTT = ST that market participants have about ST is generated. Brody et al. (2007, 2008) adopt
Brownian bridge and gamma bridge as information processes; Hoyle et al. (2011) use more general
Levy bridges for such information processes.
The rest of this paper is organized as follows. Section 2 provides more detailed background on
stable and tempered stable processes as well as on saddlepoint methods. In Section 3, we explain
some useful properties of a tempered stable Levy bridge and derive its approximate PDF. Then we
develop a bridge sampling scheme later in the section. In Section 4, numerical tests are conducted
and reported under finite variation tempered stable processes. Stratified sampling for variance
reduction is considered as well. Section 5 concludes.
2. Preliminaries
2.1. Stable and tempered stable processes
A Levy process (Xt)t≥0 is a stochastically continuous process with stationary and independent
increments starting at 0. For every t, Xt has an infinitely divisible distribution. Conversely, if µ is an
infinitely divisible distribution then there exists a Levy process (Xt)t≥0 such that the distribution
of X1 equals µ. We first define a stable distribution on the positive real line and the Levy process
corresponding to the stable distribution. The general definition of a stable process is given in EC.1.3
of the e-companion to this paper; here we focus on the case 0<α< 1.
Kim and Kim: Simulation of tempered stable Levy bridges6 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
Definition 1. Let µ be an infinitely divisible probability measure on (R+,B(R+)). It is called
α-stable, denoted by S(α, c), with an index of stability 0<α< 1 and c > 0 if it has the CHF
exp
(∫R
(eiux− 1
)ν(dx)
), u∈R, (1)
with the Levy measure of the form
ν(dx) =c
x1+α1(0,∞)(x)dx. (2)
The corresponding nonnegative Levy process Xt is called a stable (Levy) subordinator, whose
distribution is S(α, ct) for each t > 0.
The PDF of S(α, c) is only known as an infinite series of the following form:
fS(α,c)(x) =1
π (−cΓ(−α))1/α
∑n∈N
(−1)n−1
n!sin(nπα)Γ(nα+ 1)
(x
(−cΓ(−α))1/α
)−nα−1
. (3)
For some α, such as α= 1/3 or 1/2, the closed form expression is known in terms of some special
functions. Nevertheless, an exact simulation method for a stable distribution S(α, c) exists through
the representation (Chambers et al. 1976, Kawai and Masuda 2011):
S(α, c) =(− cΓ(−α)
)1/α sin(αU +ϑ)
(cosU)1/α
(cos((1−α)U −ϑ)
E
) 1−αα
, (4)
where ϑ = πα/2, U has a Unif(−π/2, π/2) distribution, and E is exponentially distributed with
mean 1 independently of U .
Now, let us consider an exponentially tempered version of a stable subodinator, that is, ν(dx) =
eθxνS(dx) for some θ that satisfies∫|x|≥1
eθxνS(dx)<∞ where νS(dx) is the Levy measure (2).
Definition 2. An infinitely divisible distribution η on (R+,B(R+)) is called one-sided tempered
stable distribution, denoted by TS(α,λ, c), with parameters c,λ ∈ (0,∞) and α ∈ [0,1) if its CHF
is given by (1) with the Levy measure
ν(dx) =c
x1+αe−λx1(0,∞)(x)dx.
The Levy process (Xt)t≥0 associated with η is called a tempered stable (Levy) subordinator.
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 7
Clearly, the distribution of Xt is given by TS(α,λ, ct) for each t > 0. The parameter c represents
the intensity of jumps, and α determines the relative importance of small jumps for the trajectories
of the process. Here, α= 0 case is included for more generality as in Cont and Tankov (2004). The
new exponential tempering parameter λ controls the decay rate of large jumps. The properties of a
one-sided tempered distribution relative to a stable distribution have been investigated (for exam-
ple, see Rosinski 2007). A one-sided tempered stable distribution also admits a series expansion of
the PDF on the positive real line, by the following relation to the PDF fS(α,c)(x):
fTS(α,λ,c)(x) = exp[−λx− cΓ(−α)λα
]fS(α,c)(x). (5)
Based on this relation, TS(α,λ, c) can be simulated exactly by the acceptance-rejection sampling,
which is described in Algorithm 1.
Algorithm 1 Exact sampling algorithm for a tempered stable subordinator [SSR]
1: repeat
2: Generate X ∼ S(α, ct) from (4) and an independent U ∼Unif(0,1)
3: until U ≤ e−λX .
4: return X
Algorithm 1, which we call simple stable rejection (SSR), has the complexity of O (exp (tλ/α)). Its
performance deteriorates quickly for large t, λ, or small α. Devroye (2009) develops an exact double
rejection method that is uniformly bounded in complexity over all parameter ranges to address
such possible low acceptance rates. Recently, Hofert (2011) suggests a fast rejection algorithm that
obtains a logarithmic improvement in complexity O (tλ/α) in comparison to SSR.
A tempered stable subordinator can be naturally extended to the whole real line.
Definition 3. For fixed parameters c+, c−, λ+, λ− ∈ (0,∞) and α+, α− ∈ [0,1), an infinitely
divisible distribution η on (R,B(R)) is called a tempered stable distribution, denoted by
TS(α+, λ+, c+;α−, λ−, c−) , if η = η+ ∗ η−, where η+ = TS(α+, λ+, c+) and η− = υ with υ =
Kim and Kim: Simulation of tempered stable Levy bridges8 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
TS(α−, λ−, c−) and denoting the dual of υ by υ(B) = υ(−B) for B ∈ B(R). Its Levy measure ν is
given by
ν(dx) =
(c+
x1+α+ e−λ+x1(0,∞)(x) +
c−
|x|1+α−e−λ
−|x|1(−∞,0)(x)
)dx.
We call the Levy process associated with η a tempered stable process.
Remark 1. In Cont and Tankov (2004), generalized tempered stable processes for parameters
c+, c−, λ+, λ− ∈ (0,∞) and α+, α− < 2 are defined. For α+, α− ∈ [0,1), which we consider in this
paper, X is a finite-variation process making infinitely many jumps in a finite horizon. For α+, α− ∈
[1,2), we have an infinite-variation process. As a special case, when c+ = c−, α+ = α−, it is called
a CGMY process in Carr et al. (2002).
For more information on tempered stable processes, we refer the reader to Kuchler and Tappe
(2011), Cont and Tankov (2004), Sato (1999) and the references therein.
Remark 2. The term ‘tempered stable process’ sometimes refers to a more general class of pro-
cesses that contains the version in Remark 1 as a subclass. For example, see Rosinski (2007) and
Grabchak (2012). In this broader context, the tempered stable processes that we consider corre-
spond to ‘exponentially tilted stable processes’.
2.2. Saddlepoint approximations for conditional probabilities
Daniels (1954) introduces the saddlepoint method to statistics in order to approximate the PDF
of the mean of n i.i.d. random variables. In what follows, we present the result in the case n= 1
but for multivariate random vectors.
Suppose X = (X1, · · ·,Xm)> is a random vector with a non-degenerate distribution in Rm. With
all continuous components Xi, let f(x) be the PDF at each x ∈ Rm. With χ the support of the
random vector X, we define its joint moment generating function (MGF) as
M(u) = E[exp
(u>X
)]=
∫χ
exp(u>x
)f(x)dx
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 9
for all values of u = (u1, · · ·, um)∈Rm such that the integral converges. We assume that the maximal
convergence set S for M contains a neighborhood of 0∈Rm. Clearly, S is a convex set. Let Iχ ⊂Rm
denote the interior of the convex hull of χ and K(u) be the cumulant generating function (CGF)
of f defined by K(u) = lnM(u). Then, the saddlepoint approximation to unknown f(x) for the
continuous vector X is given in Iχ ⊂Rm as
f(x) =1
(2π)m/2|K ′′(u)|1/2exp
(K(u)− u>x
), x∈ Iχ (6)
where u is the unique solution in S to the m-dimensional saddlepoint equation K ′(u) = x.
A natural approach to obtain a saddlepoint approximation for a conditional PDF is to use two
saddlepoint approximations, one for the joint PDF and another for the marginal as in (6). The idea
is introduced in Daniels (1958) and the explicit formula is presented in Barndorff-Nielsen and Cox
(1979). To be specific, let (X,Y) be a random vector having a non-degenerate distribution in Rm
with dim(X) = mx, dim(Y) = my, and mx +my = m. With all components continuous, suppose
there is a joint PDF f(x,y) with support (x,y)∈ χ⊂Rm. The double saddlepoint approximation
for f(y|x), the conditional PDF of Y at y given X = x, is expressed as
f(y|x) = (2π)my/2{|K ′′(u, v)||K ′′uu(u0,0)|
}−1/2
× exp[{K(u, v)− u>x− v>y
}−{K(u0,0)− u>0 x
}](7)
for (x,y)∈ Iχ. Here, the m-dimensional saddlepoint (u, v) solves the set of m equations K ′(u, v) =
(x,y), and u0 is the mx-dimensional saddlepoint for the denominator that solves K ′(u0,0) = x.
Here, K ′ (K ′u) is the gradient with respect to both components u and v (u only), and K ′′ (K ′′uu)
is the corresponding Hessian.
For the comprehensive treatments of saddlepoint approximations, good references are Butler
(2007) and Jensen (1995).
3. Bridge Sampling Methods
In this section, we study the bridge process of a stochastic process (Xt(α,λ, c))t≥0 associated with
TS(α,λ, c), and we call it a tempered stable Levy bridge. Note that gamma bridge and IG bridge
Kim and Kim: Simulation of tempered stable Levy bridges10 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
correspond to α= 0 and α= 1/2, respectively. Let us first describe some of interesting properties
of a tempered stable Levy bridge. Later we will propose an efficient bridge sampling algorithm. All
proofs developed in this section are deferred to EC.1 of the e-companion.
It is well known that a tempered stable subordinator possesses the following scaling property:
For every r > 0, rXt(α,λ, c) has the same law as Xrαt(α,λ/r, c). Accordingly, the resulting bridge
process preserves the scaling property as well and we record it in the next proposition.
Proposition 1. Let Xt(α,λ, c) be a tempered stable subordinator. For any r > 0, we have
(Xt|XT = z)d=
(1
rXrαt(α,λ/r, c)
∣∣∣∣XrαT (α,λ/r, c) = rz
).
Another interesting feature for such a process is that it is identical in law to the bridge process
for a stable subordinator, again named a stable Levy bridge.
Lemma 1. The distribution of a stable Levy bridge is equal to that of its tempered version.
Thus throughout the rest of the paper, our illustration will be based on a tempered stable process
for which any bridge sampling method is applicable to stable Levy bridges as well. Additionally,
we can use known distributional features of a stable Levy bridge for its tempered counterpart. For
instance, we are able to compute conditional moments for the latter process, based on the theory
of multi-dimensional stable processes. In the following proposition, we prove that the conditional
expectation given initial and end points is simply a linear interpolation of those points.
Proposition 2. Let Xt(α,λ, c) be a tempered stable subordinator. For 0< t1 < t2, given Xt2 = z
the conditional expectation is linear in z, that is, E[Xt1 |Xt2 = z] = (t1/t2)z.
3.1. Approximate conditional PDF
Thanks to the scaling property presented above, it is sufficient to consider tempered stable Levy
subordinators with the moment condition E[Xt] = t, from which we have a two-parameter fam-
ily. For notational convenience, we denote Xt(α,κ) for the two-parameter tempered stable Levy
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 11
subordinator associated with a one-sided tempered stable distribution, say TS(α,κ), whose Levy
measure is
ρ(x)dx=1
Γ(1−α)
(1−ακ
)1−αe−(1−α)x/κ
x1+αdx,
and the MGF is given by
E[euXt ] = etl(u), ∀u∈(−∞, 1−α
κ
)where the Laplace exponent l(·) is
l(u) =1
α
(1−ακ
)1−α [(1−ακ
)α−(
1−ακ−u)α]
. (8)
In this parametrization, λ= (1−α)/κ and c= λ1−α/Γ(1−α). The first four central moments are
computed as K1 = E[Xt] = t, K2 = Var[Xt] = κt, K3 = (2− α)κ2t/(1− α), and K4 = (3− α)(2−
α)κ3t/(1−α)2.
Consider a two-dimensional random vector (Xt2 ,Xt1) with 0< t1 < t2, where Xt =Xt(α,κ) has
the CGF tl(u) as in (8). Then the joint CGF K(u, v) of (Xt2 ,Xt1) is computed as follows:
K(u, v) = lnE[euXt2+vXt1
]= lnE
[e(u+v)Xt1+uXt2−t1
](by stationary increment property)
= t1l(u+ v) + (t2− t1)l(u) (by independent increment property)
=
t2λα− t2−t1
αλ1−α (λ−u)
α− t1αλ1−α (λ−u− v)
α, α∈ (0,1);
− t1κ
log(1− u+v
κ
)− t2−t1
κlog(1− u
κ
), α= 0,
(9)
with λ= (1−α)/κ. Note that K(u, v) is well-defined in an open neighborhood of (0,0).
In order to derive an approximate conditional PDF ofXt1 given Xt2 = z, denoted by fXt1 |Xt2 (x|z),
we apply the double saddlepoint density approximation (7).
Theorem 1. Let Xt be a tempered stable Levy subordinator associated with TS(α,κ), α ∈ [0,1),
and κ> 0. Then the saddlepoint approximation fXt1 |Xt2 (x|z) for the conditional PDF fXt1 |Xt2 (x|z)
conditioned on Xt2 = z, where 0< t1 < t2, is as follows. If α∈ (0,1), then
fXt1 |Xt2 (x|z) =1
N(z;α,κ; t1, t2)· 1√
2πκ
(t1(t2− t1)
t2
) 12(1−α)
(z
(z−x)x
) 2−α2(1−α)
× exp
[−(1−α)2
ακ
{t1
11−α
xα
1−α+
(t2− t1)1
1−α
(z−x)α
1−α− t2
11−α
zα
1−α
}], (10)
Kim and Kim: Simulation of tempered stable Levy bridges12 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
where N(z;α,κ; t1, t2) is a normalizing constant. If α= 0, then for some normalizing constant N ′
fXt1 |Xt2 (x|z) =1
N ′(z;α,κ; t1, t2)· t
t2κ + 1
22
(t2− t1)t2−t1κ + 1
2 tt1κ + 1
21
· 1√2πκ
(z−x)t2−t1κ −1x
t1κ −1z1− t2κ .
Recall the two special cases of α = 0 (gamma bridge) and α = 1/2 (IG bridge) whose exact
conditional densities fXt1 |Xt2 (x|z) are known from Ribeiro and Webber (2003, 2004):
fGBXt1 |Xt2
(x|z) =1
Beta(t1κ, t2−t1
κ
)x t1κ −1(z−x)t2−t1κ −1z1− t2κ , (11)
and
f IGBXt1 |Xt2
(x|z) =1√2πκ· t1(t2− t1)
t2
(z
(z−x)x
)3/2
exp
[− 1
2κ
(t1
2
x+
(t2− t1)2
z−x− t2
2
z
)]. (12)
Somewhat surprisingly, the exact densities and the approximate densities from Theorem 1 coincide
in these special cases.
Remark 3. An unscaled version of the approximate conditional PDF fXt1 |Xt2 (x|z) associated
with TS(α,λ, c) can be easily computed by simply replacing κ with (1−α)(Γ(1−α)c)− 1
(1−α) . This
indicates that the conditional law depends only on the parameters α and c, as expected from the
scaling property.
Remark 4. The same procedure can be conducted for the CGF of a stable process, and it gives
the equivalent result as in Theorem 1. This is also expected from Lemma 1.
Remark 5. The double saddlepoint approximation above agrees with the conditional density
fXt1 (x)fXt2−t1 (z−x)/fXt2 (z), where we use the one-dimensional single saddlepoint approximation
(6) for the PDF fXt(x) with the CGF K(u) = tl(u) as in (8):
fXt(x)∝ 1√2πκ· t
12(1−α)
x2−α
2(1−α)
exp
[(1−α)t
ακ− (1−α)x
κ− (1−α)2
ακ· t
1(1−α)
xα
(1−α)
].
Note that the double saddlepoint approximation is determined from the associated joint CGF while
the one-dimensional version is derived directly from the univariate CGF. The two approaches are
not always seen to agree. However, it is not difficult to see that the two conditional approximate
PDFs are the same for Levy processes due to their stationary and independent increment properties.
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 13
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8
9
x
co
ditio
na
l d
en
sity f
(x|z
)
α=0 (Gamma)
α=0.2
α=0.5 (IG)
α=0.8
(i)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
x
co
ditio
na
l d
en
sity f
(x|z
)
α=0 (Gamma)
α=0.2
α=0.5 (IG)
α=0.8
(ii)
Figure 1 Approximate conditional PDF fXt1|Xt2
(x|z) given z = 1 with t2 = 1/2 and κ= 0.2505: (i) t1 = 1/3, and
(ii) t1 = 1/4 (symmetric case).
Remark 6. An extension of Theorem 1 to a multivariate random vector is given in EC.2.1 of the
e-companion.
Figure 1 shows the graphs of the approximate conditional PDF fXt1 |Xt2 (x|z) for z = 1 in α
where parameters are given as t2 = 1/2 and κ = 0.2505. The graphs for α = 0,1/2 are the exact
conditional densities of gamma bridge and IG bridge. As seen in Figure 1(i), the density achieves
its maximum on [z/2, z) when t2− t1 ≤ t1; we can also observe from the explicit form (10) that it
attains the maximum on (0, z/2] when t2− t1 > t1. In the right panel (ii), on the other hand, we see
the symmetric shape of conditional densities when t1 = t2/2. The graphs give us a clue in choosing
proposal densities for the acceptance-rejection algorithm that we design in Section 3.2.
Even though an error analysis of saddlepoint approximation is difficult, it is still possible to test
the accuracy of the approximation in Theorem 1 numerically. The exact conditional PDF can be
obtained from Lemma 1 and the PDF (3) of a stable process. With the new parametrization (α,κ),
the PDF (3) is re-written as
fXt(u) =1
π
∑n∈N
(−1)n−1 sin(nπα)Γ(nα+ 1)
n!
((1−ακ
)1−αt
α
)nu−nα−1. (13)
Our MATLAB implementation retains enough terms in the infinite sum in (13) so that there is
essentially no difference in numerical outcomes. Figure 2 compares two conditional PDFs where
Kim and Kim: Simulation of tempered stable Levy bridges14 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
the parameters are given as α= 0.3, κ= 0.2505, z = 1, t1 = 1/4, and t2 = 1/2. The average of the
relative errors with respect to the true conditional PDF∣∣∣1− fXt1 |Xt2 (x|z)/fXt1 |Xt2 (x|z)
∣∣∣ over the
range of x, i.e. (0, z), is 0.9%.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
co
ditio
na
l d
en
sity f
(x|z
)
Exact PDF
SPA PDF
Figure 2 Comparison of the approximate conditional PDF with the exact PDF for parameters α= 0.3, κ= 0.2505,
z = 1, t1 = 1/4, and t2 = 1/2.
We repeat the same exercise for different κ and t1/t2 values as they turn out to be important
factors that affect the performance of the approximation based on extensive numerical tests. Figure
3(i) and (ii) plot the average and maximum of relative errors over κ and t1/t2, respectively. While
κ varies, the other parameters are fixed as done in Figure 2. When we compute the relative errors,
we set the step size of x equal to 0.001. While t1/t2 varies, κ is set to be 1 and the rest of the
parameters remain the same.
In Figure 3(i), the average relative errors are less than 4% but the maximum errors tend to
increase as κ increases. On the other hand, in Figure 3(ii) it is shown that the averages are more or
less small (less than 4%) in the most of the region except for the extreme cases where t1/t2 is close
to 0 or 1. This implies that sampling at middle points would produce the highest accuracy. This
phenomenon naturally leads us to the idea of using a bisection method when generating the entire
trajectories of a process of interest in order to guarantee good performance, and this is exactly
what we propose in Algorithm 4.
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 15
Another important parameter under consideration is α, however, the behavior of the series rep-
resentation is quite unstable for the parameter domain (0,1), making the overall comparison of two
approaches impractical. Nevertheless, we observe from many individual trials that the saddlepoint
method yields a high accuracy near α= 1/2.
0.5 1 1.5 2 2.50
0.05
0.1
0.15
0.2
κ
Rela
tive e
rror
Maximum of RE
Average of RE
(i)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.1
0.2
0.3
0.4
t1/t
2
Rela
tive e
rror
Maximum of RE
Average of RE
(ii)
Figure 3 Average and maximum relative errors with α= 0.3, z = 1, and t2 = 1/2: (i) κ varies with t1 = 1/4 (ii)
t1/t2 varies with κ= 1.
3.2. Sampling algorithms
To sample from (Xt1 |Xt2 = z), utilizing fXt1 |Xt2 (x|z) in Theorem 1, we propose an acceptance-
rejection algorithm that uses the known PDFs of the gamma and IG bridges as proposal densities.
More precisely, when α < 1/2, we let our proposal density be the conditional PDF of a gamma
process (α= 0) of (11). If α> 1/2, then we choose to use the conditional PDF of an IG process (α=
1/2) of (12). For these proposal densities, an acceptance-rejection method becomes implementable
thanks to the following result.
Kim and Kim: Simulation of tempered stable Levy bridges16 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
Proposition 3. The ratio of fXt1 |Xt2 (x|z) relative to the gamma bridge density (11) if α∈ (0,1/2)
or to the IG bridge density (12) if α∈ (1/2,1) is uniformly bounded on (0, z).
The ratios of fXt1 |Xt2 (x|z) and the proposal densities appear as constant times some auxiliary
functions, say g1(x) and g2(x) as in EC.1.5. Let us denote the respective upper bounds of those
functions by Cg1 and Cg2 . We first begin with the bridge sampling algorithm for IG subordinators,
introduced in Ribeiro and Webber (2003) in Algorithm 2. Then in Algorithm 3, our sampling
algorithm for tempered stable Levy bridges is presented.
Algorithm 2 Inverse Gaussian Bridge sampling algorithm [IGB]
1: Generate Q∼ χ2(1)
2: Compute s1 = t2−t1t1
+Q zκ2t21− zκ
2t1(t2−t1)
√4Q (t2−t1)3
t1zκ+ (t2−t1)2
t21Q2 and s2 = (t2−t1)2
t21s1
3: Generate Ber∼Bernoulli(p1) with p1 = t2−t1t1
(1 + s1)÷ t2t1
( t2−t1t1
+ s1)
4: Set R←Ber · s1 + (1−Ber) · s2 and X← z1+R
5: return X
In applying Algorithm 3 which we call [TSLB] hereafter, it is crucial to compute tight upper
bounds Cg1 and Cg2 for efficient simulation. Unfortunately, analytical computations for the maxima
of g1(x) and g2(x) on the interval (0, z) turn out to be very difficult albeit their explicit expressions.
Any numerical methods to compute Cg1 and Cg2 can be applied. One straightforward method is to
compute the maximum values at discrete points on the half interval [0, t2/2] or [t2/2, t2], depending
on the relative size of t1 versus t2, and then we can interpolate them. A bisection method can be
an alternative to find a smaller interval that contains the maximum. Numerical experiments show
that the efficiency of such computational methods for Cg1 and Cg2 varies according to parameter
settings.
We may further consider tabulating those upper bounds in terms of z and δ = t1/t2 via the
scaling property of a bridge process in Proposition 1. More specifically, suppose that we have a
table of Cg1 or Cg2 given parameter set (α,λ, c) at t2 = 1. For general t2, let r= t−1/α2 . Then(
Xδt2(α,λ, c)∣∣Xt2(α,λ, c) = z
)d=
(1
rXrαδt2(α,λ/r, c)
∣∣∣∣ 1
rXrαt2(α,λ/r, c) = z
)
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 17
Algorithm 3 Tempered Stable Levy Bridge sampling algorithm [TSLB]
1: if 0<α< 12
then
2: repeat
3: Compute Cg1 given z, t1, t2
4: Generate B ∼Beta(t1κ, t2−t1
κ
)and U ∼Unif(0,1)
5: Set X← zB
6: until U ≤ g1(X)/Cg1
7: return X
8: else if 12<α< 1 then
9: repeat
10: Compute Cg2 given z, t1, t2
11: Generate Y from [IGB] and U ∼Unif(0,1)
12: Set X← Y
13: until U ≤ g2(X)/Cg2
14: return X
15: end if
=
(1
rXδ(α,λ/r, c)
∣∣∣∣X1(α,λ/r, c) = rz
). (14)
Since the parameter λ does not affect the bridge process, one can sample from (Xδ|X1 = rz) by
using an interpolated bound from the table and then multiply it by t1/α2 . When t1 = t2/2, which
commonly happens in financial applications, it suffices to compute a one-dimensional table for z
only.
Remark 7. In order to improve the efficiency of [TSLB], we may approximate the bridge process
as the conditional expectation by Proposition 2 when z is very small. Since 0 ≤ (Xt1 |Xt2 = z) ≤
z, the error |(Xt1 |Xt2 = z)− zt1/t2| does not exceed z. If z is very small and additionally time
points t1 and t2 are small, the samples from the gamma bridge and IG bridge proposals may
Kim and Kim: Simulation of tempered stable Levy bridges18 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
not be distinguishable from zero numerically. Therefore, sampling from [TSLB] is relatively time-
consuming, and using the conditional expectation could be a reasonable alternative for speed-up.
The expected number of iterations Mi, (i= 1,2) of [TSLB] can be computed as
(i) 0<α< 12:
M1 = supx∈(0,z)
fXt1 |Xt2 (x|z)fGBt1|t2(x|z)
= Beta
(t1κ,t2− t1κ
)zt2κ −1Cg1
N,
(ii) 12<α< 1:
M2 = supx∈(0,z)
fXt1 |Xt2 (x|z)f IGBt1|t2(x|z)
=t2√
2πκ
t1(t2− t1)z−
32 exp
(− t22
2κz
)Cg2N
.
Here, the normalizing constant N is given by
N =
∫ z
0
(x(z−x)
)−1− p2exp
[− 1
p(1 + p)κ
{t1+p1
xp+
(t2− t1)1+p
(z−x)p
}]dx.
Figure 4 plots three-dimensional shaded surfaces of M1 and M2 over a rectangular grid (t1/t2, κ)
such that 0< t1/t2 < 1 with step size 0.01 and 0<κ< 2 with step size 0.05. Other parameters are
fixed as z = 1 and t2 = 1; α= 0.3 for M1 and 0.7 for M2.
00.2
0.40.6
0.81
0
0.5
1
1.5
20
2
4
6
8
10
12
t1 / t
2
κ
Expecte
d n
um
ber
of itera
tions
(i)
00.2
0.40.6
0.81
0
0.5
1
1.5
21
2
3
4
5
t1 / t
2
κ
Expecte
d n
um
ber
of itera
tions
(ii)
Figure 4 Expected number of iterations of [TSLB] algorithm as t1/t2 and κ vary where z = 1, t2 = 1: (i) M1 with
α= 0.3 (ii) M2 with α= 0.7.
First of all, Mi rapidly increases as t1/t2 goes to the extremes 0 or 1. In these cases, an existing
sequential simulation algorithm is recommended, considering that the accuracy of the approximate
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 19
conditional PDF is low as well. A guideline for the specific choice of such an algorithm is briefly
described in EC.3. However, except for these extremes, the number of iteration of [TSLB] is mostly
less than 2 in expectation. In particular, when t1/t2 = 1/2, the average M1 and M2 over κ ∈ (0,2)
with step size 0.05 are 1.2658 and 1.3636, respectively.
The behaviors of Mi’s with respect to other parameters κ and α are also observed. In the same
figure, we see that the expected number of iterations tends to increase as κ increases, however, the
growth rate is quite low compared to their sensitivities with respect to t1/t2 near extremes. As for
the stability index α, Figure 5 shows the average values of Mi over κ ∈ (0,2) with step size 0.05
versus the parameter α. As expected from Figure 1, Mi’s increase in α but again they stay below
2 in a large region of α.
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.451
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
α
Avera
ged M
1 o
ver
κ
(i)
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.951
1.5
2
2.5
3
3.5
4
4.5
α
Avera
ged M
2 o
ver
κ
(ii)
Figure 5 Averaged Mi of [TSLB] algorithm over κ ∈ (0,2) as α varies, where z = 1, t2 = 1 and t1/t2 = 1/2 are
fixed: (i) M1 with 0.05<α< 0.45 (ii) M2 with 0.55<α< 0.95.
In principle, saddlepoint methods for conditional distributions are applicable to any Levy pro-
cess as long as its Laplace transform is available. Additionally, if a process of interest belongs
to some parametric class, one of which has a closed form PDF, then a non-Gaussian base would
likely result in excellent performance for the process as mentioned in EC.2.2. However, in most
cases, it is difficult to obtain the saddlepoint explicitly or even to approximate it. Moreover, often
Kim and Kim: Simulation of tempered stable Levy bridges20 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
the complexity of the formula (7) makes the task of sampling from the conditional PDF highly
nontrivial.
3.3. Discussion on performance of the proposed method
For the rest of this section, we make a short discussion on the relative performance of [TSLB]
in comparison to existing simulation algorithms for tempered stable distributions. In particular,
we consider the three algorithms mentioned below Algorithm 1. Following the guideline in EC.3,
one can find the best performing algorithm for a given set of parameters. This combined exact
algorithm is denoted by hybrid sequential algorithm, or HSQ in short.
Let Xt be a tempered stable Levy subordinator associated with TS(α,κ). We generate the same
number of sample paths (Xt1 ,Xt2 , · · · ,Xtd) at given time points {t1, . . . , td} with constant step size
∆ = ti+1− ti. When simulating a trajectory via bridge sampling (BS), we first generate the terminal
value Xtd using HSQ or the numerical Laplace inversion of a tempered stable distribution. At the
given end point, we fill the intermediate samples by bisecting time points. Algorithm 4 describes
the BS procedure with a possible tabulation where a time grid with d= 2m for some integer m is
given.
We compare computational times to generate the same number of trajectories of (X∆, · · · ,Xd∆)
for a fixed ∆. We refer the reader to EC.3 for detailed numerical outcomes. To sum up, we observe
that the HSQ algorithm is more efficient than BS without tabulation for generating a given number
of points in general. However, bridge sampling can be an attractive alternative when we tabulate
bounding constants, and the computational burden of this tabulation is small. In our experiments,
BS is comparable in speed to HSQ without losing its high accuracy. Furthermore, BS will boost its
strength when variance reduction techniques or low-discrepancy methods can be adopted. More-
over, the advantages of BS stand out in settings where we need to fill in intermediate values, as is
illustrated in the next section.
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 21
Algorithm 4 Dyadic generation of a sample path of a tempered stable Levy subordinator via
[TSLB] without or with tabulation
1: At given time points 0 = t0 < t1 < · · ·< td with d= 2m,
2: X0← 0 and generate Xtd ∼ TS(α, (1−α)/κ, td((1−α)/κ)(1−α)/Γ(1−α))
3: for l← 1 to m do
4: n← 2m−l
5: for j← 1 to 2l−1 do
6: i← (2j− 1)n
7: if the pre-computed table of [zvec;Cvec] bounds for (X1/2|X1) is given then
8: Set r← (ti+n− ti−n)−1/α
9: Find C that is interpolated with r(Yti+n −Yti−n) given the table [zvec,Cvec]
10: Set Cgi←C for i= 1,2 in line 3 and 10 of [TSLB]
11: Generate Y from [TSLB] with t1 = 1/2, t2 = 1, and z = r(Yti+n −Yti−n)
12: Set Y ← Y/r
13: else
14: Set t1 = ti− ti−n, t2 = ti+n− ti−n and z = Yti+n −Yti−n
15: Generate Y from [TSLB] with t1, t2, and z
16: end if
17: Set Xti←Xti−n +Y
18: end for
19: end for
4. Option Pricing in Levy Models
In this section, we consider option pricing problems when the underlying asset price dynamics
is governed by tempered stable Levy processes. Poirot and Tankov (2006) show that under an
appropriate equivalent probability measure a tempered stable process becomes a stable process.
This provides a fast Monte Carlo algorithm for computing the expectation of European contingent
Kim and Kim: Simulation of tempered stable Levy bridges22 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
claims under a tempered stable process. This method, however, does not allow access to the entire
trajectory of the process, and thus we need a different sample-path generation method to deal with
path-dependent financial contracts.
More specifically, in Section 4.1, we extend the adaptive sampling scheme developed in Becker
(2010) to finite variation tempered stable processes for pricing path-dependent options. This leads
to remarkable efficiency gains in terms of computational costs. Significant amount of variance
reduction is observed by stratified sampling in Section 4.2.
4.1. Adaptive bridge sampling under finite variation tempered stable (CGMY) processes
The asset price is assumed to follow a geometric Levy process:
St = S0 exp[(w+ r)t+Xt
]where r is a risk free interest rate and w is the constant chosen so that the discounted value of a
portfolio invested in the asset is a martingale. One popular model of asset price dynamics based on
Levy models is two-sided tempered stable processes associated with TS(α+, λ+, c+; α−, λ−, c−) in
Definition 3. Such a process, say Xt, can be written as the difference of two independent Levy pro-
cesses, that is, Xt =X+t −X−t where X+
t and X−t are tempered stable subordinators associated with
TS(α+, λ+, c+) and TS(α−, λ−, c−), respectively. This includes the widely known finite-variation
CGMY processes (Carr et al. 2002). In the six-parameter setting, it is further constrained that
α+ = α− and c+ = c−.
Avramidis and L’Ecuyer (2006) develop efficient Monte Carlo algorithms for pricing path-
dependent options under the VG model, based on the representation of a VG process as the
difference of two increasing gamma processes. (Hence the name difference-of-gamma bridge sam-
pling or simply DGBS.) The authors obtain a pair of estimators whose expectations are monotone
in increasing resolution, and the true option price lies in between. When unbiased price estimates
are expensive to obtain, this pair provides upper and lower bounds without having to simulate all
the process trajectories.
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 23
In principle, as described in Section 6 of Avramidis and L’Ecuyer (2006), this method is extend-
able to option-pricing models driven by any Levy process whose paths have finite variation. How-
ever, it is noted as a major difficulty for such an extension that there has been no bridge sampling
algorithm for a more general class of Levy processes such as CGMY processes. In this sense, we
may extend the algorithm DGBS to Xt based on our bridge sampling scheme. A direct extension,
however, seems undesirable since BS loses its relative merits with respect to HSQ as the monitoring
step size decreases to zero. Nevertheless, we tested this idea and confirmed the accuracy of BS
through the pricing of path-dependent options described below. We do not report the results in
the paper to economize on space.
Although such a direct extension of DGBS does not yield any relative advantages, bridge sam-
pling is ideally suited to adaptive versions of DGBS that can lead to large savings in simulation
costs. Becker (2010) has developed such an adaptive algorithm for VG processes, and we now
extend the algorithm to CGMY processes. The key idea of adaptive sampling is to exclude the
intervals from a given partition of the time dimension that neither contribute to the minimum nor
to the maximum of a sample path. The author then presents adaptive sampling procedures for gen-
erating the infimum and supremum of VG processes and for pricing of lookback and barrier options
described in Figures 2 and 3 in Becker (2010). Numerical studies show considerable reduction in
computational efforts and memory requirements by reducing the number of bridge sampling steps.
In this subsection, we adopt this adaptive DGBS and extend the whole idea to the option
pricing under CGMY processes. To be specific, let Xt = X+t −X−t be a CGMY process where
X+t ∼ TS(Y,G,C) and X−t ∼ TS(Y,M,C) are independent tempered stable Levy subordinators,
and let µ=w+ r where w=− lnE[
exp(X1)]
=−CΓ(−Y ){
(G−1)Y −GY + (M + 1)Y −MY}. For
t1 < t< t2, we can see that
L, µt1 +X+t1−X−t2 + (t2− t1)µ− ≤ µt+Xt ≤ µt1 +X+
t2−X−t1 + (t2− t1)µ+ ,U
by the monotonicity of the subordinators with µ− = min(0, µ) and µ+ = max(0, µ). If T is the
current set of grid points during the execution of the sampling procedure, we clearly have that
Kim and Kim: Simulation of tempered stable Levy bridges24 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
maxt∈T Xt ≤maxt∈[0,T ]Xt and mint∈T Xt ≥mint∈[0,T ]Xt. Then, for a particular interval [t1, t2] of
the current partition, if mint∈T Xt ≤ L (≤) U ≤ maxt∈T Xt holds, the interval neither contains
the minimum nor the maximum of the path. Therefore, one can exclude the interval from further
subdivisions.
In what follows, we demonstrate numerical experiments for lookback and barrier options whose
discounted payoffs are given as follows with maturity T and the monitoring interval size ∆ = T/d:
1. floating strike lookback call VL = e−rTE[ST −max
{S0, S∆, . . . , Sd∆
}];
2. up-and-in call barrier VB = e−rTE[
max{
0, ST −K}1{max{S0,S∆,...,Sd∆}>B}
].
Table 1 reports the reference parameter set, denoted by (a), that is obtained by calibrating the
model to the market prices of the options on the S&P 500 index for June 2, 2003 in Carr et al.
(2005). In addition, a risk free interest rate r= 0.0548, initial stock price S0 = 100, strike K = 100
Table 1 The results of the calibration of CGMY model on SPX (Carr et al. 2005).
Parameter set T C G M Y
(a) 1.0453 0.3251 3.7103 18.4460 0.6029
and barrier level B = 120 are used.
In the parameter set (a), we take SSR algorithm as the representative of HSQ, since it works
most efficiently. In fact, BS without tabulation cannot beat SSR if a fixed number of points per path
are generated. However, the adaptive sampling scheme greatly reduces the number of simulated
points by BS. For an illustration, see Figure 6. We plot the computing times of SSR, adaptive BS,
and adaptive BS with tabulation, as the observation time d increases. The sample size for price
estimation is fixed at 104. The computing time of SSR increases exponentially, while that of adaptive
BS grows slowly. Furthermore, adaptive BS with tabulation shows outstanding performances for
larger d values. Here, upper bounds are tabulated with the step size 0.01 for z. The speed-up from
adaptive sampling is more prominent for barrier options as shown in the right panel of Figure 6.
On the other hand, there is the associated trade-off between bias and variance. In adaptive BS,
biases are caused by saddlepoint approximation in BS and the minimum approximation due to
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 25
6 7 8 9 10 11 120
20
40
60
80
100
120
log2 (d)
Com
puting tim
e for
lookback o
ption
SSR
adaptive BS
adaptive BS w/ tabulation
(i)
6 7 8 9 10 11 120
20
40
60
80
100
120
log2 (d)
Com
puting tim
e for
barr
ier
option
SSR
adaptive BS
adaptive BS w/ tabulation
(ii)
Figure 6 Computing times (in seconds) for pricing (i) lookback and (ii) barrier options versus the observation
times d with the sample size 104 under the parameter set (a).
the adaptive scheme. The true option prices are computed using 109 Monte Carlo simulation trials
by SSR algorithm. This is used to check the biases generated from adaptive BS with and without
tabulation at that particular maturity. Figure 7 illustrates the estimated mean squared errors
(denoted by MSE, given by the sum of the squared standard error and the squared bias) versus
the computing times for the sample size 104,105 and 106 under the parameter set (a) when d= 27.
Note that biases caused by adaptive BS does not show a marked difference and that adaptive BS
dominates HSQ in terms of MSE.
4.2. Stratified sampling
Stratification is often applied to Monte Carlo simulation in option pricing, yielding successful effec-
tive speed-ups in convergence and variance reduction. For example, the use of stratified sampling
for the VG process and the NIG process with the gamma bridge and the IG bridge is studied in
Ribeiro and Webber (2003, 2004). In this section, we discuss the construction of such stratified sam-
ples for the terminal values of two-sided tempered stable processes, apply it to the path-dependent
options 1–2, and examine its effects on variance reduction.
The final value XT of the tempered stable subordinator can also be easily generated by numerical
Laplace inversion. A reliable and efficient technique is explained in Glasserman and Liu (2010) based
Kim and Kim: Simulation of tempered stable Levy bridges26 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
0 0.5 1 1.5 2 2.5 3 3.50
0.02
0.04
0.06
0.08
0.1
0.12
0.14
log10
(time)
MS
E
SSR
Adaptive BS
Adaptive BS w/ tabulation
(i)
−1 −0.5 0 0.5 1 1.5 2 2.5 30
0.02
0.04
0.06
0.08
0.1
0.12
0.14
log10
(time)
MS
E
SSR
Adaptive BS
Adaptive BS w/ tabulation
(ii)
Figure 7 Estimated mean squared errors versus computation times (in seconds) for the (i) Lookback and (ii)
Barrier options for the sample size 104,105 and 106 under the parameter set (a).
on the analysis of Abate and Whitt (1992). The sampling algorithm generates U = Unif[0,1] and
then finds x> 0 such that F (x) =U with F the distribution function of XT , which is pre-computed
and tabulated following Abate and Whitt (1992). We may use spline or linear interpolation for x.
In more detail, the distribution F with its Laplace transform ϕ is computed as
F (x)≈ hx
π+
2
π
M∑k=1
sin(hkx)
kRe(ϕ)(−ikh), h=
2π
x+uε.
Here, uε is set equal to µXT +mσXT where µXT is the mean of XT , σXT is its standard deviation,
and m is some integer suitably chosen to achieve sufficient accuracy (e.g. m= 10).
With the help of the Laplace inversion method, sample paths with stratified final values can be
generated as follows for a given number of simulation trials n and length of time steps d:
1. generate a stratified sample (ui, vi), i= 1, · · · , n, from the unit hypercube of dimension 2;
2. set X+ iT = F−1
+ (ui) and X− iT = F−1
− (vi) for each i= 1, · · · , n where F± is the distribution of
X±T respectively;
3. generate entire trajectories{X+ itj
}j=1,··· ,d−1
and{X− itj
}j=1,··· ,d−1
conditioned on the strati-
fied terminal values for each i by Algorithm 4.
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 27
In Step 1, let K1 and K2 be the numbers of intervals of equal length along each dimension and
thus each stratum of the hypercube is of the form[l1− 1
K1
,l1K1
)×[l2− 1
K2
,l2K2
), lj ∈ {1,2, · · · ,Kj} for j = 1,2.
The related sampling probabilities are 1/K1K2 for each region. A random vector (ui, vi) uniformly
distributed over this stratum for each i is simply
(ui, vi) =
(l1− 1 +U1
K1
,l2− 1 +U2
K2
)where U1,U2 are independent uniform random variables in the unit interval.
Now we present the numerical results of stratified sampling under the parameter set (a). The
sample size is n and the number of strata K1 and K2 are both set to be 10. We fix d = 26 and
the estimated standard deviations (STD) of the resulting price estimates turn out not to vary
substantially over d. Figure 8 shows a significant amount of additional variance reduction for BS
with terminal stratification. It appears that the estimated STD of all the path-dependent option
values are greatly reduced. Note that, in Figure 8, we deliberately use the log sample size (not
the computation time) on the horizontal axis and STD on the vertical axis. This is to identify the
effect of stratified sampling for the additional variance reduction.
Remark 8. Stratification can also be combined with sequential sampling by stratifying at an
initial non-zero time point. However, we find from numerical tests that there is little gain from such
initial stratification. This is anticipated since the most key features of the path in option pricing
often depends on the asset values at the maturity.
Although not reported, effective variance reductions are observed for different maturities, mon-
itoring frequencies, barrier levels, and model parameters. The effect of stratification for normal
tempered stable processes is discussed as well in EC.4.
Remark 9. In EC.5 of the e-companion, we present another application of bridge sampling. In
short, a hybrid method that benefits from both sequential and bridge sampling is proposed. It is
Kim and Kim: Simulation of tempered stable Levy bridges28 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
4 4.5 5 5.5 60
0.05
0.1
0.15
0.2
0.25
0.3
0.35
log10
(sample size)
ST
D
SSR
Adaptive BS
Adaptive BS − Stratification
(i)
4 4.5 5 5.5 60
0.05
0.1
0.15
0.2
0.25
0.3
0.35
log10
(sample size)
ST
D
SSR
Adaptive BS
Adaptive BS − Stratification
(ii)
Figure 8 Estimated standard deviation due to stratification versus the number of simulation trials for the (i)
Lookback, and (ii) Barrier options under the parameter set (a).
applied to least square Monte Carlo methods for American option valuation under subordinated
Brownian motions, in order to avoid large memory requirements. Numerical results are also reported
in EC.5.
5. Conclusion
We showed that the approximate PDF of a tempered stable subordinator conditioned on the two
observed endpoints using saddlepoint methods gives an accurate bridge distribution for a wide
range of parameters. We proposed an acceptance-rejection method to simulate tempered stable
Levy bridges with the gamma bridge and the IG bridge as proposal densities. Special properties
of the process such as scaling and linear conditional moments were exploited to enable further
speed-ups. The suggested bridge sampling algorithm was applied to the pricing of path-dependent
options for a finite-variation CGMY process. Through the suggested adaptive sampling scheme,
extensive numerical tests demonstrated the accuracy and effectiveness of bridge sampling. However,
the computational burden for evaluating upper bounds in bridge sampling still remains although
it can be alleviated via tabulation.
Diverse applications of bridge sampling are possible. One particular example that we imple-
mented was stratified sampling and we observed a large amount of efficiency gains and variance
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 29
reduction. In addition, one might combine bridge sampling with randomized quasi-Monte Carlo
method. Optimal stopping time problems are also potentially related; in particular adaptive sam-
pling via bridge sampling is plausible by choosing the next step size of the simulated process
adaptively near the stopping boundary.
There are other important research topics related to the proposed method. First, one can con-
sider an extension to infinite-variation tempered stable processes with α > 1, for which only an
approximate sequential sampling method exists. Second, it is potentially useful to have an approxi-
mate conditional density and the bridge sampling scheme for statistical inference of Levy processes.
One can investigate simulation-based solutions for missing data problems as done for diffusions, or
possible applications to the Monte Carlo EM algorithm and Bayesian methods.
Acknowledgments
The authors would like to thank Prof. Paul Glasserman for his helpful comments and the workshop partic-
ipants of Stochastic Processes and their Statistics held in 2013, Okinawa, for their feedback. The authors
are also very grateful for many valuable comments from two anonymous reviewers and the Associate Editor
which helped them improve the manuscript greatly. K. Kim’s work was supported by the Basic Science
Research Program through the National Research Foundation of Korea funded by the Ministry of Education
(NRF-2014R1A1A2054868). The author names are in alphabetical order.
References
Abate J, Whitt W (1992) The Fourier-series method for inverting transforms of probability distributions.
Queueing Systems, 10(1):5–88.
Asmussen S, Hobolth A (2012) Markov bridges, bisection and variance reduction. In Monte Carlo and
Quasi-Monte Carlo Methods 2010, edited by L. Plaskota, H. Wozniakowski, Springer-Verlag, Berlin.
Avramidis AN, L’Ecuyer P (2006) Efficient Monte Carlo and Quasi-Monte Carlo option pricing under the
variance gamma model. Management Science, 52(12):1930–1944.
Avramidis AN, L’Ecuyer P, Tremblay PA (2003) Efficient simulation of gamma and variance-gamma pro-
cesses. In Proceedings of the 2003 Winter Simulation Conference, edited by S. Chick, P. J. Sanchez, D.
Ferrin, and D. J. Morrice. IEEE Press, Piscataway, NJ, 319–326.
Kim and Kim: Simulation of tempered stable Levy bridges30 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
Barndorff-Nielsen OE, Cox DR (1979) Edgeworth and saddlepoint approximations with statistical applica-
tions. Journal of the Royal Statistical Society, Series B, 41(3):279–312.
Barndorff-Nielsen OE, Shephard N (2001) Normal modified stable processes. Theory of Probability and
Mathematical Statistics, 65:1–19.
Becker M (2010) Unbiased Monte Carlo valuation of lookback, swing and barrier options with continuous
monitoring under variance gamma models. Journal of Computational Finance, 31(4):35–61.
Beskos A, Papaspiliopoulos O, Roberts GO, Fearnhead P (2006) Exact and computationally efficient
likelihood-based estimation for discretely observed diffusion processes. Journal of the Royal Statistical
Society, Series B, 68(3):333–382.
Bladt M, Sørensen M (2014) Simple simulation of diffusion bridges with application to likelihood inference
for diffusions. Bernoulli, 20(2):645–675.
Brody DC, Hughston LP, Macrina A (2007) Beyond hazard rates: A new framework for credit-risk mod-
elling. In Advances in Mathematical Finance, edited by Elliot, R., Fu, M., Jarrow, R. and Yen, Ju-
Yi, Applied and Numerical Harmonic Analysis Series, Festschrift volume in honour of Dilip Madan,
Birkhauser/Springer.
Brody DC, Hughston LP, Macrina A (2008) Information-based asset pricing. International Journal of
Theoretical and Applied Finance, 11(1):107–142.
Butler RW (2007) Saddlepoint Approximations with Applications. Cambridge University Press, Cambridge.
Carr P, Geman H, Madan D, Yor M (2002) The fine structure of asset returns: An empirical investigation.
Journal of Business, 75(2):305–333.
Carr P, Geman H, Madan D, Yor M (2005) Pricing options on realized variance. Finance and Stochastics,
9(4):453–475.
Carr P, Madan D (2009) Saddlepoint methods for option pricing. Journal of Computational Finance, 13(1):
49–61.
Chambers JM, Mallows CL, Stuck BW (1976) A method for simulating stable random variables. Journal of
the American Statistical Association, 71(354):340–344.
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 31
Cont R, Tankov P (2004) Financial Modeling with Jump Processes. Chapman and Hall/CRC, Boca Raton,
FL.
Daniels HE (1954) Saddlepoint approximations in statistics. Annals of Mathematical Statistics, 25(4):
631–650.
Daniels HE (1958) Discussion of “The regression analysis of binary sequences” by D. R. Cox. Journal of the
Royal Statistical Society, Series B, 20(2):236–238.
Dembo A, Deuschel JD, Duffie D (2004) Large portfolio losses. Finance and Stochastics, 8(1):3–16.
Devroye L (2009) Random variate generation for exponentially and ploynomially tilted stable distributions.
ACM Transactions on Modeling and Computer Simulation, 19(4):18:1–20.
Durham GB, Gallant AR (2002) Numerical techniques for maximum likelihood estimation of continuous-time
diffusion processes. Journal of Business & Economic Statistics , 20(3):297–316.
Figueroa-Lopez JE, Houdre C (2009) Small-time expansions for the transition distributions of Levy processes.
Stochastic Processes and their Applications, 119(11):3862–3389.
Figueroa-Lopez JE, Tankov P (2014) Small-time asymptotics of stopped Levy bridges and simulation schemes
with controlled bias. Bernoulli, 20(3):1126–1164.
Glasserman P (2003) Monte Carlo Methods in Financial Engineering. Springer-Verlag, New York.
Glasserman P, Kim K (2008) Beta approximations for bridge sampling. In Proceedings of the 2008 Winter
Simulation Conference, edited by Mason SJ, Hill RR, Monch L, Rose O, Jefferson T, Fowler JW. IEEE
Press, Piscataway, NJ, 569–577.
Glasserman P, Kim K (2009) Saddlepoint approximations for affine jump-diffusion models. Journal of
Economic Dynamics & Control, 33(1):15–36.
Glasserman P, Liu Z (2010) Sensitivity estimates from characteristic funtions. Operations Research, 58(6):
1611–1623.
Gordy MB (2002) Saddlepoint approximation of CreditRisk+. Journal of Banking & Finance, 26(7):1335–
1353.
Grabchak M (2012) On a new class of tempered stable distributions: moments and regular variation. Journal
of Applied Probability, 49(4):1015–1035.
Kim and Kim: Simulation of tempered stable Levy bridges32 Article submitted to Operations Research; manuscript no. (Please, provide the manuscript number!)
Hirsa A, Madan D (2004) Pricing American options under variance gamma. Journal of Computational
Finance, 7(2):63–80.
Hofert, M (2011) Sampling exponentially tilted stable distributions. ACM Transactions on Modeling and
Computer Simulation, 22(1):3:1–11.
Hoyle E, Hughston LP, Macrina A (2011) Levy random bridges and the modelling of financial information.
Stochastic Processes and their Applications, 121(4):856–884.
Jensen JL (1995) Saddlepoint Approximations. Oxford University Press, Oxford.
Kaishev V, Dimitrova D (2009) Dirichlet bridge sampling for the variance gamma process: Pricing path-
dependent options. Management Science, 55(3):483–496.
Kawai R, Masuda H (2011) On simulation of tempered stable random variates. Journal of Computational
and Applied Mathematics, 235(8):2873–2887.
Kuchler U, Tappe S (2011) Tempered stable distributions and applications to financial mathematics. Working
paper.
Lin M, Chen R, Mykland P (2010) On generating Monte Carlo samples of continuous diffusion bridges.
Journal of the American Statistical Association, 105(490):820–838.
Lugannani R, Rice S (1980) Saddlepoint approximation for the distribution of the sum of independent
random variables. Advances in Applied Probability, 12(2):475–490.
Michael J, Schucany W, Haas R (1976) Generating random variates using transformations with multiple
roots. The American Statistician, 30(2):88–90.
Pitman J, Yor M (1982) A decomposition of Bessel bridges. Zeitschrift fur Wahrscheinlichkeitstheorie und
Verwandte Gebiete, 59(4):425–457.
Poirot J, Tankov P (2006) Monte Carlo option pricing for tempered stable (CGMY) processes. Asia-Pacific
Financial Markets, 13(4):327–344.
Ribeiro C, Webber N (2003) A Monte Carlo method for the normal inverse Gaussian option valuation model
using an inverse Gaussian bridge. Working paper.
Ribeiro C, Webber N (2004) Valuing path dependent options in the variance-gamma model by Monte Carlo
with a gamma bridge. Journal of Computational Finance, 7(2):81–100.
Kim and Kim: Simulation of tempered stable Levy bridgesArticle submitted to Operations Research; manuscript no. (Please, provide the manuscript number!) 33
Roberts GO, Stramer O (2001) On inference for partially observed nonlinear diffusion models using
Metropolis-Hastings algorithms. Biometrika, 88(3):603–621.
Rogers LCG, Zane O (1999) Saddlepoint approximations to option prices. Annals of Applied Probability,
9(2):493–503.
Rosinski J (2007) Tempering stable processes. Stochastic Processes and their Applications , 117(6):677–707.
Ruschendorf L, Woerner J (2002) Expansion of transition distributions of Levy processes in small time.
Bernoulli, 8(1):81–96.
Sato K (1999) Levy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge.
Skovgaard IM (1987) Saddlepoint expansions for conditional distributions. Journal of Applied Probability,
24(4):875–887.
Yang J, Hurd TR, Zhang X (2006) Saddlepoint approximation method for pricing CDOs. Journal of
Computational Finance, 10(1):1-20.
e-companion to Kim and Kim: Simulation of tempered stable Levy bridges ec1
E-Companion of the paper titled “Simulation of temperedstable Levy bridges and its applications”
This E-Companion consists of four sections. EC.1 provides all proofs of the results developed
in Section 3 including the proof of the main results, Theorem 1 and Proposition 3. EC.2 is then
devoted to supplementary results followed from Section 3. In EC.3, the performance of the proposed
bridge sampling method is compared to that of the existing sequential methods when generating
a fixed number of observations or a skeleton of a tempered stable subordinator. EC.4 provides
additional numerical results for path-dependent option pricing as well as terminal stratification
under normal tempered stable processes. Finally, EC.5 illustrates a hybrid sampling scheme applied
to least-square Monte Carlo method for American option pricing.
EC.1. Proofs of the results in Section 3
EC.1.1. Proof of Proposition 1
By the scaling property of Xt, we have the relationship fTS(α,λ,ct)(x) = rfTS(α,λ/r,crαt)(rx). Then,
for s < t < T ,
P[Xt ∈ dy|XT = z,Xs = x
]= P
[Xt−s(α,λ, c)∈ d(y−x)|XT−s(α,λ, c) = z−x
]=fTS(α,λ,c(t−s))(y−x)fTS(α,λ,c(T−t))(z− y)
fTS(α,λ,c(T−s))(z−x)dy
=rfTS(α,λ/r,crα(t−s))(r(y−x))rfTS(α,λ/r,crα(T−t))(r(z− y))
rfTS(α,λ/r,crα(T−s))(r(z−x))dy
= P
[1
rXrα(t−s)(α,λ/r, c)∈ d(y−x)
∣∣∣∣ 1
rXrα(T−s)(α,λ/r, c) = z−x
]= P
[1
rXrαt(α,λ/r, c)∈ dy
∣∣∣∣ 1
rXrαT (α,λ/r, c) = z,
1
rXrαs(α,λ/r, c) = x
].
�
EC.1.2. Proof of Lemma 1
For a tempered stable Levy subordinator (Xt)t≥0, let fXt1 |Xt2 (x|z) be the PDF of Xt1 conditioned
on Xt2 = z, where t2 > t1 ≥ 0. It is straightforward to get
fXt1 |Xt2 (x|z) =fTS(α,λ,ct1)(x)fTS(α,λ,c(t2−t1))(z−x)
fTS(α,λ,ct2)(z)=fS(α,ct1)(x)fS(α,c(t2−t1))(z−x)
fS(α,ct2)(z)
ec2 e-companion to Kim and Kim: Simulation of tempered stable Levy bridges
due to the relation (5). �
EC.1.3. Multi-dimensional stable processes and Proof of Proposition 2
There are several equivalent definitions of stable distribution, see Samorodnitsky and Taqqu (1994)
e.g., and here we provide one with the CHF for its use.
Definition EC.1. A random variableX is said to have a stable distribution if there are parameters
0<α< 2, σ≥ 0, −1≤ β ≤ 1, and µ∈R such that its CHF has the following form:
φX(u) =
exp
[−σα|u|α
(1− iβ sign(u) tan(πα
2))
+ iµu], if α 6= 1;
exp[−σ|u|
(1 + iβ 2
πsign(u) ln |u|
)+ iµu
], if α= 1.
(EC.1)
A multivariate stable distribution, as the distribution of a stable random vector, is described in
terms of a finite measure Γ on the unit sphere of Rd and a shift vector µ0.
Definition EC.2. A d-dimensional random vector X = (X1, · · ·,Xd) with 0<α< 2 is said to be
α-stable if there exists a finite measure Γ on the unit sphere of Rd and a vector µ0 ∈Rd such that
φX(u) =
exp
[−∫Sd|(u, s)|α
(1− i sign(u, s) tan(πα
2))
Γ(ds) + i(u, µ0)], if α 6= 1;
exp[−∫Sd|(u, s)|
(1 + i 2
πsign(u, s) ln |(u, s)|
)Γ(ds) + i(u, µ0)
], if α= 1
where (u, s) denote a inner product. Here, Γ is called a spectral measure and the pair (Γ, µ0) is
unique, called a spectral representation. For a two-dimensional stable random vector X = (X1,X2)
with µ0 = 0, called a strictly stable distribution, we have
φX(u, v) = exp
{−∫S2
|us1 + vs2|α(
1− i tan(πα
2
)sign(us1 + vs2)
)Γ(ds)
}, (EC.2)
where s = (s1, s2). Each component Xi of X has a marginal stable distribution with the CHF
(EC.1), where µ= 0,
σ= σi =
(∫S2
|si|αΓ(ds)
)1/α
,
and
β = βi =1
σαi
∫S2
|si|α sign(si)Γ(ds).
e-companion to Kim and Kim: Simulation of tempered stable Levy bridges ec3
In Hardin et al. (1991), the authors give conditions for the existence of the conditional moment
E[|X2|p|X1 = x
]for p > α and obtain an explicit form of E
[X2|X1 = x
]as a function of x, which is
given in the next theorem for a non-symmetric stable subordinator.
Theorem EC.1 (Hardin et al. (1991)). Let (X1,X2) be an α-stable process with 0<α< 1 and
a spectral measure Γ(ds) on the unit circle S2 in R2. If it holds that, for some ν > 1−α,∫S2
Γ(ds)
|s1|ν<∞, (EC.3)
then we have
E[X2|X1 = x
]=
∫S2
s2|s1|α−1sign(s1)Γ(ds)x
σα1. (EC.4)
For the proof of Proposition 2, letXt be a stable subordinator associated with S(α, c) in Definition
1. Applying its Levy measure (2), X1 has the CHF (EC.1) with µ= 0, σ= [−cΓ(−α) cos(πα/2)]1/α,
and β = 1. Then we consider a two-dimensional stable subordinator X = (Xt2 ,Xt1) with 0< t1 < t2
where Xt =Xt(α, c) for 0<α< 1. Its joint CHF is
φX(u, v) = E[
exp i(uXt2 + vXt1)]
= φXt1 (u+ v)φXt2−t1 (u)
= exp[−σα
{|u+ v|αt1 + |u|α(t2− t1)
}+iσα tan(πα/2)
{sign(u+ v)|u+ v|αt1 + sign(u)|u|α(t2− t1)
}]. (EC.5)
Matching the real part of (EC.5) with that of the spectral representation (EC.2) for X = (X1,X2) =
(Xt2 ,Xt1), we have ∫S2
|us1 + vs2|αΓ(ds) = σα{|u+ v|αt1 + |u|α(t2− t1)
}. (EC.6)
First we show that X = (Xt2 ,Xt1) satisfies the standard assumption (EC.3) in Theorem EC.1. This
is verified from the following proposition.
Proposition EC.1 (Samorodnitsky and Taqqu (1994)). The spectral measure Γα of an α-
stable random vector X is concentrated on a finite number of points on the unit sphere Sd if only if
(X1, · · ·,Xd) can be expressed as a linear transformation of independent α-stable random variables,
say X =AY, where Y1, · · ·, Yd are independent α-stable and A is a d× d matrix.
ec4 e-companion to Kim and Kim: Simulation of tempered stable Levy bridges
Since we have Xt2
Xt1
d=
Xt1 +Xt2−t1
Xt1
=
1 1
1 0
Xt1
Xt2−t1
by Proposition EC.1, the spectral measure Γ(ds) on S2 is of the form Γ(ds) =
∑n
i=1 aiδ(si1,si2)(ds)
with∑n
i=1 ai <∞. In fact, we can specify the spectral measure as follows.
Lemma EC.1. The spectral measure Γ of X = (Xt2 ,Xt1) is concentrated on two points (1,1) and
(1,0). More explicitly, Γ = σαt1δ(1,1) +σα(t2− t1)δ(1,0).
Proof. Assume that Γ = a1δ(1,1) + a2δ(1,0). Due to the uniqueness of the spectral measure, it is
enough to show that this choice of Γ is adequate. Equating the CHF (EC.5) to (EC.2) using our
choice of Γ yields a1 = σαt1 and a2 = σα(t2− t1). �
By Lemma EC.1, the standard assumption in (EC.3) is satisfied. Thus, it remains to compute
the coefficient in Proposition 2. Let σα(u, v) =∫S2|us1 + vs2|αΓ(ds). Then
1
α· ∂σ
α(u, v)
∂v=
1
α
∫S2
∂
∂v|us1 + vs2|αΓ(ds)
=
∫S2
|us1 + vs2|α−1s2 sign(us1 + vs2)Γ(ds),
so that
1
α· ∂σ
α(u, v)
∂v
∣∣∣∣u=1,v=0
=
∫S2
|s1|α−1s2 sign(s1)Γ(ds) = σα1 a= σαt2a
by (EC.4). The interchange of the order of integration and differentiation follows from the domi-
nated convergence theorem since∣∣∣∣ |us1 + (v+h)s2|α− |us1 + vs2|α
h
∣∣∣∣ ≤ 1
h
∣∣∣∣∣∣∣∣1 +hs2
us1 + vs2
∣∣∣∣α− 1
∣∣∣∣ · |us1 + vs2|α
≤ 1
h
∣∣∣∣ hs2
us1 + vs2
∣∣∣∣ · |us1 + vs2|α
= |s2||us1 + vs2|α−1
for us1 + vs2 6= 0, which is integrable by Proposition EC.1. The second inequality uses the fact
||1 + z|α− 1| ≤ |z| for all z ∈R. Then by (EC.6)
1
α· ∂σ
α(u, v)
∂v
∣∣∣∣u=1,v=0
=1
α· ∂∂v
[σα{|u+ v|αt1 + |u|α(t2− t1)
}]∣∣∣∣∣u=1,v=0
= σαt1
and this leads to a= t1/t2. This completes the proof. �
e-companion to Kim and Kim: Simulation of tempered stable Levy bridges ec5
EC.1.4. Proof of Theorem 1
First, consider the case α ∈ (0,1). Using the joint CGF (9), the saddlepoint equations K ′(u, v) =
(z,x) and K ′u(u0,0) = z have explicit solutions of the form
u =1−ακ
(1−
(t2− t1z−x
) 11−α),
v =1−ακ
((t2− t1z−x
) 11−α
−(t1x
) 11−α), and
u0 =1−ακ
(1−
(t2z
) 11−α).
Therefore, plugging this solution into the formula (7), we have the following saddlepoint approxi-
mation fXt1 |Xt2 (x|z) of fXt1 |Xt2 (x|z):
fXt1 |Xt2 (x|z) ∝ 1√2π
{|K ′′(u, v)||K ′′uu(u0,0)|
}−1/2
× exp[(K(u, v)− uz− vx
)−(K(u0,0)− u0z
)]. (EC.7)
Straightforward computations then yield the desired result. When α= 0, we also solve the saddle-
point equations with the joint CGF (9), which lead to
u= κ− t2− t1κ(z−x)
, v=t2− t1κ(z−x)
− t1κx, and u0 = κ− t2
κz.
A similar procedure using (EC.7) gives the expression in the statement. �
EC.1.5. Proof of Proposition 3
Let p = α/(1− α). First, in the case of 0 < α < 1/2 or equivalently 0 < p < 1, the ratio can be
written as
fXt1 |Xt2 (x|z)fGBXt1 |Xt2
(x|z)=B1(z;α,κ; t1, t2)
N(z;α,κ; t1, t2)g1(x),
where
B1 =1√2πκ
Beta
(t1κ,t2− t1κ
)[t1(t2− t1)
t2
] 1+p2
zp2 +
t2κ exp
(t1+p2
p(1 + p)κzp
)and
g1(x) = x−t1κ −
p2 (z−x)−
t2−t1κ − p2 exp
[− 1
p(1 + p)κ
{t1
1+p
xp+
(t2− t1)1+p
(z−x)p
}].
ec6 e-companion to Kim and Kim: Simulation of tempered stable Levy bridges
Finding the maximum of g1(x) over (0, z) is equivalent to finding the minimum of h1(y) over (0,1)
where
h1(y) = a log y+ b log(1− y) + cy−p + d(1− y)−p + e
with
a=t1κ
+p
2, b=
t2− t1κ
+p
2, c=
t1+p1
p(1 + p)κzp, d=
(t2− t1)1+p
p(1 + p)κzp, e=
(t2κ
+ p
)log z.
Note that
cy−p + d(1− y)−p
(y(1− y))−p/2= c
(1− yy
)p/2+ d
(y
1− y
)p/2≥ 2√cd,
from which we have
miny∈(0,1)
h1(y) ≥ (a∧ b) log y(1− y) + 2√cd(y(1− y))−p/2 + e
≥
2(a∧b)p
(log(p√cd
a∧b
)+ 1)
+ e, if(p√cd
a∧b
)2/p
< 14;
(a∧ b) log( 14) +√cd 2p+1 + e, otherwise.
Second, in the case of 1/2<α< 1 or equivalently p > 1, we need to find the maximum of g2(x)
over (0, z) where
g2(x) = x−p−1
2 (z−x)−p−1
2 exp
[− 1
p(1 + p)κ
{t1
1+p
xp+
(t2− t1)1+p
(z−x)p
}+
1
2κ
{t21x
+(t2− t1)2
z−x
}].
Similarly as above, we consider the minimum of h2(y) over (0,1) where
h2(y) = a log y(1− y) + by−p + c(1− y)−p− dy−1− e(1− y)−1 + f,
with
a=p− 1
2, b=
t1+p1
p(1 + p)κzp, c=
(t2− t1)1+p
p(1 + p)κzp, d=
t212κz
, e=(t2− t1)2
2κz,
and lastly f = (p− 1) log z.
Then, we observe that there exists a small ε0 > 0 such that h2(y) = a log y(1− y) +O(y−p) +
O((1− y)−p) on [ε0,1− ε0]c because y−p� 1/y and (1− y)−p� 1/(1− y) for y values near 0 and
near 1, respectively. This implies that h2(y) is bounded below on [ε0,1− ε0]c. On the other hand,
the function is bounded and continuous on [ε0,1− ε0]. Hence, the desired result follows. �
e-companion to Kim and Kim: Simulation of tempered stable Levy bridges ec7
EC.2. Supplementary results in Section 3
EC.2.1. An extension of Theorem 1
The corollary below is an extension of 1 to the multivariate random vector (Xt1 , · · ·,Xtn−1) given
Xtn = xn where 0< t1 < · · ·< tn. Its proof follows in a straightforward fashion from the fact that
its multi-dimensional CGF K(u1, · · · , un) is given by
K(u1, · · · , un) =n∑k=1
(tk− tk−1) l
(n∑i=k
ui
)with t0 = 0.
Corollary EC.1. Let Xt be a tempered stable Levy subordinator associated with TS(α,κ), α ∈
[0,1), and κ> 0. Then the saddlepoint approximation fXt1 ,···,Xtn−1|Xtn (x1, · · ·, xn−1|xn) for the con-
ditional PDF fXt1 ,···,Xtn−1|Xtn (x1, · · ·, xn−1|xn) conditioned on Xtn = xn, where 0< t1 < · · ·< tn, is
as follows:
fXt1 ,···,Xtn−1|Xtn (x1, · · ·, xn−1|xn)
∝ (2πκ)−n−1
2
(∏n
k=1(tk− tk−1)
tn
) 12(1−α)
(xn∏n
k=1(xk−xk−1)
) 2−α2(1−α)
× exp
[−(1−α)2
ακ
{n∑k=1
(tk− tk−1)1
1−α
(xk−xk−1)α
1−α− tn
11−α
xnα
1−α
}]
for α∈ (0,1), and
fXt1 ,···,Xtn−1|Xtn (x1, · · ·, xn−1|xn) ∝ (2πκ)−
n−12
ttnκ + 1
2n∏n
k=1(tk− tk−1)tk−tk−1
κ + 12
×n∏k=1
(xk−xk−1)tk−tk−1
κ −1xn1− tnκ
for α= 0, where x0 = 0 and t0 = 0.
EC.2.2. Non-Gaussian-based saddlepoint approximation
The saddlepoint expansions which involve the PDF φ and the CDF Φ of a standard normal random
variable are said to be Gaussian-based. However, it is known that a greater accuracy might be
obtained by saddlepoint expansions with a different distributional base rather than the standard
normal. In fact, non-Gaussian-based saddlepoint approximations, developed in Wood et al. (1993),
ec8 e-companion to Kim and Kim: Simulation of tempered stable Levy bridges
are known to attain better precision for some heavy-tailed distributions for which the Gaussian-
based method does not work properly in certain settings.
In what follows, we present an IG-based saddlepoint approximation for tempered stable processes.
Although this is a reasonable choice given the known density form (12) at α= 1/2, the IG-based
conditional density (EC.12) does not stand out as being more efficient. The average and maximum
errors are almost the same as those of the Gaussian-based one, while the latter allows a more
explicit form and handy algorithms.
Saddlepoint approximations that have involved a non-normal density base is mostly developed in
Wood et al. (1993). Consider the target random variable X with a PDF f(x) and a CGF K(u), and
suppose that the base function has a PDF λ(η) and a CGF L(s). Then the (λ,L)-based saddlepoint
density approximation f(x) of f(x) is given by
f(x) = λ(η)
√L′′(s(η))
K ′′(u), (EC.8)
where s(η) is the saddlepoint root of L′(s) = η and u(x) is the root of K ′(u) = x. The value η is an
optimally tilted η value chosen so that η solves
L(s(η))− s(η)η=K(u)− ux (EC.9)
in η.
The mapping x 7→ η determined by the solution in (EC.9) is a well-defined smooth bijection. The
right side of (EC.9) is a function of x that implicitly depends on x through u that solves K ′(u) = x.
The maximum of K(u)− ux is 0 and occurs when u= 0 and x= E[X]. The mapped value for E[X]
is η = L′(0) that occurs when 0 = s(η). In all the other cases, K(u)− ux < 0 and each negative
value is associated with two values of x, say x− and x+, for which x− < E[X]< x+. The mapping
associates x− 7→ η− and x+ 7→ η+ where η− < L′(0) < η+ and η−, η+ are two solutions to (EC.9).
We refer the reader to Butler (2007) for more details.
For our application, let the base function be a two-parameter family of an IG distribution
IG(µ,σ2) with its density function
λ(x;µ,σ2) =µ2/3
√2πσ
x−3/2 exp[− µ
2σ2x(x−µ)2
].
e-companion to Kim and Kim: Simulation of tempered stable Levy bridges ec9
This is in fact TS(1/2, λ, c) with λ = µ/σ2 and c = µ3/2/√
2πσ in our parameter setting. The
IG distribution is the most prominent among the base functions for heavy-tailed distributions.
Moreover, it is also a special case of a tempered stable distribution that has an exact closed formula
for the PDF, at least when α = 1/2. We follow the parameter setting and formulas in Chapter
16, Butler (2007) and in Wood et al. (1993). The scale invariance below enables us to reduce one
parameter.
Proposition EC.2 (Wood et al. (1993)). Suppose the base distribution Λa,b(z) is a location
and scale transformed CDF of Λ(z), i.e. Λa,b(z) = Λ((z−a)/b). Then, provided b > 0, the resulting
formulas do not depend on a or b.
As in Butler (2007), let µ= θ and σ2 = θ3 and then the PDF and CGF become
λ(η;θ) =1√2πη−3/2 exp
[− 1
2θ2η(η− θ)2
], L(s) =
1
θ−√
1
θ2− 2s.
With K(u) = tl(u) in (8), matching the standardized cumulants for the base function with the
target density gives the optimal choice of the parameter θ written as
θ(t, x) ={K ′′′(u)}2
3{K ′′(u)}3
(3 + ω(t, x)
√{K ′′′(u)}2
{K ′′(u)}3
)−1
=(2−α)2κx
α1−α
(1−α)2t1
1−α
(3 + ω(t, x)
(2−α)√κx
α2(1−α)
(1−α)t1
2(1−α)
)−1
(EC.10)
with
ω(t, x) = sign
(1−
(t
x
) 11−α)√√√√2(1−α)
κ
(x− t
α+
1−αα
t1
1−α
xα
1−α
).
For this optimal choice of θ, the root of (EC.9), η, can be expressed as
η(t, x) = θ(t, x) +θ(t, x)2
2
(ω(t, x)2 + ω(t, x)
√ω(t, x)2 +
4
θ(t, x)
), (EC.11)
and the saddlepoint s for the CGF L(s) is
s (η(t, x)) =1
2
(1
θ(t, x)2− 1
η(t, x)2
).
ec10 e-companion to Kim and Kim: Simulation of tempered stable Levy bridges
Then, the IG-based saddlepoint approximation fXt1 |Xt2 (x|z) for the conditional PDF fXt1 |Xt2 (x|z)
for 0< t1 < t2 can be computed from (EC.8):
fXt1 |Xt2 (x|z) ∝ 1√2πκ
(t1(t2− t1)
t2
) 12(1−α)
(z
(z−x)x
) 2−α2(1−α)
exp
−1
2
(η(t1, x)− θ(t1, x)
)2
θ(t1, x)2η(t1, x)
+
(η(t2− t1, z−x)− θ(t2− t1, z−x)
)2
θ(t2− t1, z−x)2η(t2− t1, z−x)−
(η(t2, z)− θ(t2, z)
)2
θ(t2, z)2η(t2, z)
(EC.12)
where θ(t, x) and η(t, x) are as in (EC.10) and (EC.11), respectively.
EC.3. Numerical performance in comparison to existing sequential methods
In this section, we provide numerical studies to examine the efficiency of [TSLB] algorithm by
comparing it with HSQ. To specify which algorithm is used for the hybrid scheme, we here classify
HSQ into three different algorithms: the simple stable rejection (SSR) described in Algorithm 1,
Hofert’s algorithm in Hofert (2011), and Devroye’s algorithm in Devroye (2009).
SSR and Hofert’s algorithm perform better as ∆(1 − α)/κ/α gets smaller. (Recall our
parametrization λ= (1−α)/κ.) If ∆(1−α)/κ/α< 1, SSR is the fastest algorithm and Hofert’s algo-
rithm becomes identical to SSR. On the other hand, if ∆(1−α)/κ/α > 1, Hofert’s and Devroye’s
algorithms are competitors to each other. In Hofert (2011), the author suggests a simple guideline;
use Devroye’s algorithm if ∆(1−α)/κ/α> 2.7 and Hofert’s otherwise.
As an illustration, we compare computational times to generate 104 number of trajectories
(X∆, · · · ,X8∆) with ∆ = 1 and varying α and κ in Figure EC.1. The fastest algorithm in each
parameter set (α,κ) is colored in the figure among the four algorithms: SSR, Hofert, Devroye,
and BS with tabulation. When α = 0.5, Xt becomes an inverse Gaussian process and the exact
bridge sampling (IGB) is implemented. To provide a guideline for deciding when to apply which,
the complexity of Hofert is written in each cell corresponding to α and κ. Here, in our numerical
experiments, we adopt Devroye’s algorithm for generating the terminal value in Algorithm 4 since
it performs best for a large time step.
We first note that, due to the extra burden of computing bounding constants in BS without
tabulation, this method does not compete well with Hofert’s or Devroye’s. However, BS with
e-companion to Kim and Kim: Simulation of tempered stable Levy bridges ec11
tabulation is superior as shown in most of the cells in the upper left part. More specifically, BS
with tabulation performs best where the complexity of Hofert is greater than approximately 1.5
even though Devroye’s algorithm is fastest for parameter sets with very high complexities. In
the remaining cells, SSR usually dominates all the others. Although not presented in the figure,
Hofert’s algorithm works more efficiently than Devroye’s algorithm when its complexity is less than
approximately 2.3 in our experiments. In addition, the computation times of Hofert, Devroye, BS
without tabulation, and BS with tabulation are reported in Figure EC.1 for the borderline cells
indicated by ∗.
𝛼\𝜅 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10
0.05 190 95 63.3 47.5 38 31.7 27.1 23.8 21.1 19 9.5 6.33 4.75 3.8 3.17 2.71 2.38 2.11 1.9*
0.1 90 45 30 22.5 18 15 12.9 11.3 10 9 4.5 3 2.25 1.8 1.5* 1.29 1.13 1 0.9
0.15 56.7 28.3 18.9 14.2 11.3 9.44 8.1 7.08 6.3 5.67 2.83 1.89 1.42* 1.13 0.94 0.81 0.71 0.63 0.57
0.2 40 20 13.3 10 8 6.67 5.71 5 4.44 4 2 1.33* 1 0.8 0.67 0.57 0.5 0.44 0.4
0.25 30 15 10 7.5 6 5 4.29 3.75 3.33 3 1.5* 1 0.75 0.6 0.5 0.43 0.38 0.33 0.3
0.3 23.3 11.7 7.78 5.83 4.67 3.89 3.33 2.92 2.59 2.33* 1.17 0.78 0.58 0.47 0.39 0.33 0.29 0.26 0.23
0.35 18.6 9.29 6.19 4.64 3.71 3.1 2.65 2.32 2.06 1.86* 0.93 0.62 0.46 0.37 0.31 0.27 0.23 0.21 0.19
0.4 15 7.5 5 3.75 3 2.5 2.14 1.88 1.67 1.5* 0.75 0.5 0.38 0.3 0.25 0.21 0.19 0.17 0.15
0.45 12.2 6.11 4.07 3.06 2.44 2.04 1.75 1.53 1.36* 1.22 0.61 0.41 0.31 0.24 0.2 0.17 0.15 0.14 0.12
0.5 10 5 3.33 2.5 2 1.67 1.43 1.25 1.11 1 0.5 0.33 0.25 0.2 0.17 0.14 0.13 0.11 0.1
0.55 8.18 4.09 2.72 2.05 1.64 1.36 1.17* 1.02 0.91 0.82 0.41 0.27 0.2 0.16 0.14 0.12 0.1 0.09 0.08
0.6 6.67 3.34 2.22 1.67 1.33* 1.11 0.95 0.83 0.74 0.67 0.33 0.22 0.17 0.13 0.11 0.1 0.08 0.07 0.07
0.65 5.38 2.69 1.79 1.35* 1.08 0.9 0.77 0.67 0.6 0.54 0.27 0.18 0.13 0.11 0.09 0.08 0.07 0.06 0.05
0.7 4.29 2.14 1.43* 1.07 0.86 0.71 0.61 0.54 0.48 0.3 0.21 0.14 0.11 0.09 0.07 0.06 0.05 0.05 0.04
0.75 3.34 1.67* 1.11 0.83 0.67 0.56 0.48 0.42 0.37 0.33 0.17 0.11 0.08 0.07 0.06 0.05 0.04 0.04 0.03
0.8 2.5* 1.25 0.83 0.63 0.5 0.42 0.36 0.31 0.27 0.25 0.13 0.08 0.06 0.05 0.04 0.04 0.03 0.03 0.03
0.85 1.76* 0.88 0.59 0.44 0.35 0.29 0.25 0.22 0.2 0.18 0.09 0.06 0.04 0.04 0.03 0.03 0.02 0.02 0.02
0.9 1.11* 0.56 0.37 0.28 0.22 0.19 0.16 0.14 0.12 0.11 0.06 0.04 0.03 0.02 0.02 0.02 0.01 0.01 0.01
0.95 0.53* 0.26 0.18 0.13 0.11 0.09 0.08 0.07 0.06 0.05 0.03 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.01
SSR Hofert Devroye BSBS with
tabulation
Computation times at the border *
𝛼 Hofert Devroye BSBS with
tabulation
0.05 0.36 0.38 1.62 0.28
0.1 0.41 0.43 1.68 0.3
0.15 0.36 0.45 1.93 0.25
0.2 0.33 0.47 2.23 0.24
0.25 0.35 0.49 2.51 0.2
0.3 0.53 0.54 2.57 0.21
0.35 0.41 0.53 2.64 0.23
0.4 0.38 0.52 2.61 0.25
0.45 0.35 0.5 2.87 0.29
0.55 0.27 0.48 3.65 0.23
0.6 0.33 0.51 3.7 0.22
0.65 0.33 0.5 3.62 0.21
0.7 0.35 0.5 3.6 0.21
0.75 0.48 0.51 3.71 0.23
0.8 0.59 0.54 4.11 0.22
0.85 0.41 0.48 4.25 0.23
0.9 0.24 0.41 4.38 0.26
0.95 0.15 0.29 6.98 0.26
Figure EC.1 The fastest algorithm for sampling 104 number of trajectories (X∆, · · · ,X8∆) among SSR, Hofert,
Devroye, and BS with tabulation, where the number in each cell is the complexity of Hofert. The
computation times (in seconds) in the borderline cells, indicated as ∗, are presented.
In this experiment, a bound table is generated offline by parameterizing z where the length of z
vector is set to be 300, and a bound for each realized z value is obtained by linear interpolation.
Since we bisect time points, it suffices to compute a one-dimensional table for z and we exploit the
scaling property (14). The computing time for off-line tabulation is not included in our comparison.
ec12 e-companion to Kim and Kim: Simulation of tempered stable Levy bridges
This is because the tabulation time becomes relatively negligible as the number of sample paths
increases and the computing time is approximately 0.1 seconds only. All the tests are implemented
in C language (visual studio premium 2013) on a desktop computer with processor Intel Core i5
CPU 760.
The accuracy of BS is numerically checked for the parameter sets under consideration by com-
paring the empirical distributions and Q-Q plots generated by the four algorithms. We note here
that the graphs are visually indistinguishable among the four algorithms for all the parameter
sets examined. In our applications, we examine the accuracy of BS via pricing of complex path-
dependent options.
In conclusion, BS algorithm with tabulation is an attractive candidate for generating a skeleton
of a tempered stable subordinator. Both speed and accuracy of BS are comparable to that of the
existing exact algorithms, HSQ. The specific decision rule may differ as ∆ and the number of time
steps vary. However, at least in the examined conditions, BS with tabulation works more efficiently
than the others when the complexity of Hofert is greater than approximately 1.5.
EC.4. Path-dependent option pricing under normal tempered stable processes
In this section, we assume that Xt is a normal tempered stable process, which is a time-changed
Brownian motion subordinated by a tempered stable subordinator. In other words, we define
St = S0 exp[(w+ r)t+Xt
], Xt =Xt(θ,σ,α,κ) =BYt
where Bt = θt+ σWt, Wt is a standard Wiener process, and Yt is a tempered stable subordinator
associated with TS(α,κ). Bridge sampling for this process, therefore, consists of two parts: one
for Yti ’s using an algorithm such as Algorithm 4 and one for Xti ’s using the usual Brownian
bridge sampling. For further information about Xt such as the characteristic exponent or the Levy
measure, we refer the reader to Cont and Tankov (2004) or Barndorff-Nielsen and Shephard (2001).
In the model, w is found via the characteristic exponent so that
w=− lnE[
exp(X1)]
=1−ακα
[1−
(1− κ(θ+σ2/2)
1−α
)α].
e-companion to Kim and Kim: Simulation of tempered stable Levy bridges ec13
We consider three different path-dependent options with maturity T and the monitoring interval
size ∆ = T/d:
1. floating strike lookback put
VL = e−rTE[
max{S0, S∆, . . . , Sd∆
}−Sd∆
];
2. fixed strike Asian call
VA = e−rTE[
max{
0,AT −K}], AT =
1
d+ 1
d∑k=0
Sk∆;
3. up-and-in call barrier
VB = e−rTE[
max{
0, Sd∆−K}1{max{S0,S∆,...,Sd∆}>B}
].
Two parameter sets (b) α= 0.3, κ= 0.01 and (c) α= 0.6, κ= 0.01 are chosen as representatives
of the left edge of the parameter boxes in Figure EC.1. Other reference parameters for the normal
tempered stable process are r = 0.0548, θ = −0.2859, and σ = 0.1927, which are borrowed from
Hirsa and Madan (2004). Also, S0 =K = 100, B = 120, T = 2, and d= 8. As a benchmark, the true
option prices at T = 2 are computed by SSR with the sample size 109. Table EC.1 compares the
prices of options 1–3 computed by Devroye’s algorithm, which is the fastest SQ under parameter
sets considered, and by BS with tabulation. Here, n is again the number of simulation trials and
computing times are reported in seconds.
We note that a great amount of variance reduction can be achieved by terminal stratification.
Generation of stratified samples at the final time step is similar to the procedure in Section 4.2.
Again, variance reduction is most notable for barrier options.
In addition, the variability in the options’ payoffs can be reduced by stratifying the terminal
values of normal tempered stable processes. We can generate sample paths with stratified final
values as follows for a given number of simulation trials n and length of time steps d:
1. generate a stratified sample (ui, vi), i= 1, · · · , n, from the unit hypercube of dimension 2;
2. set Y iT = F−1 (ui) for each i= 1, · · · , n where F is the distribution of YT ;
ec14 e-companion to Kim and Kim: Simulation of tempered stable Levy bridges
Table EC.1 Option prices, standard errors, and mean squared errors for the normal tempered stable process
with the reference parameter sets (b) α= 0.3, κ= 0.01 and (c) α= 0.6, κ= 0.01.
Parameter set Lookback Asian Barrier
n (b) Devroye BS-tabulation Devroye BS-tabulation Devroye BS-tabulation
104
Price 12.3817 12.3384 8.5391 8.5832 15.1380 15.1010
Std 0.1220 0.1218 0.1132 0.1131 0.2260 0.2241
Bias 0.0981 0.0548 0.0985 0.0544 0.0465 0.0835
RMSE 0.1565 0.1336 0.1501 0.1255 0.2307 0.2391
Time 0.326 0.355 0.346 0.351 0.334 0.368
105
Price 12.2518 12.2482 8.6352 8.6342 15.1930 15.1756
Std 0.0381 0.0380 0.0357 0.0358 0.0715 0.0714
Bias 0.0318 0.0354 0.0024 0.0034 0.0085 0.0089
RMSE 0.0497 0.0520 0.0359 0.0359 0.0720 0.0720
Time 3.203 3.262 3.132 3.322 3.239 3.528
106
Price 12.2761 12.2890 8.6437 8.6439 15.1923 15.1756
Std 0.0121 0.0121 0.0113 0.0113 0.0226 0.0226
Bias 0.0075 0.0054 0.0061 0.0063 0.0078 0.0089
RMSE 0.0142 0.0133 0.0129 0.0130 0.0239 0.0243
Time 33.633 35.312 33.754 35.426 33.742 35.341
n (c) Devroye BS-tabulation Devroye BS-tabulation Devroye BS-tabulation
104
Price 12.3516 12.1396 8.5587 8.6092 15.0157 15.1626
Std 0.1206 0.1207 0.1146 0.1123 0.2270 0.2242
Bias 0.0699 0.1421 0.078 0.0274 0.1669 0.0199
RMSE 0.1394 0.1864 0.1386 0.1156 0.2818 0.2251
Time 0.3473 0.251 0.347 0.253 0.335 0.264
105
Price 12.2909 12.3018 8.6412 8.6184 15.1455 15.1750
Std 0.0381 0.0381 0.0359 0.0358 0.0715 0.0716
Bias 0.0092 0.0202 0.0046 0.0182 0.0370 0.0075
RMSE 0.0392 0.0431 0.0362 0.0402 0.0805 0.0720
Time 3.286 2.4415 3.365 2.543 3.315 2.471
106
Price 12.2706 12.2668 8.6425 8.6289 15.1941 15.1881
Std 0.0121 0.0121 0.0113 0.0113 0.0226 0.0226
Bias 0.0110 0.01482 0.0058 0.0078 0.0115 0.0055
RMSE 0.0164 0.0191 0.0128 0.0137 0.0253 0.0232
Time 33.2544 25.7301 33.654 25.801 33.679 25.728
e-companion to Kim and Kim: Simulation of tempered stable Levy bridges ec15
3. set X iT = θY i
T + σ√Y iTZ
i where Zi = Φ−1 (vi) for Φ standard normal cumulative distribution
function;
4. generate entire trajectories{Y itj
}j=1,··· ,d−1
at tj by Algorithm 4 and{X itj
}j=1,··· ,d−1
at Y itj
by
Brownian bridge sampling conditioned on the stratified terminal values for each i
Taking the equiprobable strata K1 =K2 = 10, Figure EC.2 shows the significant amount of variance
reduction for BS with terminal stratification. It appears that the estimated STDs of all the path-
dependent option values are greatly reduced, and in particular, it is most significant for barrier
option.
4 5 60
0.05
0.1
0.15
0.2
0.25
log10
(sample size)
STD
4 5 60
0.05
0.1
0.15
0.2
0.25
log10
(sample size)
STD
4 5 60
0.05
0.1
0.15
0.2
0.25
log10
(sample size)
STD
(b) Devroye
(b) BS − Strat
(c) Devroye
(c) BS − Strat
(b) Devroye
(b) BS − Strat
(c) Devroye
(c) BS − Strat
(b) Devroye
(b) BS − Strat
(c) Devroye
(c) BS − Strat
Figure EC.2 Additional reduction of the estimated standard deviation due to stratification versus the number of
simulation trials n for the Lookback, Asian, Barrier options from left to right under the parameter
sets (b) and (c).
EC.5. American option pricing under normal tempered stable processes
In this section, we consider the same asset model based on normal tempered stable processes as in
the previous section. Before further discussion, we first present the backward generation algorithm
of normal tempered stable processes in Algorithm 5 which we will utilize together with American
option pricing. However, it is worth noting that dyadic generation is preferable in terms of efficiency
and accuracy if applicable.
ec16 e-companion to Kim and Kim: Simulation of tempered stable Levy bridges
Algorithm 5 Backward generation of a sample path of a normal tempered stable process via the
Brownian bridge and [TSLB]
1: At given time points 0 = t0 < t1 < · · ·< td,
2: Y0← 0;X0← 0
3: Generate Ytd ∼ TS(α, (1−α)/κ, td((1−α)/κ)(1−α)/Γ(1−α)) and Z ∼N(0,1)
4: Set Xtm← θYtm +σ√YtmZ
5: for i← 1 to d do
6: Generate Ytd−i from [TSLB] with t1 = td−i, t2 = td−i+1, and z = Yt2 .
7: Generate Z ∼N(
0,(Ytd−i+1
−Ytd−i )Ytd−iYtd−i+1
σ2)
8: Set Xtd−i←Ytd−iYtd−i+1
Xtd−i+1+Z
9: end for
American option pricing is a challenging task due to its early exercise feature. It is often approx-
imated by assuming that the option can be exercised at finitely many time points. Then at each
time step, the intrinsic value, the payoff if exercised at that time step, and the continuation value,
the conditional expected payoff if not exercised, are compared to determine an optimal early exer-
cise policy. Such comparisons are done backward in time. The LSMC methodology developed by
Longstaff and Schwartz (2001) is one of the most widely adopted methods to estimate the contin-
uation value. The description of LSMC, which utilizes least square regression, is referred to the
original paper of Longstaff and Schwartz (2001). The convergence of LSMC is rigorously proved in
Stentoft (2004).
From the computational point of view, one practical concern is a data management problem
because the procedure requires to generate and store the n×d matrix of the simulated paths with
sample size n and number of time steps d. Its purpose is to evaluate the continuation value, which
is compared with the intrinsic value, in each step of a backward dynamic program. This often
limits the accuracy of price estimates given computational capacity. However, the difficulty can be
resolved if bridge sampling of the underlying asset process is available. In this case, the comparison
e-companion to Kim and Kim: Simulation of tempered stable Levy bridges ec17
of the intrinsic value and the continuation value at each step can be performed by storing simulated
data only at two time points. This is due to the fact that the LSMC needs only the current asset
price to estimate the continuation value. Recently, Pellegrino and Sabino (2015) combine diffusion
bridge construction with the LSMC, which enables a large size simulation without storing the
whole trajectories of an underlying asset.
We therefore adopt our bridge sampling to enhance the LSMC based American option pricing
under normal tempered stable processes. However, it should be recalled that the accuracy of the
proposed bridge sampling deteriorates when the time ratio is close to 0 or 1. This could cause
problems in applying Algorithm 5 when the discrete exercise-time points are close together. Thus,
we instead propose a hybrid sampling method that combines HSQ and BS in order to reduce the
computational burden in implementing the LSMC. The central idea of this approach is to strike
the balance between the unbiasedness of HSQ and the efficiency of BS. More precisely, for a given
time grid (t1, · · · , td) and a computational budget, one generates and stores the n× ds matrix of
the underlying paths at the time grid (tdb , t2db , · · · , tdsdb) where ds = d/db is the number of time
points to sample from HSQ. For each k = ds − 1 to 0, the intermediate values between tkdb and
t(k+1)db are filled recursively by Algorithm 5. Therefore, the total memory requirement is reduced
by approximately a factor of db. See Figure EC.3 for an illustration.
∙∙∙
∙∙∙
∙∙∙
Sequential sampling
Backward bridge sampling
0t bdt 2 bdt ( 1)s bd dt s bd dt
Figure EC.3 The hybrid construction of a sample path at d time points with ds number of HSQ and db number
of BS where d= ds× db.
We check the benefit of hybrid sampling in the following numerical example. The valuation of an
American put option is considered with payoff max(0,K−ST ), strike K = 100 and maturity T = 1.
Early exercise is possible only at 50 equally distributed time points including the maturity. The
ec18 e-companion to Kim and Kim: Simulation of tempered stable Levy bridges
basis functions in the LSMC are the first nine Laguerre polynomials. This setting is adopted from
Longstaff and Schwartz (2001). Note that the number of basis functions is selected not to have rank
deficiency when computing the inverse matrix in the procedure of LSMC. We continue to assume
r= 0.0548, θ=−0.2859, and σ = 0.1927. We let the volatility of tempered stable subordinator be
the same as that of the VG model, i.e., κ= 0.2505 (Avramidis and L’Ecuyer 2006, Kaishev and
Dimitrova 2009). Lastly, α= 0.3 and we denote this parameter setting by (d). Numerical tests for
American puts are implemented in MATLAB (version R 2013a).
Table EC.2 compares the classical LSMC method (denoted by LSMC) and the LSMC combined
with hybrid sampling (denoted by LSMC-hybrid) while varying the initial stock price S0 when the
sample size n= 105 and ds = 5. The true prices are estimated by LSMC with 108 simulation trials
using SSR.
Table EC.2 American option prices, estimated STDs, and MSEs for the normal tempered stable process
computed using LSMC and LSMC-hybrid when n= 105, d= 50, and ds = 5 under the parameter set (d).
S0 80 90 100
LSMC LSMC-hybrid LSMC LSMC-hybrid LSMC LSMC - hybrid
Price 19.9354 19.8542 11.9069 11.8894 7.3206 7.1996
STD 0.0098 0.0082 0.0343 0.0336 0.0319 0.0312
Bias 0.0323 0.0488 0.0442 0.0616 0.0654 0.0557
MSE 0.0011 0.0024 0.0031 0.0049 0.0053 0.0041
Table EC.3 compares the same option prices, varying κ and σ. Here, we report the relative
differences (PLSMC − PLSMC-hybrid)/PLSMC × 100 in percentage, instead of computing true prices,
where Pmethod denotes the option price computed by the method. For various κ and σ values, we
notice that the percentage differences of the put prices from LSMC and LSMC-hybrid are between
0.1% and 0.9%.
The optimal ds and db may differ as parameters change. Under the parameter setting (d), we
let d= 256 and do the same experiments by increasing ds from 1 to 32. From this exercise, it is
e-companion to Kim and Kim: Simulation of tempered stable Levy bridges ec19
Table EC.3 American option prices, estimated STDs, and percentage differences for the normal tempered
stable process computed using LSMC and LSMC-hybrid when n= 105, d= 50, and ds = 5 varying κ and σ under
the parameter set (d)
σ 0.5 0.75 1 1.25
κ LSMC LSMC-hybrid LSMC LSMC-hybrid LSMC LSMC-hybrid LSMC LSMC-hybrid
0.5
Price 16.4483 16.2975 24.5177 24.6316 33.9261 33.8523 44.6802 44.6236
STD 0.0632 0.0631 0.0816 0.0817 0.0952 0.0949 0.1029 0.1026
RD(%) 0.9168 0.4648 0.2174 0.1266
1
Price 15.7662 15.6868 23.2012 23.3266 33.1135 33.056 48.0758 48.3182
STD 0.0671 0.0672 0.0835 0.083 0.0948 0.0946 0.0988 0.0985
RD(%) 0.5031 0.5402 0.1737 0.5043
Table EC.4 American option prices, estimated STDs, and MSEs for the normal tempered stable process
computed using LSMC and LSMC-hybrid when n= 105 and d= 256, varying ds under the parameter set (d)
S0 80 90 100
LSMC LSMC-hybrid LSMC LSMC-hybrid LSMC LSMC-hybrid
ds 32 16 8 32 16 8 32 16 8
Price 19.9532 19.9525 19.9433 19.9181 11.8845 11.8840 11.8618 11.7913 7.2521 7.2540 7.2332 7.2256
STD 0.0041 0.0035 0.0035 0.0029 0.0342 0.0332 0.0339 0.0348 0.0317 0.0309 0.0311 0.0320
Bias 0.0275 0.0281 0.0374 0.0625 0.0881 0.0886 0.1108 0.1813 0.0205 0.0185 0.0394 0.0469
MSE 0.0007 0.0008 0.0014 0.0039 0.0089 0.0089 0.0134 0.0341 0.0014 0.0013 0.0025 0.0032
seen that db value for LSMC-hybrid not greater than 20 could be a reasonable choice, yielding
MSEs comparable to LSMC. We report this experiment briefly in Table EC.4. Finally, we want
to emphasize that such a gain is very important since it assures the convergence of the numerical
scheme in actual implementations. For example, if d= 256, then MATLAB cannot perform LSMC
if n> 105 while LSMC-hybrid can easily perform the task with ds = 16.
ec20 e-companion to Kim and Kim: Simulation of tempered stable Levy bridges
References
Avramidis AN, L’Ecuyer P (2006) Efficient Monte Carlo and Quasi-Monte Carlo option pricing under the
variance gamma model. Management Science, 52(12):1930–1944.
Hardin Jr C, Samorodnitsky G, Taqqu MS (1991) Nonlinear regression of stable random variables. The
Annals of Applied Probability, 1(4):481–634.
Kaishev V, Dimitrova D (2009) Dirichlet bridge sampling for the variance gamma process: Pricing path-
dependent options. Management Science, 55(3):483–496.
Longstaff, F. A., Schwartz, E. S. (2001) Valuing American options by simulation: A simple least-squares
approach. Review of Financial Studies, 14(1):113–147.
Pellegrino T, Sabino P (2015) Enhancing least squares Monte Carlo with diffusion bridges: An application
to energy facilities. Quantitative Finance, 15(5):761–772.
Samorodnitsky G, Taqqu MS (1994) Stable Non-Gaussian Random Processes. Chapman & Hall/CRC, Boca
Raton, FL.
Stentoft L (2004) Convergence of the least squares Monte Carlo approach to American option valuation.
Management Science, 50(9):1193–1203.
Wood ATA, Booth JG, Butler RW (1993) Saddlepoint approximations to the CDF of some statistics with
nonnormal limit distributions. Journal of the American Statistical Association, 88(422):680–686.