Submitted to Operations Research manuscript (Please, provide the manuscript number!) Simulation of tempered stable L´ evy bridges and its applications Kyoung-Kuk Kim Department of Industrial and Systems Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, South Korea, [email protected]Sojung Kim Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, [email protected]We consider tempered stable L´ evy 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 L´ evy 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, L´ evy 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
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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,
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
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.
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.
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:
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.
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