simulation of tempered stable l evy bridges and its...

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.


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.)


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.


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.


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 =


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


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


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


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)


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.


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


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.


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.


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

)


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


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


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


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.


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


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.


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


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


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


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.


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


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


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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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).


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.


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. �


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

}].


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. �


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),


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

].


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

).


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


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.


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].


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 ;


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


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.


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


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


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


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.


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simulation of tempered stable l evy bridges and its...

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