efficient simulation for expectations over the union of

1

Efficient Simulation for Expectations over the Union ofHalf-Spaces

DOHYUN AHN and KYOUNG-KUK KIM, Korea Advanced Institute of Science and Technology

We consider the problem of estimating expectations over the union of half-spaces. Such a problem arises in

many applications such as option pricing and stochastic activity networks. More recent applications include

systemic risk measurements of financial networks. Assuming that random variables follow a multivariate

elliptical distribution, we develop a conditional Monte Carlo method and prove its asymptotic efficiencies. We

then demonstrate the numerical performance of the proposed method in three different application areas.

CCS Concepts: •Mathematics of computing→ Probabilistic algorithms; •Computingmethodologies→ Rare-event simulation;

Additional Key Words and Phrases: conditional Monte Carlo; rare event simulation; elliptical distribution;

variance reduction

ACM Reference format:Dohyun Ahn and Kyoung-Kuk Kim. 2017. Efficient Simulation for Expectations over the Union of Half-Spaces.

ACM Trans. Model. Comput. Simul. 1, 1, Article 1 (November 2017), 20 pages.

https://doi.org/0000001.0000001

1 INTRODUCTIONIn modern stochastic models of practical interest, estimating high-dimensional probabilities or

expectations is a challenging problem, in particular, if they correspond to rare events. To enhance

the efficiency of the estimation, in the literature, a number of variance reduction techniques, such

as importance sampling, conditional Monte Carlo, splitting, and stratification, have been developed

for several specific models [Asmussen and Glynn 2007]. However, it is still difficult to efficiently

estimate multivariate probabilities or expectations on a non-convex set, for example, a union of

half-spaces.

Consider ann-dimensional random vector X = (X1, . . . ,Xn) and letA := ∪Ni=1

x ∈ Rn |a⊤i x > bi for fixed ai ∈ Rn and bi > 0, i = 1, . . . ,N . In this paper, our objective is to calculate the quantity

E[h(X)1X∈A

](1)

where h(·) is a real-valued function on A. Here, A can be geometrically interpreted as the set

outside the convex polytope ∩Ni=1

x ∈ Rn |a⊤i x ≤ bi . If h ≡ 1, then we are simply calculating the

probability P(X ∈ A). The motivation of this problem comes from systemic risk measurements

of financial networks [Eisenberg and Noe 2001; Glasserman and Young 2015]; when a number of

Authors’ address: Department of Industrial and Systems Engineering, Korea Advanced Institute of Science and Technology,

291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea. Email address of D. Ahn: [email protected], E-mail

address of K. Kim (corresponding author): [email protected].

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https://doi.org/0000001.0000001

ACM Transactions on Modeling and Computer Simulation, Vol. 1, No. 1, Article 1. Publication date: November 2017.

https://doi.org/0000001.0000001

https://doi.org/0000001.0000001

1:2 Ahn and Kim

banks are connected with interbank liabilities, [Ahn et al. 2017] shows that the default event of a

specific bank is expressed as X ∈ A. In this case, h(X) could be 1, the total loss of the system, or

any reasonable systemic risk measure. Due to a possibly high dimension, the expectation is quite

involved, and such a complexity makes the use of Monte Carlo simulation an effective alternative

to analytic computations. For large bi ’s, this belongs to the realm of rare event simulation.

This paper is hence related to several simulation-based works such as [Juneja et al. 2007], [Chan

and Kroese 2011], [Blanchet and Shi 2013a], and [Blanchet and Shi 2013b]. Those works concentrate

on rare event simulation schemes for probabilities that some functions of independent random

variables exceed increasing thresholds. In particular, the first paper proposes two simulation

methods for the tail probability of the maximum of sums of random variables: the exponential

twisting based method and the asymptotic hazard rate twisting based method. This probability is a

special case of (1) when each ai consists of 1 or 0 and all bi are equal. In the other three papers,

the authors are interested in estimating the tail probability of the sum of random variables via

conditional Monte Carlo, cross entropy, and splitting, respectively. This is the case when N = 1

and a1 is the vector of ones.

In those works, however, random variables are assumed to be independent. Since we use ra-

dial sampling without any independence assumption, our work can be compared with [Blanchet and

Rojas-Nandayapa 2011]. This paper proposes two schemes for the probability P(eX1 + · · · + eXn > b

)where X is possibly dependent. One is a conditional Monte Carlo approach when X has an elliptical

distribution, and the other is an importance sampling method when each Xi is logarithmically

long tailed. The authors also show that both estimators are asymptotically optimal under mild

assumptions. Even though their first approach shares the same basic idea with ours, there are three

significant differences between the two works. First, we consider the quantity of the form (1) for

any h(·) satisfying a mild condition, whereas their method is for the above probability. Second, due

to this difference, their estimator is not applicable to the examples in Section 3 whereas ours is

not for the log-elliptical sums. Lastly, our approach involves new analyses for the efficiency of the

proposed estimator as well as some new ideas for performance improvement.

Our contributions can be summarized as follows. As briefly mentioned above, we first develop a

conditional Monte Carlo method based on radial conditional sampling for estimating (1) when X is

elliptically distributed. To the best of the authors’ knowledge, no studies have been conducted in

this direction. Further, our approach covers most of the special cases mentioned above. Second, we

prove asymptotic efficiencies of the proposed method under a rare event setting, that is, when the

size of X gets smaller (or when bi ’s grow). Specifically, this is done when the radial random variable

follows two broad families of probability distributions. Based on this, we provide three applications:

(i) pricing of rainbow options, (ii) delay probabilities in a stochastic activity network, and (iii)

default probabilities in financial networks. In particular, the issue of systemic risk measurement in

financial networks has never been introduced in the simulation literature. We also discuss how

we can tailor our proposal to specific constraints such as nonnegativity of random variables. We

demonstrate the performance of the suggested method via numerical examples for each application.

Before we move onto the next section, let us provide two notions of efficiencies in the context of

rare event simulation. Assume that z(m) denotes the expectation of our interest wherem ∈ (0,∞)

and z(m) → 0 as m → ∞. Consider an unbiased estimator Z (m) for z(m), i.e., E[Z (m)] = z(m).

Then, it is said to be asymptotically optimal, or logarithmic efficient if

lim

m→∞

log E[Z (m)2

]log z(m)2

= 1.


Efficient Simulation for Expectations over the Union of Half-Spaces 1:3

An efficiency condition stronger than asymptotical optimality is the concept of bounded relative

error. An unbiased estimator Z (m) for z(m) has bounded relative error if

lim

m→∞

E[Z (m)2

]z(m)2

< ∞.

In addition, we introduce some notations that are used in the paper. The d-by-d identity matrix is Idwhose subscript is often omitted as long as it is clear from the context. The d-dimensional Euclidean

space is Rd with its basis vectors e1, e2, . . . , ed where ei is the i-th column of Id . Its nonnegativeorthant is Rd+. We use 1, 0 for vectors of ones and zeros in a suitable dimension, respectively. We

list basic notations as follows:

• for any vector v ∈ Rd , vi is the i-th component of v;• for any two vectors v,w ∈ Rd , v ≤ w means entry-wise inequality;

• for any two vectors v,w ∈ Rd with v ≤ w, [v,w] := x ∈ Rd | v ≤ x ≤ w;

• for any d × d matrix M (similarly for vectors), M−i is the matrix obtained by eliminating the

i-th column and row from M;

The remainder of the paper is organized as follows. Section 2 introduces our conditional Monte

Carlo method and proves its asymptotic efficiencies. In Section 3, we present three applications of

this method, and numerical results are provided for the verification of the newly developed method

for each application. Finally, Section 4 concludes the paper. All proofs can be found in the appendix.

2 CONDITIONAL MONTE CARLOThe crude Monte Carlo simulation for (1) is described in Algorithm 1. Under this crude Monte Carlo

method, the inefficiency is aggravated as the target event X ∈ A becomes rare. We therefore

develop a conditional Monte Carlo method for the random vector to be sampled from the target

region A directly. It is shown to be effective when the target event has a very small probability.

Our method is based on the distribution assumptions on X below.

Algorithm 1 Crude Monte Carlo

1: Sample X = x2: if a⊤i x > bi for some i then3: Compute h(x) and set T = h(x)4: else set T = 0

5: end if6: return T

Assumption 2.1. The random vector X has an elliptical distribution, given by X = µ +RΛΘ. Here,

µ < A is a fixed mean vector, R is a nonnegative radial random variable, Λ is an n × d matrix

such that Σ := ΛΛ⊤is positive definite, and Θ is uniformly distributed on the unit sphere in Rd

independent of R. Furthermore, the continuous density function fR , the cumulative distribution

function FR , and its inverse F−1

R for the random variable R are given.

Assumption 2.1 is mildly restrictive because elliptical distributions are a widely used class of

distributions in the literature [McNeil et al. 2015]. Note that we assume µ < A, and this is to

avoid unnecessary complications and to focus on the rare event simulation. Otherwise, the event

X ∈ A is generally not rare, and the use of variance reduction techniques is usually not necessary.

In addition, as it is true for Gaussian and Student t-distribution, fR , FR , and F−1

R for the random

variable R are usually known for popular elliptical distributions.


1:4 Ahn and Kim

Fig. 1. Graphical representation of µ + RΛθ |R ≥ 0, Ac and A.

2.1 Conditional Monte Carlo: IntroductionOur first and natural approach is to evaluate the expectation by conditioning on Θ. In the special

case of h ≡ 1, this leads to an estimator with smaller variance as long as one has an analytic method

of evaluation for P(µ + RΛθ ∈ A) when Θ = θ :

Var(1X∈A

)= Var

(E[1X∈A |Θ

] )+ E

[Var

(1X∈A |Θ

) ]≥ Var

(E[1X∈A |Θ

] )= Var

(P(µ + RΛΘ ∈ A|Θ

) ).

In this case, our unbiased estimator to P(X ∈ A) is

1

m

m∑i=1

P(µ + RΛθ i ∈ A

)based on a sample of sizem, θ 1, . . . ,θm. This method of calculating probabilities is known as

directional simulation in the mechanical engineering literature. An interested reader is referred to

[Bjerager 1988] or [Ditlevsen et al. 1988] for instance.

The challenging part is to find the region for R such that µ + RΛθ ∈ A for Θ = θ , say R(θ ). We

observe that the complement of A, denoted by Ac, can be seen as a convex polytope of the form

Ac =

N⋂i=1

x ∈ Rn |a⊤i x ≤ bi

.

Since µ ∈ Ac, as illustrated in Figure 1, the ray µ + RΛθ belongs to A if the value of R exceeds

a point where the ray from µ intersects the boundary of Ac. In other words, one can prove that

µ + RΛθ ∈ A if and only if R ∈(r (θ ),∞

)where

r (θ ) =

mini |a⊤i Λθ>0 (bi − a⊤i µ)/a⊤i Λθ , if a⊤i Λθ > 0 for some i;

∞, otherwise.

For the rest of this section, we simply use the notation R(θ ) to denote the interval (r (θ ),∞) for a

fixed θ .



Theorem 2.2. Suppose that Assumption 2.1 holds. Then, an unbiased estimator for (1) is given by

h(µ + RΛΘ

)P(R ∈ R(Θ)|Θ

)where R has the conditional distribution (R |R ∈ R(Θ),Θ). The ratio of the second moments of thisestimator and the crude Monte Carlo estimator is bounded above by

max

|θ |=1

P(R ∈ R(θ )

).

The above result on the second moments is a conservative one. For each sampled Θ = θ , theefficiency gain from the conditional sampling is

E[h(µ + RΛθ

)2P

(R ∈ R(θ )

)2Θ = θ ]

E[h(X)21X∈A |Θ = θ

] = P(R ∈ R(θ )

).

Hence, the average expected gain on each sampled Θ = θ becomes E [P(R ∈ R(Θ)|Θ)] = P(X ∈ A).

We also note that, by Assumption 2.1, we can sample R from (R |R ∈ R(θ )) by setting

R = F−1

R

(FR (r (θ )) +U

[1 − FR (r (θ ))

] )(2)

whereU is uniformly distributed in (0, 1).

When h ≡ 1, it is not necessary to sample R, and the estimator in Theorem 2.2 simply becomes

P(R ∈ R(Θ)|Θ) which certainly is a conditional Monte Carlo estimator. For general h, this estimator

is not a typical conditional Monte Carlo estimator as it is not conditional expectation of (1) given Θ.

Nevertheless, we put more emphasis on the idea of conditioning on Θ and thus call this approach

conditional Monte Carlo consistently no matter what h we take.

2.2 Conditional Monte Carlo: Efficient AlgorithmWhen the right hand side bi of A are large, X ∈ A is a rare event. Thus, if we replace bi withmbi , then the event becomes rarer asm → ∞. More generally, instead of increasing bi , we considerthe following sequence of diminishing random vectors:

Xm =1

mµ +

1

ma RΛΘ, m = 1, 2, . . .

for some a > 0. Here, µ,R,Λ,Σ, and Θ are assumed as in Assumption 2.1. Given Assumption 2.1, it

is known that the density of X is given by fX(x) = |Σ|−1/2д((x − µ)⊤ Σ−1 (x − µ)) for some function

д. This gives us the density of Xmgiven by

fm(x) =ma

|Σ|1/2

д

(m2a

(x −

1

mµ

)⊤Σ−1

(x −

1

mµ

)).

Now, we take a change of measure so that Xmhas mean zero under a new measure. This means

that

E[h(Xm)1Xm ∈A

]= E0

[h(Xm)

fm(Xm)

f 0

m(Xm)1Xm ∈A

]where the superscript of E0

denotes the probability measure P0under which Xm

is distributed as

m−aRΛΘ with density f 0

m . Then, we can write the unbiased estimator in Theorem 2.2 as

hm(RΛΘ

)P(R ∈maR(Θ)|Θ

), (3)

where R is distributed as R conditional on R ∈maR(Θ) and the function hm is

hm(s) = h(m−as)д((s −ma−1µ)⊤Σ−1(s −ma−1µ)

)д (s⊤Σ−1s)

.


1:6 Ahn and Kim

Algorithm 2 Conditional Monte Carlo

1: Sample Θ = θ from the uniform distribution on the unit sphere

2: Find R(θ ) = r ≥ 0|rΛθ ∈ A

3: if R(θ ) is nonempty then4: Sample R from (R |R ∈maR(θ )) by (2)

5: Compute hm(RΛθ ) and set T = hm(RΛθ )P(R ∈maR(θ )|θ )6: else set T = 0


Theorems 2.3 and 2.4 show the efficiency of this estimator for the cases where R has aWeibull-like

distribution and a regularly varying distribution, respectively:

• Weibull-like distribution: fR (r ) = α1rβ1e−α2r β2

for some α1,α2, β2 > 0 and β1;

• Regularly varying distribution: limr→∞ fR (tr )/fR (r ) = t−ρ for some index ρ > 1.

Theorem 2.3. Suppose that Assumption 2.1 holds, h(·) is bounded away from 0 on A, and thereexists δ > 0 such that h(rΛθ ) ≤ eδ r

β2 for all (r ,θ ) ∈ R+ × Rn with rΛθ ∈ A and |θ | = 1. If R has a

Weibull-like distribution, then the unbiased estimator (3) is asymptotically optimal, i.e.,

lim

m→∞

log E[hm

(RΛΘ

)2

P(R ∈maR(Θ)|Θ

)2

]log E

[hm

(RΛΘ

)P(R ∈maR(Θ)|Θ

)]2= 1.

Theorem 2.4. Suppose that Assumption 2.1 holds, h(·) is bounded away from 0 on A, and we haveE0

[h(X)21X∈A

]< ∞. If R has a regularly varying distribution with index ρ > 1, then the unbiased

estimator (3) has bounded relative error, i.e.,

lim

m→∞

E[hm

(RΛΘ

)2

P(R ∈maR(Θ)|Θ

)2

]E[hm

(RΛΘ

)P(R ∈maR(Θ)|Θ

)]2< ∞.

Weibull-like distributions in Theorem 2.3 include some light-tailed distributions when β2 ≥ 1 as

well as some heavy-tailed distributions when β2 < 1. One example is the case where X follows an

n-dimensional normal distribution. In this case, it is known that R2has a chi-square distribution

with n degrees of freedom. Then, it is easy to see that fR (r ) = Krn−1e−r2/2

for some K > 0.

On the other hand, another prominent example among heavy-tailed distributions is a regularly

varying distribution. Theorem 2.4 can be applied to every regularly varying distribution including

the case where X has an n-dimensional t-distribution with ν degrees of freedom. In this case,

R2/n ∼ F (n,ν ), so fR (r ) = Krn−1(n2 +νr 2)−(n+ν )/2for some K > 0. In the next section, we will show

how we can tailor this proposed method to specific applications and how much variance reduction

is achieved.

Even if we do not shift the mean µ to zero, the efficiency results in Theorems 2.3 and 2.4 are

still preserved for the estimator (3). However, when we deal with nonnegative random variables as

in Sections 3.2 and 3.3, the mean shift is helpful in simplifying tasks and reducing computational

burdens considerably; sampling Θ in particular. Furthermore, under a rare event setting, we observe

a great improvement in efficiencies as the mean vector Xmconverges to zero.



3 APPLICATIONSIn this section, we introduce three different examples to which the conditional Monte Carlo scheme

in the previous section can be applied. For each example, we present numerical results comparing

with crude Monte Carlo. The numerical results in this section are based on the assumption that Xis independent. The cases where X is correlated are presented in the online supplement. We use

a personal computer with Intel Core i5-4690 CPU and 8 GB RAM. All implementations are done

using MATLAB R2014b.

For each result, two efficiency measures are reported: variance ratio (VR) and efficiency ratio

(ER). The former is the ratio of the variance of the crude Monte Carlo estimator to that of the

new estimator. The latter is obtained by multiplying VR by the ratio of the running time of the

crude Monte Carlo scheme to that of the new scheme. Even though ER can be considered as a

better estimate in terms of the efficiency, it is greatly dependent on various components such as

languages, computer performance, programming skills and so on. Therefore, we regard VR as a

significant performance indicator and report it together.

In the following subsection, we illustrate how our method is directly utilized for pricing rainbow

options. In Section 3.2, we present a modified scheme specialized to the case when X is nonnegative

and estimate the probability of large delay in stochastic activity networks. Lastly, in Section 3.3,

we provide another extension of our method to the case when X is truncated to a hypercube of

the form [0, c] for some c ∈ Rn+ and compute the default probability of an institution in a financial

network.

3.1 Pricing Rainbow OptionsRainbow options constitute one main category of multi-asset options. The payoff structure of these

options includes, for example, the best or worst of underlying assets and cash at expiry, and calls or

puts on the maximum or minimum of underlying assets. Due to the multidimensional complexity,

rainbow options are generally priced using Monte Carlo methods. In this subsection, we compute

the price of European call-on-max options whose payoff at expiry can be written as

Payoff = max

(max

(S1(T ), . . . , Sn(T )

)− K , 0

)where Si (t) is the price of i-th underlying asset at time t , K is the strike price, and T is the

time of maturity. We note that it is similar to compute the price of European best-of-assets-or-

cash options and European put-on-min options whose payoffs are max(S1(T ), . . . , Sn(T ),K) andmax(K − min(S1(T ), . . . , Sn(T )), 0), respectively.

We assume that Si (t) follows a geometric Brownian motion. Then, for all i , we have

Si (T ) = Si (0) exp

( (r − σ 2

i /2

)T + σi

√TXi

)where r is the risk-free rate, σi is the volatility of i-th asset, and X ∼ N (0,C) with correlation

matrix C. Thus, Si (T ) > K is equivalent to

σi√TXi > log

(K/Si (0)

)−(r − σ 2

i /2

)T (4)

for all i . Define A1 := ∪ni=1

x ∈ Rn |σi√Txi > log

(K/Si (0)

)−(r − σ 2

i /2

)T . Then, the price of this

option is

e−rT E[(

max

(S1(T ), . . . , Sn(T )

)− K

)1X∈A1

].

As described in [Ouwehand and West 2006], there are various types of rainbow options, but no

special Monte Carlo scheme for rainbow options pricing is reported in the literature. Instead, one

may apply any reasonable variance reduction techniques such as control variate method. In Table 1,


1:8 Ahn and Kim

K Method Estimate Standard Dev. Time (sec) VR ER

150

Crude 5.73 4.79 × 10−3

128 - -

Conditional 5.73 4.32 × 10−3

553 1.2 0.3Control Variate 5.73 3.55 × 10

−3150 1.8 1.6

Combination 5.73 3.15 × 10−3

679 2.3 0.4

250

Crude 9.47 × 10−2

7.13 × 10−4

124 - -

Conditional 9.40 × 10−2

3.18 × 10−4

484 5.0 1.3Control Variate 9.43 × 10

−23.78 × 10

−4146 3.6 3.0

Combination 9.39 × 10−2

1.36 × 10−4

564 27 6.0

350

Crude 2.74 × 10−3

1.30 × 10−4

125 - -


2.24 × 10−5

434 34 10

Control Variate 2.76 × 10−3

4.83 × 10−5

145 7.2 6.2Combination 2.76 × 10

−35.01 × 10

−6477 671 176

450

Crude 1.18 × 10−4

3.14 × 10−5

133 - -


1.85 × 10−6

424 289 90

Control Variate 1.26 × 10−4

6.39 × 10−6

157 24 20

Combination 1.24 × 10−4

2.10 × 10−7

453 22, 272 6, 520

Table 1. Estimates of prices of European call-on-max options using Algorithm 1, Algorithm 2, a control variatemethod, and the combination of Algorithm 2 and the control variate method when its underlying assetsfollow uncorrelated geometric brownian motions.

we compare the crude method, Algorithm 2, and one control variate method. The control variate we

take here is a single e−rT (Si (T )−K)+, whose expectation is computed via the Black-Scholes formula.

This somewhat naive choice yields the sample correlation as much as 98% between the target payoff

and the control variate for the test cases in Table 1. One can consider linear combinations of such

control variates, but one key point here is that we can combine control variates with conditional

Monte Carlo to have further variance reduction. This is possible because the region in which

Si (T ) > K is a subset of A1.

Along the same vein, our conditional Monte Carlo can be applied on top of other variance

reduction techniques. For instance, multiple control variates may be utilized. In our numerical tests,

they work extemely well. One caveat is that the estimator is biased as the optimal coefficient vector

is estimated from simulated samples.

Table 1 shows the estimates of this price based on four methods whenX1, . . . ,Xn are uncorrelated.

For the implementation, we run 107replications for each result, and the parameters we used are

n = 10, T = 0.5, r = 0.05, σ = (0.05, 0.1, . . . , 0.5), and S(0) = (104, 103, . . . , 95). The control variate

is the call option on S10. Since X has mean 0 in this example, increasingK is equivalent to increasing

m with a = 1, which means that the event A1 becomes rarer. Thus, we observe the efficiency

improvement by increasing the strike price K : 150, 250, 350, and 450.

The table shows that the conditional Monte Carlo method is computationally heavier than the

crude one, considering the difference of running times between two methods. This difference can

also be observed in other experiments of this paper. Such phenomenon mainly comes from the

slow computation of F−1

R and FR in (2) using MATLAB functions such as chi2cdf, chi2inv, fcdf,and finv. In particular, when R is light-tailed, a method of [Derflinger et al. 2010] can serve as one

possible approach to quickly generate R. This fast alternative is not exact but effective when the

distribution has light tails.



Fig. 2. A stochastic activity network with 7 nodes and 10 edges

Despite the slow computation, the effectiveness of the conditional Monte Carlo method makes

it widely dominant over the crude one. Also, it performs better than the control variate method

when target estimates are small, and it becomes far more efficient when it is combined with the

control variate method. Clearly, as K goes up, the price estimate decreases, and the VR and ER

estimates increase. A closer look at the difference of the VR and ER estimates between strike prices

gives that the increment from K = 350 to K = 450 is larger than the increment from K = 250 to

K = 350. Also, it is bigger than the increment from K = 150 to K = 250. This implies that those

estimates grow nonlinearly with strike prices, which still holds in the correlated case as reported

in the online supplement.

3.2 Stochastic Activity NetworksAnother application of this method arises in stochastic activity networks [Nelson 2013]. Each

network represents a project composed of several combinations of steps and activities. All activities

have nonnegative stochastic durations X1, . . . ,Xn , and all predetermined combinations Cj of

those activities should be completed at the end. Figure 2 illustrates an example of a stochastic

activity network with 7 nodes and 10 edges. In this figure, nodes represent project steps and

edges are activities with durations X1, . . . ,X10. The combinations of activities are C1 = 1, 4, 8, 10,

C2 = 1, 4, 9, C3 = 2, 5, 8, 10, C4 = 2, 5, 9, C5 = 3, 6, 8, 10, C6 = 3, 6, 9, and C7 = 3, 7, 10.

A common interest in the literature is the estimation of the probability of large delay, P(T >b), since such delay can cause various costs and expenses in managing the project. Here, T :=

maxj∑

i ∈Cj Xi denotes the overall duration which is the longest path through the network, and

b is a threshold of this project. We define A2 := ∪j x ∈ Rn+ |∑

i ∈Cj xi > b. Then, since activitydurations are assumed to be nonnegative, we have

P(T > b

)= P

(X ∈ A2 |X ∈ Rn+

)=

P(X ∈ A2

)ρ

where ρ := P(X ∈ Rn+). A resulting estimator would be biased if we were to estimate the numerator

and the denominator separately. Since the essence of this paper lies in computing P(X ∈ A2), we

assume that ρ is given as a constant and focus on estimating the numerator.

Like this example, when the target set A is a subset of Rn+, we can further reduce the computa-

tional time by ignoring any sample θ such that Λθ is not in Rn+. A better way of doing this is to

sample θ uniformly on the intersection of the set θ |Λθ ≥ 0, |θ | = 1. The simplest case is when


1:10 Ahn and Kim

Σ = In . In this case, when θ = (θ1, . . . ,θn) is generated, we can simply use (|θ1 |, |θ2 |, . . . , |θn |) for arandom directional vector in Rn+. We then divide the final estimate by 2

n. However, in general we

shall implement a sequential importance sampling (IS) method introduced in Remark 3.1 to sample

θ such that Λθ ≥ 0. See Algorithm 3. Since we consider uncorrelated cases in the numerical results

in Sections 3.2 and 3.3, we simply take the absolute value of the direction as above instead of using

the sequential IS. In the online supplement, we report the results of correlated cases using the

sequential IS. This sequential IS causes a delay in the computation, but Algorithm 3 still dominates

the crude Monte Carlo scheme.

Algorithm 3Modified Conditional Monte Carlo

1: Sample Θ = θ from the uniform distribution on the set θ |Λθ ≥ 0, |θ | = 1 and compute the

likelihood ratio l(θ ) using a sequential IS2: Find R(θ ) = r ≥ 0|rΛθ ∈ A

3: if R(θ ) is nonempty then4: Sample R from (R |R ∈maR(θ )) by (2)

5: Compute hm(RΛθ ) and set T = hm(RΛθ )P(R ∈maR(θ )|θ )l(θ )6: else set T = 0


Remark 3.1. Note that since Λ = (Λi j ) is a lower triangular matrix and Θ d= Z/|Z| where

Z = (Z1, . . . ,Zn)⊤ ∼ N (0, In),

P(ΛΘ ≥ 0) = P(Λ11Z1 ≥ 0, Λ21Z1 + Λ22Z2 ≥ 0, · · · , Λn1Z1 + · · · + ΛnnZn ≥ 0).

Using this result, steps 1 and 2 in Algorithm 3 can be done by the following procedure:

(1) Sample Z1 = z1 from p1(·) := ϕ(·)/P(Λ11Z1 ≥ 0) and set l1(z1) = ϕ(z1)/p1(z1) = P(Λ11Z1 ≥ 0).

(2) For k = 2, . . . ,n,• sample Zk = zk from

pk (·|z1, . . . , zk−1) := ϕ(·)/P(Λk1z1 + · · · + Λ(k−1)(k−1)zk−1 + ΛkkZk ≥ 0)

• compute

lk (z1, . . . , zk ) = lk−1(z1, . . . , zk−1)

∏ki=1

ϕ(zi )

pk (zk |z1, . . . , zk−1)∏k−1

i=1ϕ(zi )

= lk−1(z1, . . . , zk−1)P(Λk1z1 + · · · + Λ(k−1)(k−1)zk−1 + ΛkkZk ≥ 0).

(3) Set θ = z/|z| and l(θ ) = ln(z1, . . . , zn)

where z = (z1, . . . , zn)⊤and ϕ(·) is the probability density function of the standard normal distribu-

tion.

[Juneja et al. 2007] propose the asymptotic hazard rate twisting and exponential twisting methods

to estimate the probability of large delay for independent activities. Table 2 compares our new

scheme and those methods with crude Monte Carlo when activities are uncorrelated and normally

distributed. In this experiment, we utilize the example illustrated in Figure 2. For the numerical

results, we implement each algorithm to compute P(X ∈ A2) and divide it by ρ. We assume

that µi = 0.5 for all i and Σ is a diagonal matrix whose diagonal entries are all 0.36. We run 107

replications for each result and use the following parameters: n = 10, a = 0.05, and b = 5. In order



m Method Estimate Standard Dev. Time (sec) VR ER

1

Crude 4.01×10−2

1.96×10−4

118 - -

Conditional 4.02×10−2

4.25×10−5

799 21 3.1

Asymp. HRT 4.00×10−2

1.52×10−4

98 1.7 2.0

Exp. Twist. 4.02×10−2

4.65×10−5

103 18 20

2

Crude 3.82×10−3

1.48×10−4

117 - -


3.47×10−6

799 1,822 267


5.13×10−5

97 8.3 10

Exp. Twist. 3.99×10−3

9.34×10−6

101 251 292

5

Crude 5.63×10−4

1.23×10−4

117 - -


5.12×10−7

796 57,641 8,461


1.40×10−5

97 78 93

Exp. Twist. 4.46×10−4

1.81×10−6

100 4,633 5,439

10

Crude 2.00×10−4

9.98×10−5

116 - -


1.77×10−7

791 319,674 46,846


6.06×10−6

98 271 320

Exp. Twist. 1.28×10−4

6.44×10−7

100 24,066 27,921

Table 2. Estimates of probabilities of large delay in stochastic activity networks using Algorithm 1, Algorithm 3,asymptotic hazard rate twisting, and exponential twisting methods when activities are uncorrelated andnormally distributed.


1

Crude 4.96×10−2

2.10×10−4

119 - -


2.15×10−5

571 96 20

2

Crude 2.49×10−2

3.98×10−4

118 - -


8.13×10−6

575 2,393 490

5

Crude 2.18×10−2

8.39×10−4

118 - -


5.99×10−6

573 19,611 4,021

10

Crude 1.94×10−2

1.06×10−3

118 - -


5.13×10−6

574 42,809 8,774

Table 3. Estimates of probabilities of large delay in stochastic activity networks using Algorithm 1 andAlgorithm 3 when activities are uncorrelated and t-distributed.

to see the efficiency improvement in the rare event setting, each algorithm is implemented with

m = 1, 2, 5, and 10. Increasingm means that the event of interest becomes rarer.

In Table 3, we consider the case when activities are uncorrelated and t-distributed. In this

example, we exclude the schemes of [Juneja et al. 2007] due to the inapplicability of those schemes

to t-distributed activities. Here, the degree of freedom for the multivariate t-distribution is set equal

to 8. The rest of the settings are the same with the previous experiment.

In both cases, as in the first application, VR and ER estimates increase asm gets larger in all the

instances. In case of the VR estimate, the conditional Monte Carlo method thoroughly dominates

the crude Monte Carlo and two other methods. In case of the ER estimate, our method is still

dominant over the crude one and the asymptotic hazard rate twisting method. Also, it is better

than the exponential twisting method when the probability estimates are small. In addition, since


1:12 Ahn and Kim

the schemes of [Juneja et al. 2007] cannot be applied to the case of correlated activities, for this

case, we compare our scheme only with the crude one and make similar observations which are

reported in the online supplement.

We further note that ρ is calculated by MATLAB functions mvncdf and mvtcdf in Tables 2 and 3

respectively because the estimated errors are appeared to be negligible. This results from the fact

that ρ is not in the rare event setting. In fact, it converges to a positive constant P(ΛΘ ≥ 0) whichis 1/2

nin the uncorrelated case. However, when n is large enough, a more effective alternative

such as [Botev 2017] can be utilized in order to estimate ρ.

3.3 Financial NetworksA financial system is composed of a number of financial institutions and payment obligations

among those institutions. [Eisenberg and Noe 2001] describe the system as a directed network with

nodes that represent banks and edges that correspond to the obligations. Recently, [Ahn et al. 2017]

consider the Eisenberg-Noe framework for systemic risk with random shocks. Based on the shock

amplification due to the network structure, they find the region for the shock vector that makes a

specific bank default and conduct the analysis on default probabilities in financial networks.

The modeling framework consists of the following ingredients:

• n: the number of financial institutions in the network;

• ci ,di : the outside asset value and the outside liability of node i , respectively;• pi j : the payment from node i to j where pii = 0;

• Π = (πi j ): the relative liabilities matrix, where πi j is the proportion of node i’s obligation to

node j defined by

πi j =

pi j/(di +

∑j,i pi j ), if di +

∑j,i pi j > 0;

0, otherwise;

• wi := (ci +∑

j,i pji ) − (di +∑

j,i pi j ): the initial net worth (book value) of node i;• Xi ∈ [0, ci ]: a random shock to the outside asset ci .

In this framework, they show that the default probability of institution i is

P(max

ξ ∈Qξ⊤

(X − w) > 0

)where Q is the set of extreme points of the polytope (ζ⊤, 1)⊤ ∈ Rn+ |(I − Π−i )ζ ≤ π i

−i . Here, πi

is the ith column of Π. An interested reader is referred to [Ahn et al. 2017] for more details. We

define A3 := ∪ξ ∈Q x ∈ [0, c]|ξ⊤x > ξ⊤w. Since we assume X ∈ [0, c], this probability is equal to

P(X ∈ A3 |X ∈ [0, c]

)=

P(X ∈ A3

)ρ

where ρ := P(X ∈ [0, c]). As in the previous subsection, we concentrate on the computation of

P(X ∈ A3) assuming that ρ is given.

For fixed θ , we define

ro(θ ) := min

i |(Λθ )i,0

[max

ci − µi(Λθ )i

,−µi(Λθ )i

].

Then, we observe that µ + RΛθ ∈ [0, c] is equivalent to R ∈ [0, ro(θ )]. Hence, µ + RΛθ ∈ A3 if and

only if 0 < r (θ ) < ro(θ ) and R ∈ Ro(θ ) := (r (θ ), ro(θ )]. If we replace R(·) by Ro(·), then it is easy to

see that Theorems 2.2 to 2.4 still hold and that Algorithm 3 can be applied to this case.

We now present our computational results on estimating the default probability of institution

n. Note that we are also able to compute other risk measures such as expected loss given default



µ = 0.005cm Method Estimate Standard Dev. Time (sec) VR ER

1

Crude 1.50×10−2

7.72×10−4

128 - -


1.77×10−5

829 1,900 293

10

Crude 4.49×10−3

6.41×10−4

124 - -


7.42×10−6

788 7,468 1,175

100

Crude 1.11×10−3

3.35×10−4

123 - -


2.87×10−6

739 13,645 2,274

1,000

Crude 2.05×10−4

1.45×10−4

123 - -


8.67×10−7

690 27,820 4,960


1

Crude 2.53×10−2

4.47×10−4

124 - -


2.73×10−5

826 267 40

10

Crude 4.86×10−3

5.98×10−4

123 - -


7.72×10−6

784 5,999 944

100

Crude 1.18×10−3

3.42×10−4

124 - -


2.89×10−6

738 14,016 2,346

1,000

Crude 2.04×10−4

1.44×10−4

123 - -


8.68×10−7

690 27,628 4,921

Table 4. Estimates of the default probability of institution 10 using Algorithm 1 and Algorithm 3 when thereare multivariate normal random shocks which are uncorrelated.

using our scheme. In our experiments, the financial networks introduced in [Acemoglu et al. 2015]

are considered. They are convex combinations of a regular ring network and a symmetric regular

complete network. More precisely, it is assumed that po :=∑

j,i pji =∑

j,i pi j and that outside

liabilities are the same, i.e., d := d1 = · · · = dn . Here, po represents the liability of a bank to other

banks in the system. Then, the relative liabilities matrix is then given by

Π =po

d + po

((1 − γ )(en e1 · · · en−1) +

γ

n − 1

(J − I))

for γ ∈ [0, 1] where J is the matrix of ones. It represents a ring network when γ = 0 and stands for

a complete network when γ = 1.

The parameters used in this example are n = 10, γ = 0.5, d = 6, po = 3, and

c =(6.1, 6.4, 6.4, 6.3, 6.6, 6.1, 6.4, 6.3, 6.2, 6.7

)⊤.

Tables 4 and 5 are based on the assumption that the uncorrelated shock vector X is normally

distributed and t-distributed, respectively. For each i , the mean µi is 0.5% or 1.5% of the outside

asset value ci whereas Σ is a diagonal matrix whose i-th diagonal element is the square of 4% of

ci . Note that the proposed algorithm yields estimates for P(X ∈ A3). In order to produce proper

estimates, we also compute ρ using MATLAB functions mvncdf and mvtcdf, respectively. Finaloutcomes are given as the ratio of the two. Furthermore, we set a = 0.05 andm = 1,10, 100, and1, 000. We run the same number of replications (10

7) for both algorithms as before.

In Tables 4 and 5, by implementing Algorithm 3, we illustrate its efficiency compared to the crude

Monte Carlo. First, we see that both VR and ER estimates are mostly better when µ is small. The

event of hittingA3 becomes rarer as µ gets smaller. We achieve more significant improvements from


1:14 Ahn and Kim


1

Crude 3.57×10−2

1.19×10−3

143 - -


2.32×10−5

845 2,647 446

10

Crude 2.48×10−2

1.53×10−3

127 - -


1.73×10−5

873 7,787 1,135

100

Crude 1.38×10−2

1.19×10−3

122 - -


1.10×10−5

873 11,656 1,630

1,000

Crude 7.67×10−3

8.86×10−4

123 - -


6.60×10−6

871 18,054 2,542


1

Crude 3.30×10−2

4.97×10−4

124 - -


2.28×10−5

873 474 67

10

Crude 2.50×10−2

1.41×10−3

122 - -


1.75×10−5

872 6,487 909

100

Crude 1.39×10−2

1.18×10−3

121 - -


1.10×10−5

888 11,463 1,568

1,000

Crude 7.77×10−3

8.91×10−4

122 - -


6.60×10−6

886 18,221 2,508

Table 5. Estimates of the default probability of institution 10 using Algorithm 1 and Algorithm 3 when thereare multivariate t random shocks which are uncorrelated.

the conditional Monte Carlo asm increases for fixed µ. For instance, in Table 4, when µ = 0.015c,the VR estimate shoots up from 267 to 27,628 and the ER estimate goes up from 40 to 4,921 asmincreases from 1 to 1, 000. When X is correlated, the efficiency of the conditional method is still

significantly better than that of the crude method. For this case, we refer the reader to the online

supplement.

4 CONCLUSIONIn this article, we proposed a conditional Monte Carlo scheme to estimate expectations over

the union of half-spaces when a random vector X follows a multivariate elliptical distribution.

Conditioning on a directional component of X, our procedure shifts the mean to the origin and

samples its radial component from the interval which corresponds to the target region. This method

can be applied to the case when X is dependent as well as the case when it is independent. Under a

regime that the size of X gets smaller, we proved that, first, the suggested estimator is asymptotically

optimal when the radial component has a Weibull-like distribution and, second, that it has bounded

relative error when the radial component follows a regularly varying distribution. Therefore, a

large number of probability distributions can be covered by our scheme.

Based on the method, diverse applications are possible. One particular example that we first

implemented was the computation of the price of European call-on max options. We observed

increasing amounts of both variance reduction and efficiency gains as the strike price increases.

In addition, we combined our scheme with a sequential importance sampling method to sample

nonnegative directional components. This idea was applied to estimate the probability of large delay

in stochastic activity networks and to calculate the default probability of a particular institution in



financial networks. Numerical results pointed out that our scheme still has a bigger computational

burden than the other compared methods in spite of using the sequential importance sampling.

However, thanks to a large amount of variance reduction, in almost all cases, the newly developed

method was shown to be more efficient than the others.

We found that this paper is not without any limitations. We hence note three other interesting

topics related to the suggested method that need to be investigated in the future. Firstly, we

set the target region as the union of half-spaces, but one can consider an extension to arbitrary

extreme sets for which only a bivariate approach exists [Drees and de Haan 2015]. Secondly,

efficient computational methods for some families of probability distributions other than elliptical

distributions can be explored. Last but not least, to address nonnegative random variables, we used

the truncated elliptical random variables in Section 3.2. But it could be possible to develop another

technique to estimate the quantity (1) assuming that X follows a log-elliptical distribution, which

would be a generalization of [Blanchet and Rojas-Nandayapa 2011]. Despite those limitations, we

hope that this work sheds new light on the rare event simulation in a multivariate setting.

A PROOFS OF THEOREMSProof of Theorem 2.2. We observe that

E[h(µ + RΛΘ)P

(R ∈ R(Θ)|Θ

) ]= E

[E[h(µ + RΛΘ)P

(R ∈ R(Θ)|Θ

) R ∈ R(Θ),Θ] ]

= E[P(R ∈ R(Θ)|Θ

)E[h(µ + RΛΘ)|R ∈ R(Θ),Θ

] ].

Since R is distributed as R conditional on R ∈ R(Θ),Θ, we can replace R with R. By unconditioning

the last expression, it is equal to

E[E[h(µ + RΛΘ)1R∈R(Θ) |Θ

] ]= E

[h(X)1X∈A

].

Now, for the second moment, we compute as follows:

E[h(µ + RΛΘ)2P

(R ∈ R(Θ)|Θ

)2

]= E

[P(R ∈ R(Θ)|Θ

)2E

[h(µ + RΛΘ)2 |R ∈ R(Θ),Θ

] ]= E

[P(R ∈ R(Θ)|Θ

)E[h(µ + RΛΘ)21R∈R(Θ) |Θ

] ]= E

[P(R ∈ R(Θ)|Θ

)E[h(X)21X∈A |Θ

] ].

By taking the supremum of P(R ∈ R(Θ)|Θ) over the unit sphere, we can see that the ratio of the

second moments of the estimators is bounded above by the supremum.

Proof of Theorem 2.3. For simplicity of exposition, we divide the proof into 4 steps.

Step 1 We first make the following observation. We denote the density functions of Θ and R by

fΘ(θ ) and fR (r ), respectively, and define

φm(x) :=

m2a

(x −

1

mµ

)⊤Σ−1

(x −

1

mµ

) 1

2

so that

fm(x) =ma

|Σ|1/2

д(φm(x)2

).

Also, by [Fang et al. 1990], the relationship between д(·) and fR (·) is

fR (r ) =2πn/2

Γ(n/2)rn−1д(r 2).


1:16 Ahn and Kim

Using these results, for the second moment, we proceed as follows:

E[hm

(RΛΘ

)2

P(R ∈maR(Θ)|Θ

)2

]= E

[P(R ∈maR(Θ)|Θ

)E[hm

(RΛΘ

)2

1R∈maR(Θ) |Θ] ]

=

∫fΘ(θ )dθ

∫ ∞

mar (θ )fR (r )dr

∫ ∞

mar (θ )hm (rΛθ )2 fR (r )dr

=m2a∫

fΘ(θ )dθ

∫ ∞

r (θ )fR (m

aτ )dτ

∫ ∞

r (θ )hm (maτΛθ )2 fR (maτ )dτ

∼m2a∫

fΘ(θ )dθ

∫ ∞

r (θ )fR (m

aτ )dτ

∫ ∞

r (θ )h(τΛθ )2

fR (φm(τΛθ ))2

fR (maτ )dτ . (5)

Since

hm(maτΛθ ) = h(τΛθ )

(maτ

φm(τΛθ )

)n−1 fR (φm(τΛθ ))fR (maτ )

∼ h(τΛθ )fR (φm(τΛθ ))fR (maτ )

uniformly on the compact set (τ ,θ ) ∈ R+ × Rn |τΛθ ∈ A, |θ | = 1 asm increases, the asymptotic

equivalence (5) holds.

Step 2 Let ϵ > 0 be small enough. Then, there existsmo > 0 such that for allm ≥ mo and for all

(τ ,θ ) ∈ R+ × Rn satisfying τΛθ ∈ A and |θ | = 1,

maτ (1 − ϵ) ≤ φm(τΛθ ) ≤ maτ (1 + ϵ), (maτ )β1−β2+1 ≤ e(ϵ/2)(maτ )β2

, and h(τΛθ ) ≤ e(ϵ/2)(maτ )β2

.

Then, form ≥ mo , we have

the right-hand side of (5)

≤ m2a∫

fΘ(θ )dθ

∫ ∞

r (θ )fR (m

aτ )dτ

∫ ∞

r (θ )eϵ (m

aτ )β2fR (m

aτ (1 − ϵ))2

fR (maτ )dτ

=m2a∫

fΘ(θ )dθ

∫ ∞

r (θ )α1(m

aτ )β1e−α2(maτ )β2

dτ

∫ ∞

r (θ )α1(1 − ϵ)

2β1 (maτ )β1e−2α2(1−ϵ )β2−α2−ϵ (maτ )β2

dτ

≤

∫fΘ(θ )dθ

∫ ∞

r (θ )

α1

(α2 − ϵ/2)β2

ψ 1

m(τ )dτ

∫ ∞

r (θ )

α1(1 − ϵ)2β1

β22α2(1 − ϵ)β2 − α2 − 3ϵ/2ψ 2

m(τ )dτ

= κ1

∫fΘ(θ )dθ

∫ ∞

r (θ )ψ 1

m(τ )dτ

∫ ∞

r (θ )ψ 2

m(τ )dτ

= κ1

∫fΘ(θ )e

−2α2(1−ϵ )β2−ϵ (mar (θ ))β2

dθ . (6)

Here, the following auxiliary functions and constant are used:

ψ 1

m(τ ) :=ma(α2 − ϵ/2)β2(maτ )β2−1e−(α2−ϵ/2)(maτ )β2

,

ψ 2

m(τ ) :=ma2α2(1 − ϵ)β2 − α2 − 3ϵ/2β2(m

aτ )β2−1e−2α2(1−ϵ )β2−α2−3ϵ/2(maτ )β2

,

and

κ1 =α2

1(1 − ϵ)2β1

β2

2(α2 − ϵ/2)2α2(1 − ϵ)β2 − α2 − 3ϵ/2

.



Step 3 Let us consider the multidimensional Laplace-type integral

J (λ) :=

∫ΩG(x)e−λF(x)dx

where λ > 0, Ω ⊂ Rn , G(·) is continuous, and F (·) has continuous second-order partial derivatives

in a neighborhood of xo := argminx∈Ω F (x). The multidimensional Laplace’s approximation in

[Wong 2001] states that if J (λ) converges absolutely for all λ ≥ λo , xo is the unique minimizer of

F (·), and the Hessian matrix (∂2 f

∂xi∂x j

) x=xo

is positive definite, then

J (λ) ∼ CoG(xo)e−λF(xo )λp(n)

where Co is a positive constant and p(n) = −n/2 if xo is a critical point; −(n + 1)/2 otherwise.

We know that the polytope Achas a finite number No ≤ N of facets. Suppose a⊤i x ≤ bi ,

i = 1, . . . ,No , be facet-defining inequalities. Let us split the range of the integral in the right-hand

side of (6) into (No + 1) regions, say Ω0,Ω1, . . . ,ΩNo . Here, Ω0 := ∩Ni=1

θ |a⊤i Λθ ≤ 0 and Ωk is the

region of θ where kth facet and the direction vector Λθ intersect for k = 1, . . . ,No . By Section 2.1,

for all θ ∈ Ω0, r (θ ) = ∞, and for all θ ∈ Ωk , k = 1, . . . ,No ,

r (θ ) =bk − a⊤k µ

a⊤k Λθ.

Let θk = argminθ ∈Ωkr (θ ) for each k . Then, it is easy to see that θk is the unique minimizer on

Ωk , r∗(·)β2

has continuous second-order partial derivatives in a neighborhood of θk , and its its

Hessian matrix at θk is positive definite for each k . Hence, using the multidimensional Laplace’s

approximation, we have

the right-hand side of (6) ∼ κ1

No∑k=1

C1

k fΘ(θk )e−2α2(1−ϵ )β2−ϵ (mar ∗(θ k ))β2

maβ2pk (n)

∼ κ1C1

ko fΘ(θko )e−2α2(1−ϵ )β2−ϵ (mar ∗(θ ko ))

β2

maβ2pko (n)

where C1

k > 0,k = 1, . . . ,No , pk (n) = −n/2 or −(n + 1)/2, and ko := argmink r∗(θk ).

Step 4 Similarly, the following relationship can be found for the first moment.

E[hm

(RΛΘ

)P(R ∈maR(Θ)|Θ

)]= E

[hm (RΛΘ) 1R∈maR(Θ)

]=

∫fΘ(θ )dθ

∫ ∞

mar (θ )hm (rΛθ ) fR (r )dr

∼ma∫

fΘ(θ )dθ

∫ ∞

r (θ )h(τΛθ )fR (φm(τΛθ ))dτ .

The asymptotic equivalence is based on similar arguments in Step 1.


1:18 Ahn and Kim

Next, we use the assumption that h is bounded away from 0 on A. Let us denote its positive

lower bound by L. Then, for largem, the last term in the above computations is bounded below by

maL

∫fΘ(θ )dθ

∫ ∞

r (θ )fR (m

aτ (1 + ϵ))dτ

=maL

∫fΘ(θ )dθ

∫ ∞

r (θ )α1(1 + ϵ)

β1 (maτ )β1e−α2(1+ϵ )β2 (maτ )β2

dτ

≥ κ2

∫fΘ(θ )dθ

∫ ∞

r (θ )maβ2(α2(1 + ϵ)

β2 + ϵ)(maτ )β2−1e−(α2(1+ϵ )β2+ϵ )(maτ )β2

dτ

= κ2

∫fΘ(θ )e

−(α2(1+ϵ )β2+ϵ )(mar (θ ))β2

dθ . (7)

where κ2 := Lα1β−1

2(1 + ϵ)β1/(α2(1 + ϵ)

β2 + ϵ). The inequality holds because there existsm1 > 0

such that (maτ )β1−β2+1 ≥ e−ϵ (maτ )β2

for allm,τ ,θ satisfyingm ≥ m1, τΛθ ∈ A and |θ | = 1. This is

similar Step 2.

Now we apply the multidimensional Laplace’s approximation as in Step 3 so that we obtain

the right hand side of (7) ∼ κ2

No∑k=1

C2

k fΘ(θk )e−(α2(1+ϵ )β2+ϵ )(mar ∗(θ k ))β2

maβ2pk (n)

∼ κ2C2

ko fΘ(θko )e−(α2(1+ϵ )β2+ϵ )(mar ∗(θ ko ))

β2

maβ2pko (n)

where pk (n) and ko are as defined in Step 3, and C2

k , k = 1, . . . ,No , are positive constants.

Therefore, we finally arrive at

L := lim

m→∞

log E[hm

(RΛΘ

)2

P(R ∈maR(Θ)|Θ

)2

]log E

[hm

(RΛΘ

)P(R ∈maR(Θ)|Θ

)]2

≥α2(1 − ϵ)

β2 − ϵ

α2(1 + ϵ)β2 + ϵ.

Since ϵ is arbitrary, L ≥ 1. Then, the result holds since L ≤ 1 is trivial.

Proof of Theorem 2.4. Since R follows a regularly varying distribution, limr ↑∞ fR (tr )/fR (r ) =t−ρ for some index ρ > 1 uniformly on intervals of the form (to ,∞), to > 0 by [Resnick 2008]. This

implies that there exists a point z ≥ 0 such that fR (·) decreases on [z,∞). Let ϵ > 0 be small enough.

Then, according to the proof of Theorem 2.3, we have the following relationships for the second

moment:

E[hm

(RΛΘ

)2

P(R ∈maR(Θ)|Θ

)2

]∼m2a

∫fΘ(θ )dθ

∫ ∞

r (θ )fR (m

aτ )dτ

∫ ∞

r (θ )h(τΛθ )2

fR (φm(τΛθ ))2

fR (maτ )dτ

≤ m2a∫

fΘ(θ )dθ

∫ ∞

r (θ )fR (m

aτ )dτ

∫ ∞

r (θ )h(τΛθ )2

fR (maτ (1 − ϵ))2

fR (maτ )dτ

∼m2a∫

fΘ(θ )dθ

∫ ∞

r (θ )fR (m

a)τ−ρdτ

∫ ∞

r (θ )h(τΛθ )2 fR (ma)τ−ρ (1 − ϵ)−2ρ

dτ

=m2a fR (ma)2(1 − ϵ)−2ρ

(max

θ

∫ ∞

r (θ )τ−ρdτ

) ∫fΘ(θ )dθ

∫ ∞

r (θ )h(τΛθ )2τ−ρdτ .



It is easy to see that the assumption E0

[h(X)21X∈A

]< ∞ implies∫

fΘ(θ )dθ

∫ ∞

r (θ )h(τΛθ )2τ−ρdτ < ∞.

On the other hand, the squared first moment becomes

E[hm

(RΛΘ

)P(R ∈maR(Θ)|Θ

)]2

∼m2a∫

fΘ(θ )dθ

∫ ∞

r (θ )h(τΛθ )fR (φm(τΛθ ))dτ

2

≥ L2m2a∫

fΘ(θ )dθ

∫ ∞

r (θ )fR (m

aτ (1 + ϵ))dτ

2

∼ L2m2a∫

fΘ(θ )dθ

∫ ∞

r (θ )fR (m

a)τ−ρ (1 + ϵ)−ρdτ

2

= L2m2a fR (ma)2(1 + ϵ)−2ρ

∫fΘ(θ )dθ

∫ ∞

r (θ )τ−ρdτ

2

.

Therefore, the ratio of the two is clearly bounded from above.

ACKNOWLEDGMENTSThe authors would like to thank Paul Glasserman and Wanmo Kang for their helpful comments.

The authors also appreciate valuable feedback from the referees which helped them improve the

manuscript. This work was supported by the Basic Science Research Program through the National

Research Foundation of Korea funded by the Ministry of Education (NRF-2016R1D1A1B03930772).

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Online Supplement to

Efficient Simulation for Expectations over the Union of Half-Spaces

Dohyun Ahn∗, Kyoung-Kuk Kim†

Korea Advanced Institute of Science and Technology

June 2017

In this online supplement, we repeat the numerical experiments in Section 3 for the cases where

X is correlated. We use the same number of replications for each case and use the same parameters

as in Section 3 except for the correlation matrix.

We consider a random vector consisting of two groups of random variables. Each pair of

random variables has a correlation, but there is no correlation between two groups. We construct

the correlation matrix C, given below, to describe this situation, where X1, X2, X4, X5, X7, X9,

and X10 compose one group and the rest of the variables form the other group. The results are

exhibited in Tables 1 to 5. Since they have the same patterns as the results in Section 3, we omit

the explanation.

C =

1 0.2488 0 0.1420 0.0893 0 0.0184 0 0.0431 −0.0022

0.2488 1 0 0.0720 0.1623 0 0.0086 0 0.2835 0.0623

0 0 1 0 0 0.0040 0 −0.0293 0 0

0.1420 0.0720 0 1 0.1373 0 0.0508 0 −0.0148 −0.0527

0.0893 0.1623 0 0.1373 1 0 −0.0602 0 −0.0327 0.2909

0 0 0.0040 0 0 1 0 0.0571 0 0

0.1484 0.0086 0 0.0508 −0.0602 0 1 0 0.0592 0.0104

0 0 −0.0293 0 0 0.0571 0 1 0 0

0.0431 0.2835 0 −0.0148 −0.0327 0 0.0592 0 1 −0.0519

−0.0022 0.0623 0 −0.0527 0.2909 0 0.0104 0 −0.0519 1

∗Department of Industrial and Systems Engineering, E-mail: [email protected]†Department of Industrial and Systems Engineering, E-mail: [email protected]

1

K Method Estimate Standard Dev. Time (sec) VR ER

150

Crude 5.71 4.78× 10−3 136 - -

Conditional 5.72 4.32× 10−3 530 1.2 0.3

Control Variate 5.71 3.55× 10−3 151 1.8 1.6

Combination 5.72 3.16× 10−3 687 2.3 0.5

250

Crude 9.39× 10−2 7.10× 10−4 135 - -

Conditional 9.36× 10−2 3.16× 10−4 464 5.1 1.5

Control Variate 9.40× 10−2 3.76× 10−4 149 3.6 3.2

Combination 9.39× 10−2 1.37× 10−4 579 27 6.2

350

Crude 2.78× 10−3 1.28× 10−4 135 - -

Conditional 2.75× 10−3 2.19× 10−5 412 34 11

Control Variate 2.76× 10−3 4.78× 10−5 150 7.1 6.4

Combination 2.76× 10−3 5.15× 10−6 479 613 173

450

Crude 1.08× 10−4 2.70× 10−5 134 - -

Conditional 1.23× 10−4 1.76× 10−6 381 235 83

Control Variate 1.20× 10−4 5.18× 10−6 149 27 25

Combination 1.24× 10−4 2.20× 10−7 439 15, 117 4, 619

Table 1: Estimates of prices of European call-on-max options using Algorithm 1, Algorithm 2, a

control variate method, and the combination of Algorithm 2 and the control variate method when

its underlying assets follow correlated geometric brownian motions.


1Crude 6.23×10−2 2.21×10−4 119 - -

Conditional 6.25×10−2 6.84×10−5 2,416 10 0.5

2Crude 9.54×10−3 1.90×10−4 113 - -

Conditional 9.63×10−3 8.54×10−6 2,042 496 27

5Crude 1.73×10−3 1.57×10−4 112 - -

Conditional 1.67×10−3 1.95×10−6 2,174 6,529 337

10Crude 6.80×10−4 1.29×10−4 113 - -

Conditional 6.14×10−4 8.67×10−7 2,039 21,995 1,214

Table 2: Estimates of probabilities of large delay in stochastic activity networks using Algorithm 1

and Algorithm 3 when activities are correlated and normally distributed.

2


1Crude 7.49×10−2 2.37×10−4 115 - -

Conditional 7.54×10−2 3.60×10−5 2,182 43 2.3

2Crude 4.17×10−2 4.19×10−4 114 - -

Conditional 4.18×10−2 1.53×10−5 1,999 751 43

5Crude 3.50×10−2 7.68×10−4 115 - -

Conditional 3.47×10−2 1.12×10−5 1,857 4,707 291

10Crude 2.96×10−2 9.06×10−4 114 - -

Conditional 3.02×10−2 9.50×10−6 1,866 9,080 555

Table 3: Estimates of probabilities of large delay in stochastic activity networks using Algorithm 1

and Algorithm 3 when activities are correlated and t-distributed.

µ = 0.005c


1Crude 1.78×10−2 5.95×10−4 140 - -

Conditional 1.77×10−2 1.85×10−5 2,081 1,035 69

10Crude 5.70×10−3 4.82×10−4 141 - -

Conditional 5.21×10−3 7.52×10−6 2,038 4,111 285

100Crude 1.73×10−3 2.77×10−4 138 - -

Conditional 1.48×10−3 2.84×10−6 1,988 9,501 658

1,000Crude 4.47×10−4 1.41×10−4 138 - -

Conditional 3.22×10−4 8.34×10−7 1,943 28,799 2,053

µ = 0.015c


1Crude 2.92×10−2 3.81×10−4 139 - -

Conditional 2.97×10−2 2.88×10−5 2,087 176 12

10Crude 6.09×10−3 4.54×10−4 134 - -

Conditional 5.57×10−3 7.85×10−6 2,027 3,342 221

100Crude 1.78×10−3 2.78×10−4 126 - -

Conditional 1.49×10−3 2.85×10−6 1,983 9,475 602

1,000Crude 4.47×10−4 1.41×10−4 127 - -

Conditional 3.22×10−4 8.34×10−7 1,939 28,708 1,884

Table 4: Estimates of the default probability of institution 10 using Algorithm 1 and Algorithm 3

when there are multivariate normal random shocks which are correlated.

3

µ = 0.005c


1Crude 4.21×10−2 9.20×10−4 132 - -

Conditional 4.25×10−2 2.53×10−5 2,123 1,324 82

10Crude 2.91×10−2 1.10×10−3 132 - -

Conditional 2.87×10−2 1.81×10−5 2,108 3,690 230

100Crude 1.74×10−2 8.79×10−4 140 - -

Conditional 1.67×10−2 1.14×10−5 2,113 5,927 392

1,000Crude 8.96×10−3 6.33×10−4 147 - -

Conditional 9.25×10−3 6.83×10−6 2,117 8,592 598

µ = 0.015c


1Crude 4.11×10−2 4.46×10−4 145 - -

Conditional 4.14×10−2 2.62×10−5 2,115 289 20

10Crude 2.89×10−2 1.02×10−3 149 - -

Conditional 2.88×10−2 1.84×10−5 2,109 3,084 218

100Crude 1.74×10−2 8.74×10−4 143 - -

Conditional 1.67×10−2 1.15×10−5 2,117 5,811 394

1,000Crude 8.99×10−3 6.34×10−4 150 - -

Conditional 9.26×10−3 6.83×10−6 2,112 8,612 612

Table 5: Estimates of the default probability of institution 10 using Algorithm 1 and Algorithm 3

when there are multivariate t random shocks which are correlated.

4

efficient simulation for expectations over the union of

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