Efficient Simulation for Expectations over the Union of Half-Spaces

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1 Efficient Simulation for Expectations over the Union of Half-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; Computing methodologies 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 Efficient Simulation for Expectations over the Union of Half-Spaces. ACM Trans. Model. Comput. Simul.,, Article (November 207), 20 pages. INTRODUCTION In 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 an n-dimensional random vector X = (X,..., X n ) and let A := i= N {x Rn ai x > b i } for fixed a i R n and b i > 0, i =,..., N. In this paper, our objective is to calculate the quantity E [ ] h(x) {X A } () where h( ) is a real-valued function on A. Here, A can be geometrically interpreted as the set outside the convex polytope N i= {x Rn a i x b i }. If h, 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 200; Glasserman and Young 205]; when a number of Authors address: Department of Industrial and Systems Engineering, Korea Advanced Institute of Science and Technology, 29 Daehak-ro, Yuseong-gu, Daejeon 344, Republic of Korea. address of D. Ahn: dohyun.ahn@kaist.ac.kr, address of K. Kim (corresponding author): kkim328@kaist.ac.kr. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. 207 Copyright held by the owner/author(s). Publication rights licensed to Association for Computing Machinery. XXXX-XXXX/207/-ART $ ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

2 :2 Ahn and Kim banks are connected with interbank liabilities, [Ahn et al. 207] shows that the default event of a specific bank is expressed as {X A}. In this case, h(x) could be, 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 b i 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 20], [Blanchet and Shi 203a], and [Blanchet and Shi 203b]. 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 () when each a i consists of or 0 and all b i 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 = and a is the vector of ones. In those works, however, random variables are assumed to be independent. Since we use radial sampling without any independence assumption, our work can be compared with [Blanchet and Rojas-Nandayapa 20]. This paper proposes two schemes for the probability P ( e X + + e X n > 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 X i 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 () 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 () 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 b i 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 where m (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 log E [ Z(m) 2] lim m log z(m) 2 =. ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

3 Efficient Simulation for Expectations over the Union of Half-Spaces :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 E [ Z(m) 2] lim m z(m) 2 <. In addition, we introduce some notations that are used in the paper. The d-by-d identity matrix is I d whose subscript is often omitted as long as it is clear from the context. The d-dimensional Euclidean space is R d with its basis vectors e, e 2,..., e d where e i is the i-th column of I d. Its nonnegative orthant is R d +. We use, 0 for vectors of ones and zeros in a suitable dimension, respectively. We list basic notations as follows: for any vector v R d, v i is the i-th component of v; for any two vectors v, w R d, v w means entry-wise inequality; for any two vectors v, w R d with v w, [v, w] := {x R d 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 CARLO The crude Monte Carlo simulation for () is described in Algorithm. 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 Crude Monte Carlo : Sample X = x 2: if a i x > b i for some i then 3: Compute h(x) and set T = h(x) 4: else set T = 0 5: end if 6: return T Assumption 2.. 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 R d independent of R. Furthermore, the continuous density function f R, the cumulative distribution function F R, and its inverse FR for the random variable R are given. Assumption 2. is mildly restrictive because elliptical distributions are a widely used class of distributions in the literature [McNeil et al. 205]. 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, f R, F R, and FR for the random variable R are usually known for popular elliptical distributions. ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

4 :4 Ahn and Kim Fig.. Graphical representation of {µ + RΛθ R 0}, A c and A. 2. Conditional Monte Carlo: Introduction Our first and natural approach is to evaluate the expectation by conditioning on Θ. In the special case of h, 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 ( ) {X A } = Var (E [ {X A } Θ ]) + E [Var ( {X A } Θ )] (E [ {X A } Θ ]) Var = Var ( P ( µ + RΛΘ A Θ )). In this case, our unbiased estimator to P(X A) is m P ( µ + RΛθ i A ) m i= based on a sample of size m, {θ,..., θ m }. This method of calculating probabilities is known as directional simulation in the mechanical engineering literature. An interested reader is referred to [Bjerager 988] or [Ditlevsen et al. 988] 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 A c, can be seen as a convex polytope of the form A c = N i= { x R n ai x b } i. Since µ A c, as illustrated in Figure, the ray µ + RΛθ belongs to A if the value of R exceeds a point where the ray from µ intersects the boundary of A c. In other words, one can prove that µ + RΛθ A if and only if R ( r(θ), ) where { min{i r(θ) = a i Λθ >0} (b i ai 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. ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

5 Efficient Simulation for Expectations over the Union of Half-Spaces :5 Theorem 2.2. Suppose that Assumption 2. holds. Then, an unbiased estimator for () 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 this estimator and the crude Monte Carlo estimator is bounded above by max P( R R(θ) ). θ = The above result on the second moments is a conservative one. For each sampled Θ = θ, the efficiency gain from the conditional sampling is E [h ( µ + RΛθ ) 2 ( ) 2 ] P R R(θ) Θ = θ E [ h(x) 2 {X 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., we can sample R from (R R R(θ)) by setting ( R = FR F R (r(θ)) + U [ F R (r(θ)) ]) (2) where U is uniformly distributed in (0, ). When h, 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 () 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 Algorithm When the right hand side b i of A are large, {X A} is a rare event. Thus, if we replace b i with mb i, then the event becomes rarer as m. More generally, instead of increasing b i, we consider the following sequence of diminishing random vectors: X m = m µ + RΛΘ, m =, 2,... ma for some a > 0. Here, µ, R, Λ, Σ, and Θ are assumed as in Assumption 2.. Given Assumption 2., it is known that the density of X is given by f X (x) = Σ /2 д((x µ) Σ (x µ)) for some function д. This gives us the density of X m given by f m (x) = ma Σ /2д (m (x 2a m ) µ Σ (x m )) µ. Now, we take a change of measure so that X m has mean zero under a new measure. This means that E [ [ h(x m ] ) {X m A } = E 0 h(x m ) f m(x m ] ) fm(x 0 m ) {X m A } where the superscript of E 0 denotes the probability measure P 0 under which X m is distributed as m a RΛΘ with density fm. 0 Then, we can write the unbiased estimator in Theorem 2.2 as ( ) ( ) h m RΛΘ P R m a R(Θ) Θ, (3) where R is distributed as R conditional on R m a R(Θ) and the function h m is h m (s) = h(m a s) д ( (s m a µ) Σ (s m a µ) ) д (s Σ. s) ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

6 :6 Ahn and Kim Algorithm 2 Conditional Monte Carlo : 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 then 4: Sample R from (R R m a R(θ)) by (2) 5: Compute h m ( RΛθ) and set T = h m ( RΛθ)P(R m a R(θ) θ) 6: else set T = 0 7: end if 8: return T Theorems 2.3 and 2.4 show the efficiency of this estimator for the cases where R has a Weibull-like distribution and a regularly varying distribution, respectively: Weibull-like distribution: f R (r) = α r β e α 2r β2 for some α, α 2, β 2 > 0 and β ; Regularly varying distribution: lim r f R (tr)/f R (r) = t ρ for some index ρ >. Theorem 2.3. Suppose that Assumption 2. holds, h( ) is bounded away from 0 on A, and there exists δ > 0 such that h(rλθ) e δr β2 for all (r, θ) R + R n with rλθ A and θ =. If R has a Weibull-like distribution, then the unbiased estimator (3) is asymptotically optimal, i.e., log E lim m log E [ ( ) 2 ( ) ] 2 h m RΛΘ P R m a R(Θ) Θ [ ( ) ( )] 2 =. h m RΛΘ P R m a R(Θ) Θ Theorem 2.4. Suppose that Assumption 2. holds, h( ) is bounded away from 0 on A, and we have E 0 [ h(x) 2 ] {X A } <. If R has a regularly varying distribution with index ρ >, then the unbiased estimator (3) has bounded relative error, i.e., E lim m E [ ( ) 2 ( ) ] 2 h m RΛΘ P R m a R(Θ) Θ [ ( ) ( )] 2 <. h m RΛΘ P R m a R(Θ) Θ Weibull-like distributions in Theorem 2.3 include some light-tailed distributions when β 2 as well as some heavy-tailed distributions when β 2 <. One example is the case where X follows an n-dimensional normal distribution. In this case, it is known that R 2 has a chi-square distribution with n degrees of freedom. Then, it is easy to see that f R (r) = Kr n e r 2 /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, R 2 /n F(n,ν), so f R (r) = Kr n (n 2 +νr 2 ) (n+ν )/2 for 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 X m converges to zero. ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

7 Efficient Simulation for Expectations over the Union of Half-Spaces :7 3 APPLICATIONS In 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 X is independent. The cases where X is correlated are presented in the online supplement. We use a personal computer with Intel Core i CPU and 8 GB RAM. All implementations are done using MATLAB R204b. 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 R n + and compute the default probability of an institution in a financial network. 3. Pricing Rainbow Options Rainbow 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 ( S (T ),..., S n (T ) ) ) K, 0 where S i (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-orcash options and European put-on-min options whose payoffs are max(s (T ),..., S n (T ), K) and max(k min(s (T ),..., S n (T )), 0), respectively. We assume that S i (t) follows a geometric Brownian motion. Then, for all i, we have ( (r S i (T ) = S i (0) exp σ 2 i /2 ) ) T + σ i TXi where r is the risk-free rate, σ i is the volatility of i-th asset, and X N (0, C) with correlation matrix C. Thus, S i (T ) > K is equivalent to σ i TXi > log ( K/S i (0) ) ( r σ 2 i /2) T (4) for all i. Define A := n i= {x Rn σ i Txi > log ( K/S i (0) ) ( r σi 2/2) T }. Then, the price of this option is [( e rt E max ( S (T ),..., S n (T ) ) ) ] K {X A }. 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, ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

8 :8 Ahn and Kim K Method Estimate Standard Dev. Time (sec) VR ER Crude Conditional Control Variate Combination Crude Conditional Control Variate Combination Crude Conditional Control Variate Combination Crude Conditional Control Variate Combination , 272 6, 520 Table. Estimates of prices of European call-on-max options using Algorithm, Algorithm 2, a control variate method, and the combination of Algorithm 2 and the control variate method when its underlying assets follow 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 (S i (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. 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 S i (T ) > K is a subset of A. 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 shows the estimates of this price based on four methods when X,..., X n are uncorrelated. For the implementation, we run 0 7 replications for each result, and the parameters we used are n = 0, T = 0.5, r = 0.05, σ = (0.05, 0.,..., 0.5), and S(0) = (04, 03,..., 95). The control variate is the call option on S 0. Since X has mean 0 in this example, increasing K is equivalent to increasing m with a =, which means that the event A becomes rarer. Thus, we observe the efficiency improvement by increasing the strike price K: 50, 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 FR and F R 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. 200] can serve as one possible approach to quickly generate R. This fast alternative is not exact but effective when the distribution has light tails. ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

9 Efficient Simulation for Expectations over the Union of Half-Spaces :9 Fig. 2. A stochastic activity network with 7 nodes and 0 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 = 50 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 Networks Another application of this method arises in stochastic activity networks [Nelson 203]. Each network represents a project composed of several combinations of steps and activities. All activities have nonnegative stochastic durations X,..., X n, and all predetermined combinations {C j } of those activities should be completed at the end. Figure 2 illustrates an example of a stochastic activity network with 7 nodes and 0 edges. In this figure, nodes represent project steps and edges are activities with durations X,..., X 0. The combinations of activities are C = {, 4, 8, 0}, C 2 = {, 4, 9}, C 3 = {2, 5, 8, 0}, C 4 = {2, 5, 9}, C 5 = {3, 6, 8, 0}, C 6 = {3, 6, 9}, and C 7 = {3, 7, 0}. 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 := max j i C j X i denotes the overall duration which is the longest path through the network, and b is a threshold of this project. We define A 2 := j {x R n + i C j x i > b}. Then, since activity durations are assumed to be nonnegative, we have P ( T > b ) = P ( X A 2 X R n ) P ( ) X A 2 + = ρ where ρ := P(X R n +). 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 A 2 ), 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 R n +, we can further reduce the computational time by ignoring any sample θ such that Λθ is not in R n +. A better way of doing this is to sample θ uniformly on the intersection of the set {θ Λθ 0, θ = }. The simplest case is when ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

10 :0 Ahn and Kim Σ = I n. In this case, when θ = (θ,..., θ n ) is generated, we can simply use ( θ, θ 2,..., θ n ) for a random directional vector in R n +. 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. 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 3 Modified Conditional Monte Carlo : Sample Θ = θ from the uniform distribution on the set {θ Λθ 0, θ = } and compute the likelihood ratio l(θ) using a sequential IS 2: Find R(θ) = {r 0 rλθ A} 3: if R(θ) is nonempty then 4: Sample R from (R R m a R(θ)) by (2) 5: Compute h m ( RΛθ) and set T = h m ( RΛθ)P(R m a R(θ) θ)l(θ) 6: else set T = 0 7: end if 8: return T Remark 3.. Note that since Λ = (Λ ij ) is a lower triangular matrix and Θ d = Z/ Z where Z = (Z,..., Z n ) N (0, I n ), P(ΛΘ 0) = P(Λ Z 0, Λ 2 Z + Λ 22 Z 2 0,, Λ n Z + + Λ nn Z n 0). Using this result, steps and 2 in Algorithm 3 can be done by the following procedure: () Sample Z = z from p ( ) := ϕ( )/P(Λ Z 0) and set l (z ) = ϕ(z )/p (z ) = P(Λ Z 0). (2) For k = 2,..., n, sample Z k = z k from p k ( z,..., z k ) := ϕ( )/P(Λ k z + + Λ (k )(k ) z k + Λ kk Z k 0) compute ki= ϕ(z i ) l k (z,..., z k ) = l k (z,..., z k ) p k (z k z,..., z k ) k i= ϕ(z i) = l k (z,..., z k )P(Λ k z + + Λ (k )(k ) z k + Λ kk Z k 0). (3) Set θ = z/ z and l(θ) = l n (z,..., z n ) where z = (z,..., z n ) and ϕ( ) is the probability density function of the standard normal distribution. [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 A 2 ) and divide it by ρ. We assume that µ i = 0.5 for all i and Σ is a diagonal matrix whose diagonal entries are all We run 0 7 replications for each result and use the following parameters: n = 0, a = 0.05, and b = 5. In order ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

11 Efficient Simulation for Expectations over the Union of Half-Spaces : m Method Estimate Standard Dev. Time (sec) VR ER Crude Conditional Asymp. HRT Exp. Twist Crude Conditional , Asymp. HRT Exp. Twist Crude Conditional ,64 8,46 Asymp. HRT Exp. Twist ,633 5,439 Crude Conditional ,674 46,846 Asymp. HRT Exp. Twist ,066 27,92 Table 2. Estimates of probabilities of large delay in stochastic activity networks using Algorithm, Algorithm 3, asymptotic hazard rate twisting, and exponential twisting methods when activities are uncorrelated and normally distributed. m Method Estimate Standard Dev. Time (sec) VR ER Crude Conditional Crude Conditional , Crude Conditional ,6 4,02 0 Crude Conditional ,809 8,774 Table 3. Estimates of probabilities of large delay in stochastic activity networks using Algorithm and Algorithm 3 when activities are uncorrelated and t-distributed. to see the efficiency improvement in the rare event setting, each algorithm is implemented with m =, 2, 5, and 0. Increasing m 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 as m 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 ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

12 :2 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) which is /2 n in the uncorrelated case. However, when n is large enough, a more effective alternative such as [Botev 207] can be utilized in order to estimate ρ. 3.3 Financial Networks A financial system is composed of a number of financial institutions and payment obligations among those institutions. [Eisenberg and Noe 200] describe the system as a directed network with nodes that represent banks and edges that correspond to the obligations. Recently, [Ahn et al. 207] 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; c i,d i : the outside asset value and the outside liability of node i, respectively; p ij : the payment from node i to j where p ii = 0; Π = (π ij ): the relative liabilities matrix, where π ij is the proportion of node i s obligation to node j defined by { pij /(d π ij = i + j i p ij), if d i + j i p ij > 0; 0, otherwise; w i := (c i + j i p ji) (d i + j i p ij): the initial net worth (book value) of node i; X i [0,c i ]: a random shock to the outside asset c i. In this framework, they show that the default probability of institution i is ( ) P max ξ (X w) > 0 ξ Q where Q is the set of extreme points of the polytope {(ζ, ) R n + (I Π i )ζ π i i }. Here, π i is the ith column of Π. An interested reader is referred to [Ahn et al. 207] for more details. We define A 3 := ξ Q {x [0, c] ξ x > ξ w}. Since we assume X [0, c], this probability is equal to P ( X A 3 X [0, c] ) = P( ) X A 3 where ρ := P(X [0, c]). As in the previous subsection, we concentrate on the computation of P(X A 3 ) assuming that ρ is given. For fixed θ, we define [ { }] ci µ i µ i r o (θ) := min max,. {i (Λθ ) i 0} (Λθ) i (Λθ) i Then, we observe that µ + RΛθ [0, c] is equivalent to R [0, r o (θ)]. Hence, µ + RΛθ A 3 if and only if 0 < r(θ) < r o (θ) and R R o (θ) := (r(θ), r o (θ)]. If we replace R( ) by R o ( ), 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 ρ ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

13 Efficient Simulation for Expectations over the Union of Half-Spaces :3 µ = 0.005c m Method Estimate Standard Dev. Time (sec) VR ER Crude Conditional , Crude Conditional ,468,75 00 Crude Conditional ,645 2,274,000 Crude Conditional ,820 4,960 µ = 0.05c m Method Estimate Standard Dev. Time (sec) VR ER Crude Conditional Crude Conditional , Crude Conditional ,06 2,346,000 Crude Conditional ,628 4,92 Table 4. Estimates of the default probability of institution 0 using Algorithm and Algorithm 3 when there are multivariate normal random shocks which are uncorrelated. using our scheme. In our experiments, the financial networks introduced in [Acemoglu et al. 205] are considered. They are convex combinations of a regular ring network and a symmetric regular complete network. More precisely, it is assumed that p o := j i p ji = j i p ij and that outside liabilities are the same, i.e., d := d = = d n. Here, p o represents the liability of a bank to other banks in the system. Then, the relative liabilities matrix is then given by p ( o Π = ( γ )(e n e e n ) + γ ) d + p o n (J I) for γ [0, ] where J is the matrix of ones. It represents a ring network when γ = 0 and stands for a complete network when γ =. The parameters used in this example are n = 0, γ = 0.5, d = 6, p o = 3, and c = ( 6., 6.4, 6.4, 6.3, 6.6, 6., 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 µ i is 0.5% or.5% of the outside asset value c i whereas Σ is a diagonal matrix whose i-th diagonal element is the square of 4% of c i. Note that the proposed algorithm yields estimates for P(X A 3 ). In order to produce proper estimates, we also compute ρ using MATLAB functions mvncdf and mvtcdf, respectively. Final outcomes are given as the ratio of the two. Furthermore, we set a = 0.05 and m =,0, 00, and, 000. We run the same number of replications (0 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 hitting A 3 becomes rarer as µ gets smaller. We achieve more significant improvements from ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

14 :4 Ahn and Kim µ = 0.005c m Method Estimate Standard Dev. Time (sec) VR ER Crude Conditional , Crude Conditional ,787,35 00 Crude Conditional ,656,630,000 Crude Conditional ,054 2,542 µ = 0.05c m Method Estimate Standard Dev. Time (sec) VR ER Crude Conditional Crude Conditional , Crude Conditional ,463,568,000 Crude Conditional ,22 2,508 Table 5. Estimates of the default probability of institution 0 using Algorithm and Algorithm 3 when there are multivariate t random shocks which are uncorrelated. the conditional Monte Carlo as m increases for fixed µ. For instance, in Table 4, when µ = 0.05c, the VR estimate shoots up from 267 to 27,628 and the ER estimate goes up from 40 to 4,92 as m increases from to, 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 CONCLUSION In 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 ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

15 Efficient Simulation for Expectations over the Union of Half-Spaces :5 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 205]. 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 () assuming that X follows a log-elliptical distribution, which would be a generalization of [Blanchet and Rojas-Nandayapa 20]. Despite those limitations, we hope that this work sheds new light on the rare event simulation in a multivariate setting. A PROOFS OF THEOREMS Proof 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) {R R(Θ)} Θ ]] = E [ ] h(x) {X A }. Now, for the second moment, we compute as follows: [ E h(µ + RΛΘ) 2 P ( R R(Θ) Θ ) ] 2 = E [P ( R R(Θ) Θ ) 2 [ E h(µ + RΛΘ) 2 R R(Θ), Θ ]] = E [P ( R R(Θ) Θ ) E [ h(µ + RΛΘ) 2 {R R(Θ)} Θ ]] [ = E P ( R R(Θ) Θ ) E [ h(x) 2 {X 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 We first make the following observation. We denote the density functions of Θ and R by f Θ (θ) and f R (r), respectively, and define φ m (x) := {m (x 2a m ) µ Σ (x m )} 2 µ so that f m (x) = ma ( Σ /2д φ m (x) 2). Also, by [Fang et al. 990], the relationship between д( ) and f R ( ) is f R (r) = 2π n/2 Γ(n/2) r n д(r 2 ). ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

16 :6 Ahn and Kim Using these results, for the second moment, we proceed as follows: [ ( ) 2 ( ) ] 2 E h m RΛΘ P R m a R(Θ) Θ [ ( ) [ ( ]] 2 = E P R m a R(Θ) Θ E h m RΛΘ) {R m a R(Θ)} Θ = f R (r)dr h m (rλθ) 2 f R (r)dr Since = m 2a m 2a ( h m (m a τ Λθ) = h(τ Λθ) m a m a τ φ m (τ Λθ) f R (m a τ )dτ f R (m a τ )dτ m a ) n f R (φ m (τ Λθ)) f R (m a τ ) h m (m a τ Λθ) 2 f R (m a τ )dτ h(τ Λθ) 2 f R (φ m (τ Λθ)) 2 f R (m a dτ. (5) τ ) h(τ Λθ) f R (φ m (τ Λθ)) f R (m a τ ) uniformly on the compact set {(τ,θ) R + R n τ Λθ A, θ = } as m increases, the asymptotic equivalence (5) holds. Step 2 Let ϵ > 0 be small enough. Then, there exists m o > 0 such that for all m m o and for all (τ,θ) R + R n satisfying τ Λθ A and θ =, m a τ ( ϵ) φ m (τ Λθ) m a τ ( + ϵ), (m a τ ) β β 2 + e (ϵ/2)(ma τ ) β2, and h(τ Λθ) e (ϵ/2)(ma τ ) β2. Then, for m m o, we have the right-hand side of (5) m 2a f R (m a τ )dτ = m 2a = κ = κ α (m a τ ) β e α 2(m a τ ) β2 dτ α (α 2 ϵ/2)β 2 ψ m(τ )dτ ψ m(τ )dτ e ϵ(ma τ ) β 2 f R (m a τ ( ϵ)) 2 f R (m a dτ τ ) ψ 2 m(τ )dτ α ( ϵ) 2β (m a τ ) β e {2α 2( ϵ) β 2 α 2 ϵ }(m a τ ) β2 dτ α ( ϵ) 2β β 2 {2α 2 ( ϵ) β 2 α2 3ϵ/2} ψ 2 m(τ )dτ f Θ (θ)e 2{α 2( ϵ) β 2 ϵ }(m a ) β2 dθ. (6) Here, the following auxiliary functions and constant are used: ψ m(τ ) := m a (α 2 ϵ/2)β 2 (m a τ ) β 2 e (α 2 ϵ/2)(m a τ ) β2, and ψ 2 m(τ ) := m a {2α 2 ( ϵ) β 2 α 2 3ϵ/2}β 2 (m a τ ) β 2 e {2α 2( ϵ) β 2 α 2 3ϵ/2}(m a τ ) β2, κ = α 2 ( ϵ)2β β 2 2 (α 2 ϵ/2){2α 2 ( ϵ) β 2 α2 3ϵ/2}. ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

17 Efficient Simulation for Expectations over the Union of Half-Spaces :7 Step 3 Let us consider the multidimensional Laplace-type integral J(λ) := Ω G(x)e λ F(x) dx where λ > 0, Ω R n, G( ) is continuous, and F ( ) has continuous second-order partial derivatives in a neighborhood of x o := argmin x Ω F (x). The multidimensional Laplace s approximation in [Wong 200] states that if J(λ) converges absolutely for all λ λ o, x o is the unique minimizer of F ( ), and the Hessian matrix ( 2 ) f x i x j x=x o is positive definite, then J(λ) C o G(x o )e λ F(x o) λ p(n) where C o is a positive constant and p(n) = n/2 if x o is a critical point; (n + )/2 otherwise. We know that the polytope A c has a finite number N o N of facets. Suppose a i x b i, i =,..., N o, be facet-defining inequalities. Let us split the range of the integral in the right-hand side of (6) into (N o + ) regions, say Ω 0, Ω,..., Ω No. Here, Ω 0 := N i= {θ a i Λθ 0} and Ω k is the region of θ where kth facet and the direction vector Λθ intersect for k =,..., N o. By Section 2., for all θ Ω 0, r(θ) =, and for all θ Ω k, k =,..., N o, r(θ) = b k a k µ a k Λθ. Let θ k = argmin θ Ωk r(θ) for each k. Then, it is easy to see that θ k is the unique minimizer on Ω k, r ( ) β 2 has continuous second-order partial derivatives in a neighborhood of θ k, and its its Hessian matrix at θ k is positive definite for each k. Hence, using the multidimensional Laplace s approximation, we have N o the right-hand side of (6) κ C k f Θ(θ k )e 2{α 2( ϵ) β 2 ϵ }(m a r (θ k )) β2 m aβ 2p k (n) k= κ C k o f Θ (θ ko )e 2{α 2( ϵ) β 2 ϵ }(m a r (θ ko )) β2 m aβ 2p ko (n) where C k > 0, k =,..., N o, p k (n) = n/2 or (n + )/2, and k o := argmin k r (θ k ). Step 4 Similarly, the following relationship can be found for the first moment. E [ ( ) ( )] h m RΛΘ P R m a R(Θ) Θ = E [ ] h m (RΛΘ) {R m a R(Θ)} = m a m a The asymptotic equivalence is based on similar arguments in Step. h m (rλθ) f R (r)dr h(τ Λθ)f R (φ m (τ Λθ))dτ. ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

18 :8 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 large m, the last term in the above computations is bounded below by m a L f R (m a τ ( + ϵ))dτ = m a L κ 2 = κ 2 α ( + ϵ) β (m a τ ) β e α 2(+ϵ) β 2 (m a τ ) β2 dτ m a β 2 (α 2 ( + ϵ) β 2 + ϵ)(m a τ ) β 2 e (α 2(+ϵ) β 2 +ϵ)(m a τ ) β2 dτ f Θ (θ)e (α 2(+ϵ) β 2 +ϵ)(m a ) β2 dθ. (7) where κ 2 := Lα β 2 ( + ϵ)β /(α 2 ( + ϵ) β 2 + ϵ). The inequality holds because there exists m > 0 such that (m a τ ) β β 2 + e ϵ(ma τ ) β2 for all m,τ,θ satisfying m m, τ Λθ A and θ =. This is similar Step 2. Now we apply the multidimensional Laplace s approximation as in Step 3 so that we obtain N o the right hand side of (7) κ 2 C 2 k f Θ(θ k )e (α 2(+ϵ) β 2 +ϵ)(m a r (θ k )) β2 m aβ 2p k (n) k= κ 2 C 2 k o f Θ (θ ko )e (α 2(+ϵ) β 2 +ϵ)(m a r (θ ko )) β2 m aβ 2p ko (n) where p k (n) and k o are as defined in Step 3, and C 2 k, k =,..., N o, are positive constants. Therefore, we finally arrive at [ ( ) 2 ( ) ] 2 log E h m RΛΘ P R m a R(Θ) Θ L := lim [ ( ) ( )] m 2 α 2( ϵ) β 2 ϵ log E h m RΛΘ P R m a α R(Θ) Θ 2 ( + ϵ) β. 2 + ϵ Since ϵ is arbitrary, L. Then, the result holds since L is trivial. Proof of Theorem 2.4. Since R follows a regularly varying distribution, lim r f R (tr)/f R (r) = t ρ for some index ρ > uniformly on intervals of the form (t o, ), t o > 0 by [Resnick 2008]. This implies that there exists a point z 0 such that f R ( ) 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: [ ( ) 2 ( ) ] 2 E h m RΛΘ P R m a R(Θ) Θ m 2a m 2a m 2a f R (m a τ )dτ f R (m a τ )dτ = m 2a f R (m a ) 2 ( ϵ) 2ρ (max θ f R (m a )τ ρ dτ h(τ Λθ) 2 f R(φ m (τ Λθ)) 2 f R (m a dτ τ ) h(τ Λθ) 2 f R(m a τ ( ϵ)) 2 f R (m a dτ τ ) ) τ ρ dτ h(τ Λθ) 2 f R (m a )τ ρ ( ϵ) 2ρ dτ h(τ Λθ) 2 τ ρ dτ. ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.

19 Efficient Simulation for Expectations over the Union of Half-Spaces :9 It is easy to see that the assumption E 0 [ h(x) 2 ] {X A } < implies h(τ Λθ) 2 τ ρ dτ <. On the other hand, the squared first moment becomes [ ( ) ( )] 2 { E h m RΛΘ P R m a R(Θ) Θ m 2a L 2 m 2a { L 2 m 2a { } 2 h(τ Λθ)f R (φ m (τ Λθ))dτ = L 2 m 2a f R (m a ) 2 ( + ϵ) 2ρ { Therefore, the ratio of the two is clearly bounded from above. } 2 f R (m a τ ( + ϵ))dτ } 2 f R (m a )τ ρ ( + ϵ) ρ dτ τ ρ dτ } 2. ACKNOWLEDGMENTS The 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-206RDAB ). REFERENCES D. Acemoglu, A. Ozdaglar, and A. Tahbaz-Salehi Systemic Risk and Stability in Financial Networks. American Economic Review 05, 2 (205), D. Ahn, N. Chen, and K.-K. Kim On Default Probabilities in Financial Networks. (207). working paper. S. Asmussen and P. W. Glynn Stochastic Simulation: Algorithms and Analysis. Springer. P. Bjerager Probability Integration by Directional Simulation. Journal of Engineering Mechanics 4, 8 (988), J. H. Blanchet and L. Rojas-Nandayapa. 20. Efficient Simulation of Tail Probabilities of Sums of Dependent Random Variables. Journal of Applied Probability 48A (20), J. H. Blanchet and Y. Shi. 203a. Efficient Rare Event Simulation for Heavy-Tailed Systems via Cross Entropy. Operations Research Letters 4, 3 (203), J. H. Blanchet and Y. Shi. 203b. Efficient Splitting-Based Rare Event Simulation Algorithms for Heavy-Tailed Sums. In Proceedings of the 203 Winter Simulation Conference, R. Pasupathy, S.-H. Kim, A. Tolk, R. Hill, and M. E. Kuhl (Eds.). IEEE, Z. I. Botev The Normal Law under Linear Restrictions: Simulation and Estimation via Minimax Tilting. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 79, (207), J. C. C. Chan and D. P. Kroese. 20. Rare-Event Probability Estimation with Conditional Monte Carlo. Annals of Operations Research 89, (20), Gerhard Derflinger, Wolfgang Hörmann, and Josef Leydold Random Variate Generation by Numerical Inversion when Only the Density is Known. ACM Transactions on Modeling and Computer Simulation 20, 4 (200), 8: 8:25. O. Ditlevsen, P. Bjerager, R. Olesen, and A. M. Hasofer Directional Simulation in Gaussian Processes. Probabilistic Engineering Mechanics 3, 4 (988), H. Drees and L. de Haan Estimating Failure Probabilities. Bernoulli 2, 2 (205), L. Eisenberg and T. H. Noe Systemic Risk in Financial Systems. Management Science 47, 2 (200), K.-T. Fang, S. Kotz, and K.-W. Ng Symmetric Multivariate and Related Distributions. Chapman and Hall, London, UK. P. Glasserman and H. P. Young How Likely is Contagion in Financial Networks? Journal of Banking and Finance 50 (205), S. Juneja, R. L. Karandikar, and P. Shahabuddin Asymptotics and Fast Simulation for Tail Probabilities of Maximum of Sums of Few Random Variables. ACM Transactions on Modeling and Computer Simulation 7, 2 (2007), 35. ACM Transactions on Modeling and Computer Simulation, Vol., No., Article. Publication date: November 207.