We analyze the computational problem of estimating financial risk in a nested simulation. In this approach,

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1 MANAGEMENT SCIENCE Vol. 57, No. 6, Jue 2011, pp iss eiss doi /msc INFORMS Efficiet Risk Estimatio via Nested Sequetial Simulatio Mark Broadie Graduate School of Busiess, Columbia Uiversity, New York, New York 10027, Yipig Du Idustrial Egieerig ad Operatios Research, Columbia Uiversity, New York, New York 10027, Ciamac C. Moallemi Graduate School of Busiess, Columbia Uiversity, New York, New York 10027, We aalyze the computatioal problem of estimatig fiacial risk i a ested simulatio. I this approach, a outer simulatio is used to geerate fiacial scearios, ad a ier simulatio is used to estimate future portfolio values i each sceario. We focus o oe risk measure, the probability of a large loss, ad we propose a ew algorithm to estimate this risk. Our algorithm sequetially allocates computatioal effort i the ier simulatio based o margial chages i the risk estimator i each sceario. Theoretical results are give to show that the risk estimator has a faster covergece order compared to the covetioal uiform ier samplig approach. Numerical results cosistet with the theory are preseted. Key words: simulatio; decisio aalysis; risk; risk maagemet; sequetial aalysis History: Received February 24, 2010; accepted Jauary 9, 2011, by Gérard Cacho, stochastic models ad simulatio. Published olie i Articles i Advace April 29, Itroductio The measuremet ad maagemet of risk is a icreasigly importat fuctio at fiacial istitutios. A primary goal of risk measuremet is to esure that baks ad other fiacial firms have sufficiet capital reserves i relatio to their holdigs ad ivestmet activities. The recet failures of large ad small ivestmet ad commercial baks highlight the eed for better modelig ad computatio of fiacial risk measures. Risk measuremet is typically divided ito two stages: sceario geeratio ad portfolio revaluatio. Sceario geeratio refers to the samplig of risk factors over a give time horizo. This first (or outer) stage is ofte performed with Mote Carlo simulatio, especially whe more realistic models with a large umber of correlated risk factors are used. Portfolio revaluatio refers to the computatio of the portfolio value at the risk time horizo give a particular sceario of risk factors. Ofte the portfolio cotais derivative securities with oliear payoffs that, i cojuctio with more realistic fiacial models, require Mote Carlo simulatio for this secod (or ier) stage. Thus, i realistic applicatios, the risk measuremet calculatio ivolves a two-level ested Mote Carlo simulatio. Because ested Mote Carlo simulatio ca represet a prohibitive computatioal challege, various approximatio approaches are ofte employed. The focus of our paper is o algorithmic improvemets of the direct ested Mote Carlo simulatio approach, so that risk computatio ca be doe o portfolios of derivative securities with more realistic multifactor fiacial models. I this paper, we cosider what is perhaps the most basic risk measure, the probability that the future portfolio value falls below a prespecified threshold, i other words, the probability of a large loss. Whe aalytical formulas are available for the portfolio revaluatio step, a primary challege of a sigle-level Mote Carlo simulatio is to reduce the variace of the simulatio risk estimator. I the ested settig, simulatio is also used for the portfolio revaluatio step, ad additioal sources of variability are itroduced. The secod level of simulatio itroduces bias ito the computatio, ad hece both bias ad variace eed to be balaced ad reduced to miimize the total error i the simulatio risk estimate. The problem of estimatig the probability of a loss via ested simulatio was first aalyzed by Lee (1998) ad Lee ad Gly (2003), ad was subsequetly cosidered by Gordy ad Jueja (2010). These authors primarily cosidered ad aalyzed uiform ested simulatio estimators. Such estimators employ 1172

2 Maagemet Sciece 57(6), pp , 2011 INFORMS 1173 a costat umber of ier samples across portfolio revaluatio calculatios, thus allocatig computatioal effort uiformly across all scearios. They demostrate that, asymptotically, the bias of a uiform estimator is a fuctio of the umber of ier samples used i each portfolio revaluatio, whereas the variace of a uiform estimator is a fuctio of the umber of outer scearios. They characterize the asymptotically optimal uiform estimator. This estimator balaces a limited computatioal budget betwee usig may outer scearios, to lower variace, ad usig may ier samples i each sceario, to lower bias, i a way that miimizes the overall mea squared error (MSE) amog the class of uiform estimators. This paper seeks to exploit the fact that accurate portfolio revaluatio is ot equally importat across all scearios. Nested simulatio ca be made much more efficiet by allocatig computatioal effort ouiformly across scearios. Nouiform estimators have bee suggested previously by others i a umber of cotexts (e.g., Lee ad Gly 2003; Lesevski et al. 2004, 2007; Gordy ad Jueja 2008; La et al. 2010). Here, we propose ad aalyze a ovel class of ouiform estimators based o the idea of allocatig additioal effort to scearios with a greater expected margial chage to the risk measure. Specifically, the mai cotributios of this paper are as follows: 1. We propose a ouiform ested simulatio algorithm for estimatig the probability of a loss. Our algorithm proceeds by allocatig the ier stage samples for portfolio revaluatio i a sequetial fashio. At each time step, it myopically selects the sceario where oe additioal ier stage sample will have the greatest margial impact to the estimated loss probability. This algorithm is simple to implemet ad icurs miimal computatioal overhead. 2. We provide a aalysis that demostrates the lower asymptotic bias of our approach. Give m ier stage samples i each sceario, a uiform ested estimator has a asymptotic bias of order m 1. We aalyze a simplified variatio of our ouiform estimator ad demostrate that with a average of m ier stage samples per sceario, the asymptotic bias is of order m 2+ for all positive. Hece, for the same overall umber of samples, the ouiform estimator reduces bias by a order of magitude. This theoretical aalysis builds o ideas from sequetial hypothesis testig ad highlights the relatioship betwee our ouiform estimatio algorithm ad classical sequetial hypothesis testig. 3. We provide a aalysis that demostrates the lower asymptotic MSE of our approach. Give a computatioal budget of k, the optimal uiform ested estimator results i a asymptotic MSE of order k 2/3. Because ouiform samplig provides a lower bias for the same umber of ier stage samples, some of this computatioal savigs ca be used for the geeratio of additioal outer scearios to lower variace. We show that our ouiform method has a asymptotic MSE of order k 4/5+ for all positive. Furthermore, we demostrate a practical implemetatio of our ouiform estimator that adaptively balaces bias (ier samplig) ad variace (outer sceario geeratio). 4. We demostrate the practical beefits of our method via umerical experimets. Numerical experimets demostrate that the performace of our ouiform ested estimatio algorithm is up to two orders of magitude better tha competig methods. Hece, we illustrate that the results achievable i practice are cosistet with the gais suggested by the theory. The rest of this paper is orgaized as follows. Sectio 1.1 cotais a brief literature review. The problem setup ad otatio are give i 2. Results for uiform ier stage samplig are reviewed i 3. A sequetial ouiform algorithm is motivated ad preseted i 4 ad a theoretical aalysis is give i 5. Sectio 6 gives a practically implemetable adaptive versio of the sequetial algorithm, ad umerical results are provided i 7. Cocludig remarks are give i 8, ad proofs are provided i the appedix Literature Review Overviews of fiacial risk measuremet ad maagemet are give i Crouhy et al. (2000), Jorio (2006), ad McNeil et al. (2006). There is a large literature o the properties of alterative risk measures (see, e.g., Artzer et al. 2000, Rockafellar ad Uryasev 2002, Föllmer ad Schied 2002). Variace reductio techiques to improve first stage samplig are give i Glasserma et al. (2000, 2002). Most closely related to our work is that of Lee (1998) ad Lee ad Gly (2003), who cosider the problem of estimatig the probability of a large loss ad aalyze ested simulatio estimators ad their covergece properties uder uiform ier stage samplig. They cosider two settigs, where the uderlyig sceario space is either cotiuous or discrete. 1 They establish that, give a total computatioal budget of k, the optimal uiform ested estimator results o a asymptotic MSE of order k 2/3 i the cotiuous case ad k 1 log k i the discrete case. Idepedetly, Gordy ad Jueja (2010) also cosider estimatig the probability of large loss i the cotiuous case, uder a differet set of assumptios. They 1 I this paper, we will cosider oly cotiuous sceario spaces. Note that the theory is qualitatively differet i the discrete case versus the cotiuous case.

3 1174 Maagemet Sciece 57(6), pp , 2011 INFORMS also cosider two additioal risk measures (value at risk ad expected shortfall). For each of these three risk measures, they derive asymptotic bias ad variace results for uiform secod stage samplig. This allows them to derive the optimal allocatio of effort betwee first ad secod stage samplig ad derive the optimal asymptotic MSE of order k 2/3. They also propose a jackkife procedure for reducig bias. The idea of ouiform ested estimatio of risk measures dates back to at least the work of Lee ad Gly (2003). I the discrete case, they idetify a class of ouiform ested estimators for the probability of a large loss with asymptotic MSE of order k 1 log k. I this settig, the ouiform estimator achieves the same asymptotic covergece as the uiform estimator, but with a better costat. Lesevski et al. (2004, 2007) propose a ouiform ested estimator for a related discrete problem: they estimate the worst case expected loss across a fiite set of scearios. They are able to develop cofidece itervals for their estimatio procedure. La et al. (2007, 2010) ad Liu et al. (2010) exted this work to the case of estimatig expected shortfall. Cotemporaeous with the preset work, Gordy ad Jueja (2008) suggest a broad class of ouiform estimators for estimatig the probability of a loss large, as i the preset settig. Their descriptio is rather geeral, however, whereas we provide a cocrete algorithm. Note that some of the ouiform estimators i this prior literature have similarities to the ouiform estimator that we propose; we discuss these i 4. Critically, however, oe of this prior work is able to establish theoretically that a ouiform estimator coverges at a faster asymptotic order tha is possible with uiform estimators. There are some coectios betwee ested simulatio to estimate risk ad rakig ad selectio (R&S) procedures that search for the best amog a fiite umber of systems. For a overview of rakig ad selectio, see Kim ad Nelso (2005) ad the book by Che ad Lee (2010). Each R&S system correspods to a outer sample, ad samplig a performace measure from a system correspods to a ier sample. May R&S procedures rely o myopic rules to determie a allocatio of ier samples (e.g., Frazier et al. 2008), ad the spirit of our procedure is similar. R&S typically cosiders a fiite ad small umber of systems, whereas our outer samplig draws from a ifiite ad ofte multidimesioal domai. The R&S objective of fidig the best performig system is also differet tha estimatig a risk measure across ad rage of first stage outcomes. Fially, also of iterest is the work of Liu ad Staum (2010); they explore a alterative approach based o stochastic krigig for estimatig a risk measure. Hog ad Jueja (2009) cosider the beefits of kerel smoothig i risk estimatio. Su et al. (2011) cosider ested simulatio i the cotext of estimatig coditioal variace. 2. Problem Formulatio: Nested Simulatio Cosider the problem of measurig the risk of a portfolio of securities at some future time t = (the risk horizo), from the perspective of a observer at time t = 0. Deote the curret portfolio value by X 0. The value of the portfolio at time, X, is i geeral a radom variable ad thus is ot kow at time 0. We assume, however, that there is a probabilistic model for the ucertaity betwee times 0 ad. I particular, suppose that is a set of possible future scearios or risk factors. Each sceario icorporates sufficiet iformatio so as to determie all assets prices at time. Thus, i each sceario, the portfolio has value X. The mark-to-market loss of the portfolio at time i sceario is give by 2 L X 0 X A risk measure is a fuctioal that quatifies the risk of the radom variable L by a scalar L. Some commo examples of risk measures iclude value at risk ad coditioal value at risk. I this paper, we will focus o what is perhaps the most basic risk measure, the probability of a large loss; that is, give a threshold c, we are iterested i estimatig the probability of the loss L exceedig c. Deote the resultig probability by PL c. To estimate the loss probability, we face two challeges. First, typically, the space of possible scearios is quite large, if ot ifiite. Thus, oe approach is to approximate the distributio of the loss radom variable L with a empirical distributio obtaied by Mote Carlo samplig. This is referred to as the outer level (or first stage) of the simulatio. I particular, if 1 are idepedet ad idetically distributed (i.i.d.) samples draw accordig to the physical (or real-world) distributio of, the we ca approximate the loss probability by 1 Li c (1) However, eve i a sigle sceario i, it may be difficult to exactly compute the loss L i. The portfolio may cotai a collectio of complex, path-depedet securities with radom cashflows betwee times ad some fial horizo T. The, the loss L i must be estimated via a ier level (or secod stage) of Mote Carlo simulatio of the expected cashflows of 2 Without loss of geerality, we assume the portfolio has o itermediate cashflows before time, ad that the riskless rate is 0.

4 Maagemet Sciece 57(6), pp , 2011 INFORMS 1175 Figure 1 Illustratio of Uiform Samplig 1. i. Zˆ i,1. Ẑ i,m 0 T Notes. The outer stage geerates fiacial scearios 1. Coditioal o sceario i, m ier stage portfolio losses Z i 1 Z i m are geerated. Time t the portfolio over the iterval T. The ier simulatio occurs uder the risk-eutral distributio, coditioed o the sceario i. If Z i 1 Z i m are m i.i.d. samples of losses geerated accordig to this secod stage of simulatio, each with mea L i, the we ca approximate the loss L i i sceario i by ˆL i 1 m m Z i j (2) The Uiform estimator of Algorithm 1 describes a ested simulatio procedure that combies the estimates from the outer ad ier levels of simulatio i the obvious way to produce a overall estimate of the loss probability. The estimator is a fuctio of two parameters:, the umber of outer stage samples, ad m, the umber of ier stage samples. We say that this estimator samples uiformly i the sese that a costat umber of ier stage samples is used for each outer stage sceario. This procedure is illustrated i Figure 1. Algorithm 1 (Estimate the probability of a large loss usig a uiform ested simulatio. The parameter m is the umber of ier samples per sceario. The parameter is the umber of outer scearios.) 1: procedure Uiform(m ) 2: for i 1 to do 3: geerate sceario i 4: coditioed o sceario i, geerate i.i.d. samples Z i 1 Z i m of portfolio losses 5: compute a estimate of the loss i sceario i, ˆL i 1/m m j=1 Z i j 6: ed for 7: compute a estimate of the probability of a large loss, ˆ 1/ ˆL i c 8: retur ˆ 9: ed procedure 3. Optimal Uiform Samplig The Uiform estimator is a fuctio of two parameters:, the umber of scearios, ad m, the umber of ier stage samples for each sceario. This raises a obvious questio: what are the best choices j=1 for the parameters m ad? This questio has bee addressed i the work of Lee (1998) ad Gordy ad Jueja (2010). We follow the latter approach. Deote the Uiform estimate of the probability of a large loss by ˆ m Uiformm. The obvious objective is to choose parameters m so as to miimize the MSE of the estimate ˆ m, subject to the costrait of a limited budget of computatioal resources. The Uiform estimator ivolves outer sceario geeratio ad ier samplig. We will make the assumptio that the computatioal effort of this estimator is domiated by the latter. 3 Give parameters m, a total of m ier samples are geerated to compute the estimate ˆ m. Thus, give a computatioal work budget k o the total umber of ier samples, we have the optimizatio problem miimize m E ˆ m 2 subject to m k m 0 The mea squared error objective ca be decomposed ito variace ad bias terms accordig to E ˆ m 2 = E ˆ m E ˆ m 2 }{{} variace (3) + E ˆ m 2 }{{} (4) bias 2 To aalyze the asymptotic behavior of the MSE, first cosider the followig techical assumptio: 4 Assumptio 1. Deote by L the portfolio loss i sceario at time, ad deote by ˆL a estimator of the form (2) for L, based o the average of m i.i.d. ier 3 This will typically be true because the risk horizo is ofte short relative to the time horizo T of realized cashflows. I ay evet, the aalysis i this paper ca easily be exteded to accout for the computatioal effort of sceario geeratio. 4 For a alterative set of assumptios, see Lee (1998).

5 1176 Maagemet Sciece 57(6), pp , 2011 INFORMS stage samples. Assume the followig: 1. The joit probability desity fuctio p m l ˆl of L ˆL ad its partial derivatives /lp m l ˆl ad 2 /l 2 p m l ˆl exist for each m ad l ˆl. 2. For each m 1, there exist fuctios f 0 m, f 1 m, ad f 2m so that p m l ˆl f 0 m ˆl l p ml ˆl f 1 mˆl 2 l p ml ˆl 2 f 2 mˆl for all l ˆl. Furthermore, sup m ˆl r f i m ˆl dˆl < for all i = ad 0 r 4 Gordy ad Jueja (2010) establish the followig: 5 Theorem 1. Suppose that Assumptio 1 holds, ad deote by f the desity of the loss variable L. As m, the bias of the Uiform estimator asymptotically satisfies where E ˆ m = c m + Om 3/2 c c c 1 2 f ce 2 L = c (5) ad 2 is the variace of the ier stage samples i sceario. Theorem 1 directly provides a asymptotic aalysis of the bias term i the MSE (4). Theorem 1 ca immediately be employed to aalyze the variace term, as i the followig corollary: Corollary 1. Uder the coditios of Theorem 1, as m, the variace of the Uiform estimator satisfies Var ˆ m = Proof. Note that Var ˆ m = Var 1 ( 1 ˆL i c + Om 1 1 = E ˆ m 1 E ˆ m ) = 1 Var ˆL 1 c 5 I what follows, give arbitrary sequeces f N ad g N, ad a positive sequece q N, as N, we will say that f N = g N + Oq N if lim N f N g N /q N <, i.e., if the differece betwee f ad g is asymptotically bouded above by some costat multiple of q. Similarly, we will say that f N = g N + oq N if lim N f N g N /q N = 0, i.e., if the differece betwee f ad g is asymptotically domiated by every costat multiple of q. Fially, we will say that f N = g N + q N if 0 < lim if N f N g N /q N lim N f N g N /q N <, i.e., if the differece betwee f ad g is asymptotically bouded above ad below by costat multiples of q. where we have used the fact that the loss estimates ˆL i are idepedet ad idetically distributed. Applyig Theorem 1, Var ˆ m = 1 + E ˆ m 1 E ˆ m + E ˆ m 1 = + Om 1 1 Theorem 1 ad Corollary 1 provide a complete asymptotic characterizatio of the MSE of the Uiform estimator. The asymptotic variace of the estimator is determied by the umber of scearios ad decays as 1, whereas the asymptotic bias of the estimator is determied by the umber of ier stage samples per sceario m ad decays as m 1. Give a computatioal budget of a total of k ier stage samples, a aive choice of parameters m might be to sample equally i the outer ad ier stages, i.e., set m = = k 1/2. This would result i a asymptotic bias squared of order k 1, a asymptotic variace of order k 1/2, ad a overall asymptotic MSE of order k 1/2. Because the variace is asymptotically domiatig the bias squared ad determiig the MSE, the aive Uiform estimator is clearly ot optimal. Oe could do better by usig fewer ier stage samples per sceario ad icreasig the umber of scearios. To fid the optimal Uiform estimator, usig Theorem 1 ad Corollary 1, we ca approximate the miimum MSE problem (3) by the optimizatio problem miimize m 1 subject to m k m 0 This suggests optimal allocatios m = k 1/3 / + 2 c m 2 = k 2/3 ( ) 1 1/3 where (6) 2 2 c ad the optimal asymptotic mea squared error E ˆ m 2 = 3 2 k 2/3 + ok 2/3 (7) The optimal allocatios suggested by (6) ivolve, asymptotically, order k 2/3 outer stage scearios ad order k 1/3 ier stage samples per sceario. However, the optimal costat factors deped o the costat c, ad it is ot clear how to effectively estimate c a priori. As we will see i 7, the choice of these

6 Maagemet Sciece 57(6), pp , 2011 INFORMS 1177 costat factors is critical to the practical performace of a uiform estimator. Fially, it is istructive to compare the rate of covergece of the optimal Uiform estimator i a two-level ested Mote Carlo simulatio to that of a estimator of the probability of a large loss i a siglelevel Mote Carlo simulatio. I the latter case, scearios 1 are geerated. It is assumed that i each sceario i, the loss L i ca be exactly computed ad the probability is estimated via (1). Note that the estimator (1) is ubiased ad has a variace proportioal to 1. I a sigle-level simulatio, the, the amout of work is proportioal to, whereas the MSE of the estimator decays proportioal to 1. I a two-level simulatio, however, as show above, the amout of work is proportioal to k, whereas the MSE decays at best at a rate of k 2/3. This slower rate of decay is due to the bias itroduced by the ier level of simulatio. 4. Sequetial Samplig The Uiform estimator described i 2 ad 3 employs a costat umber of ier stage samples for each outer stage sample. It is ituitively clear to see that this may ot be a efficiet strategy. As a illustrative example, cosider the situatio depicted i Figure 2. Here, we wish to estimate the loss probability associated with the shaded regio. There are two outer stage scearios, 1 ad 2, associated with the portfolio losses L 1 ad L 2, respectively. These true losses are approximated, i each sceario, by the estimated losses ˆL 1 ad ˆL 2. Suppose that, uder a uiform ested simulatio, the portfolio losses estimated i each sceario are distributed accordig to the dashed probability distributios. The, it is clear that it would be advatageous to employ fewer ier stage samples at sceario 1, ad more ier stage samples at sceario 2. This is because the loss probability estimate ˆ is calculated accordig to ˆ 1 ˆL i c (8) Thus, oly the ordial positio of the estimates ˆL 1 ad ˆL 2 relative to the loss threshold c is relevat. Give the ucertaity i the estimate ˆL 1, it is fairly certai that L 1 < c, ad, ideed, this could likely be iferred usig fewer ier samples i sceario 1. Give the ucertaity i the estimate ˆL 2, the fact that L 2 c, o the other had, is much less certai. Without more ier samples i this sceario, there may be sigificat risk of misclassifyig L 2. These observatios suggest that a ouiform samplig strategy may be superior: the umber of ier samples m 1 employed at sceario 1 should be less tha the umber of ier samples m 2 employed at sceario 2. The discussio above suggests that i a sceario with a loss L that is much greater tha c or much less tha c, few ier samples are ecessary. If the loss L is close to c, however, may ier samples are ecessary. Ufortuately, a priori, it is ot clear how to do this. It is impossible to kow the value of L this is exactly what we seek to estimate via the ier Mote Carlo simulatio. We propose a procedure that simultaeously maitais estimates of the loss i each sceario while sequetially attemptig to allocate additioal ier samples across the outer scearios. We will first motivate our algorithm with a iformal justificatio, ad the give a precise descriptio. I particular, suppose that there are scearios 1. For each sceario i, suppose that m i ier samples Z i 1 Z imi have bee made, resultig i the loss estimate ˆL i mi 1/m i j=1 Z i j. This results i a overall probability of a large loss estimate ˆ give by (8). Without loss of geerality, assume that ˆL i c. Suppose we wish to perform oe additioal ier stage Figure 2 A Illustratio of the Beefits of Nouiform Samplig Probability ˆL 1 Choose m 2 large ˆL 2 L( 1 ) c L( 2 ) Loss Choose m 1 small Notes. The ucertaity i the loss ˆL 1 estimated i sceario 1 is ulikely to impact the overall probability of large loss estimate, hece the umber of ier samples m 1 i this sceario ca be chose to be small. I sceario 2, however, a large umber of ier samples m 2 should be used.

7 1178 Maagemet Sciece 57(6), pp , 2011 INFORMS Figure 3 A Illustratio of the Impact of a Additioal Sample Probability ˆL i c ˆL i Loss Add 1 sample Note. A additioal ier sample i sceario i will oly chage the overall probability of loss estimate if ˆL i moves to the opposite side of the loss threshold c. sample. If we were to perform the additioal sample i sceario i, this would result i a ew loss estimate give by ˆL i 1 m i +1 Z m i + 1 i j = 1 Z m j=1 i + 1 i mi +1 + m i m i + 1 ˆL i The additioal sample will oly impact the estimate ˆ if the ˆL i is o the opposite side of the threshold level c tha ˆL i, i.e., if ˆL i < c. This is illustrated i Figure 3. To myopically maximize the impact of the sigle additioal sample, we will seek to choose the sceario i that maximizes the probability of such a sig chage. Suppose that the additioal sample Z i mi +1 has variace i 2 2 i. Observe that P ˆL i <c = P( Z imi +1 L i < m i ˆL i c L i c ) P Z imi +1 L i < m i ˆL i c ( ) 1 1+ m2 i ˆL i 2 i c 2 (9) Here, the approximatio follows from the assumptio that m i 1, so that m i ˆL i c L i c m i ˆL i c. The iequality follows from the oe-sided Chebyshev iequality. By aalogous cosideratio of the symmetric case (where ˆL i < c), a myopic allocatio rule that seeks to maximize the probability of a sig chage estimated via the Chebyshev boud 6 (9) will choose to add the additioal ier sample i sceario i, where i m arg mi i ˆL i i c (10) i A alterative justificatio for the myopic rule (10) arises if the additioal sample Z i mi +1 is draw from a locatio-scale family of distributios, e.g., if Z i mi +1 is ormally distributed. Such a distributio is specified by a mea L i ad a variace i 2 so that ( ) z Li P Z i mi +1 < z = G 6 We thak a aoymous reviewer for suggestig this motivatio. i where G is a icreasig fuctio. I this case, P ˆL i < c P Z i mi +1 L i < m i ˆL i c ( = G m ) i ˆL i c (11) i Maximizig the probability of a sig chage accordig to (11) also results i the myopic rule (10). We call the quatity miimized i (10), m i / i ˆL i c, the error margi associated with the sceario i. The allocatio rule (10), which picks a sceario by greedily miimizig the error margi, makes ituitive sese qualitatively. It ecourages additioal ier samples at scearios that are close to the loss boudary (i.e., ˆL i c is small), scearios with few ier samples (i.e., m i is small), or scearios with sigificat variability i the portfolio losses (i.e., i is large). The Sequetial estimator of Algorithm 2 employs the allocatio rule (10). This estimator takes a triple m 0 m of iput parameters. Here, is the desired umber of outer stage scearios, m 0 is the iitial umber of ier stage samples per sceario, ad m is the desired average umber of ier stages samples per sceario at the coclusio of the algorithm. The algorithm proceeds as follows: first, scearios are geerated, ad, for each sceario, m 0 ier stage samples are performed. The remaiig m m 0 ier stage samples are allocated oe at a time i a sequetial fashio myopically, as i (10). Algorithm 2 (Estimate the probability of a large loss usig a sequetial ouiform ested simulatio. The parameter m 0 is the iitial umber of ier samples per sceario. The parameter m is the average umber of ier samples per sceario at the coclusio of the simulatio. The parameter is the umber of scearios.) 1: procedure Sequetial(m 0 m ) 2: for i 1 to do 3: geerate sceario i 4: coditioed o sceario i, geerate i.i.d. samples Z i 1 Z im 0 of portfolio losses 5: m i m 0

8 Maagemet Sciece 57(6), pp , 2011 INFORMS : ed for 7: while m i < m do 8: set i argmi i m i ˆL i c/ i, where, for each 1 i, ˆL i is the curret estimate of the loss i sceario i, ˆL i 1/m i m i j=1z ij, ad i is the stadard deviatio of the distributio of losses i sceario i 9: geerate oe additioal portfolio loss sample Z i m i +1 i sceario i 10: m i m i : ed while 12: compute a estimate of the probability of a large loss, ˆ 1/ ˆL i c 13: retur ˆ 14: ed procedure Note that the Sequetial estimator requires access to the coditioal stadard deviatio i 2 of losses i each sceario i, to compute the error margi. These are ot required for the Uiform estimator ad, moreover, are typically ot kow i practice. However, these coditioal stadard deviatios ca be estimated i a olie fashio over the course of the estimatio algorithm; we discuss such variatios i 7.4. Furthermore, the Sequetial estimator requires additioal computatioal overhead beyod that of the Uiform estimator. However, this is miimal: the oly additioal requiremet is to track scearios i order of error margi. This ca be accomplished efficietly via a priority queue data structure (see, e.g., Corme et al. 2002). With a priority queue, determiig the sceario with miimum error margi (lie 8 i Algorithm 2) ca be accomplished i costat time (i.e., i a amout of time idepedet of m ad ). Oce a ew ier sample is geerated for a sceario (lies 9 ad 10 i Algorithm 2), order log time would be required to update the priority queue data structure. I practice, this is ot sigificat. The Sequetial estimator also requires more memory tha the Uiform estimator. I particular, the Uiform estimator ca be implemeted i a way where scearios are processed oe at a time ad ever eed to be simultaeously stored i memory. Such a implemetatio would have a costat memory requiremet (i.e., idepedet of m ad ). For the Sequetial estimator, each of the outer scearios must be stored i memory over the course of the algorithm, hece the memory requiremet is of order. I practice, eve give a very large umber of scearios (e.g., millios), each of very high dimesio (e.g., thousads), this memory requiremet is well withi the reach of commodity hardware. Each ier sample may require simulatig multiple steps over a log time horizo, but the memory requiremet is miimal because all itermediate computatios are discarded, ad oly the ier sample loss is recorded. 7 The Sequetial estimator has some similarities to ouiform estimators that have bee proposed i the literature. Lee ad Gly (2003) suggest a ouiform ested estimator i the case where the sceario space is discrete. They choose the umber of ier samples m i i each sceario i so as to optimize certai large deviatio asymptotics. Usig a Gaussia approximatio as a heuristic, this results i the allocatio i 2 m i (12) L i c2 Because the loss L i i sceario i is ukow, Lee ad Gly (2003) propose a two-pass algorithm: i the first pass, a small umber of ier samples are geerated i each sceario ad are used to compute ier sample allocatios i a secod productio ru. Our Sequetial estimator differs from (12) i several fudametal ways: First, the allocatio (12) is loosely aalogous to miimizig the square of the error margi, as opposed to the error margi itself. Secod, the allocatio (12) is accomplished with multiple passes, whereas our estimator is fully sequetial. Ideed, i 5, tools from sequetial aalysis will prove fudametal i the theoretical aalysis of our estimators. Fially, ad most importatly, i the settig of Lee ad Gly (2003), ouiform samplig does ot provide a qualitatively differet rate of covergece tha uiform samplig. Give a total computatioal budget of order k, both the uiform ad ouiform methods achieve a asymptotic MSE of order k 1 log k, albeit with differet costats. As we shall see i 5, we will be able to establish theoretically that a ouiform estimator coverges at a faster asymptotic order tha is possible with uiform estimators. Gordy ad Jueja (2008) suggest a geeral class of multipass dyamic allocatio schemes for uuiform ested estimatio. Such schemes would, for example, divide the simulatio ito a sequece of J phases, where i the jth phase ier samples would oly be allocated to scearios i if ˆL i c j. Here, 1 > 2 > > J is a sequece of thresholds. Gordy ad Jueja (2008) provide some umerical evidece that such schemes may provide a sigificat improvemet over uiform estimators, but the choice of specific parameters of the algorithm (e.g., the umber of phases J or the thresholds j ) is left as a directio for future research. 7 The ouiform Threshold estimator that will be discussed i 5.1 does ot require ay additioal computatioal or memory overhead beyod that of the stadard Uiform estimator.

9 1180 Maagemet Sciece 57(6), pp , 2011 INFORMS 5. Aalysis I 4, we itroduced the ouiform Sequetial estimator ad motivated this algorithm via a iformal discussio. I this sectio, we will provide a aalysis of ouiform estimatio. We begi i 5.1 by itroducig a simplified variatio of the Sequetial estimator. This simplified estimator preserves the myopic ad ouiform behavior of the Sequetial estimator, but is more ameable to aalysis. Moreover, the simplified estimator is remiiscet of a compoud sequetial hypothesis test ad highlights coectios to the classical field of sequetial aalysis. I 5.2, we provide a asymptotic aalysis of the bias ad variace of simplified ouiform estimator. Fially, i 5.3, we discuss optimal parameter choices for the simplified ouiform estimator. We demostrate that this estimator has a asymptotic MSE of order k 4/5+, for all positive, as a fuctio of the computatioal budget k. This ca be compared to the asymptotic MSE of order k 2/3 of the optimal uiform estimator A Simplified Nouiform Estimator Aalysis of the Sequetial estimator described i 4 presets a umber of challeges. Foremost amog these is the fact that, over the course of the ested simulatio of the Sequetial estimator, the loss estimates ˆL 1 ˆL are depedet radom variables. This depedece is iduced by the myopic selectio rule (10), which, at each poit i time, simultaeously depeds upo all of the loss estimates. To make the aalysis tractable, we will cosider a modificatio of the Sequetial estimator that results i idepedet loss estimates while maitaiig the spirit of myopic ouiform samplig. I particular, recall that the Sequetial estimator takes as iput a parameter m, specifyig the desired average umber of ier samples i each sceario, ad a parameter, specifyig the desired umber of scearios. Over the course of the algorithm, m total ier stage samples will be geerated. These samples are allocated i a sequetial fashio so as to myopically miimize the error margi, m i / i ˆL i c, uiformly over 1 i. If we imagie the algorithm to be i a state where a sigificat umber of ier samples have bee geerated, i.e., m i 1 for each i, the oe would expect the error margis to be roughly costat; if ot, more ier samples would have bee geerated for the scearios with lower error margis. Oe could achieve a similar effect by fixig a threshold > 0 ad cotiuig to add ier stage samples to each sceario i util the error margi exceeds, i.e., m i i ˆL i c (13) This is precisely what is doe by the Threshold estimator of Algorithm 3. Algorithm 3 (Estimate the probability of a large loss usig a threshold-based ouiform ested simulatio. The parameter is the error margi threshold. The parameter is the umber of scearios.) 1: procedure Threshold( ) 2: for i 1 to do 3: geerate sceario i 4: set i to be the stadard deviatio of the distributio of the losses i sceario i 5: m i 0 6: repeat 7: geerate oe additioal portfolio loss sample Z imi +1 i sceario i 8: m i m i + 1 9: compute a estimate of the loss i sceario i, ˆL i 1/m i m i j=1 Z i j 10: util m i / i ˆL i c 11: ed for 12: compute a estimate of the probability of a large loss, ˆ 1/ ˆL i c 13: retur ˆ 14: ed procedure At a high level, the Sequetial ad Threshold estimators are quite similar. Both seek to ouiformly allocate ier stage samples based o miimizatio of the error margi. However, they are parameterized differetly. The Sequetial estimator takes as a iput the parameter m, which is the mea umber of ier stage samples. O the other had, the Threshold estimator takes as iput the parameter, which is the threshold for the error margi. As argued earlier, for large values of m ad, these two algorithms yield similar results. Furthermore, we will see umerical evidece for this i 7. From a practical perspective, the Sequetial estimator is more atural. I particular, if all other parameters are fixed, it is easy to choose a value for m. This parameter explicitly specifies the total umber of ier stage samples to be geerated by m, ad therefore determies the ruig time of the algorithm. Thus, we ca choose m based o the available ruig time. I the Threshold estimator, the parameter implicitly specifies the total umber of ier stage samples to be geerated, ad hece idirectly determies the ruig time. It is ot clear, however, how to make choice of a priori that esure a certai ruig time, for example. From a theoretical perspective, however, the Threshold estimator proves much more ameable to aalysis. The mai reaso is that, at ay poit durig the executio of the algorithm, the loss estimates ˆL 1 ˆL are idepedet ad idetically distributed radom variables. This i.i.d. structure will

10 Maagemet Sciece 57(6), pp , 2011 INFORMS 1181 Figure 4 A Illustratio of the Threshold Estimator S m (i) 0 m i m Note. Give a sceario i, the estimator geerates ier stage samples util the partial sum S i m crosses barriers at or. If the exit occurs through the upper barrier at, as illustrated, the sceario is declared to be a loss exceedig c. If the exit occurs through the lower barrier at, the sceario is declared ot to be a loss exceedig c. prove crucial i the aalysis of 5.2, because it allows the aalysis of the overall algorithm via the aalysis of a sigle outer stage sceario. Moreover, the Threshold estimator has aother iterestig iterpretatio. Give a threshold, cosider a sceario i with ier loss samples Z i 1 Z i 2 Examiig (13), the algorithm will geerate m i ier stage samples i this sceario, with m i = ifm > 0 S i m (14) where, for m 0, the partial sum is defied by m S i m 1 Z i j c (15) i j=1 Note that S i m m 0 is a radom walk with uit variace icremets. The, the umber of samples m i is determied by the first exit time of the radom walk from the iterval. This is illustrated i Figure 4. If the exit occurs through the upper barrier at, the ˆL i > c, ad the sceario is declared to be a loss exceedig c. If the exist occurs through the lower barrier at, the ˆL i < c, ad the sceario is declared ot to be a loss exceedig c. The iterpretatio of the threshold policy i terms of the first exit of a radom walk is remiiscet of sequetial hypothesis testig (see, e.g., Siegmud 1985). Ideed, for each sceario i, the threshold estimator is defiig a sequetial compoud hypothesis test of whether the i.i.d. uit variace radom variables Z i j c/ i have a positive or egative mea. As we show ext, techiques from sequetial aalysis will prove helpful i theoretical aalysis of our algorithm Asymptotic Aalysis Defie to be the Threshold estimate, i.e., Threshold. As i 3, we will aalyze the accuracy of this estimator by decomposig the mea squared error ito bias ad variace terms. We begi with a assumptio: Assumptio 2. Assume the followig: 1. Coditioal o a outer stage sceario i, the ier stage samples Z i 1 Z i 2 are i.i.d. ormal radom variables. Deote the stadard deviatio of these samples by i. 2. Give a sceario, defie the ormalized excess loss L c/. The, the probability desity fuctio p of, pu d P u du exists ad is cotiuously differetiable i a eighborhood of 0. The secod coditio of Assumptio 2 is a techical coditio that is remiiscet of the first coditio of Assumptio 1. The first coditio is motivated by the radom walk iterpretatio of 5.1. I particular, cosider the radom walk formed by the partial sums Sm i m 0 from (15). By the fuctioal cetral limit theorem, uder a proper scalig, this process coverges to a Browia motio, i.e., a radom walk with ormal icremets. The first coditio makes the assumptio that the uscaled radom walk also has ormal icremets. We are iterested i the accuracy of the Threshold estimator i the asymptotic regime where the resultig estimate coverges to the true value, i.e., as (may outer stage scearios) ad (may ier stage samples). Our first result is the followig theorem, which characterizes the asymptotic bias of this estimator. Theorem 2. Uder Assumptio 2, as, the asymptotic bias of the Threshold estimator satisfies E = O 2.

11 1182 Maagemet Sciece 57(6), pp , 2011 INFORMS The proof of Theorem 2 is provided i the appedix. It relies o the radom walk iterpretatio of 5.1 as well as techiques from sequetial aalysis. Specifically, expoetial martigales are used i combiatio with the optioal stoppig theorem. The followig is a immediate corollary of Theorem 2 ad provides a asymptotic expressio for the variace of the simplified sequetial estimator. Corollary 2. Uder the coditios of Theorem 2, as, the variace of the Threshold estimator satisfies Var = Proof. Note that Var = Var 1 ( 1 ˆL i c + O 2 1 = E 1 E ) = 1 Var ˆL 1 c where we have used the fact that the loss estimates ˆL i are idepedet ad idetically distributed. Applyig Theorem 2, Var = 1 + E 1 E + E 1 = + O 2 1 The total ru time of the Threshold estimator is proportioal to the total umber of ier stage samples geerated. Note, however, by the ature of the algorithm, the umber of ier samples is stochastic. Hece, defie m to be the expected umber of ier stage samples at a sigle outer stage sceario, give parameter ; that is, [ { m E if m > 0 }] m ˆL c (16) Here, the expectatio is over the sceario ad the correspodig loss estimate ˆL. The, give parameters, the Threshold estimator has expected ru time proportioal to m. The followig theorems, whose proof is give i the appedix, characterizes the rate of growth of this ru time as a fuctio of. Theorem 3. Uder Assumptio 2, as, the expected umber of ier stages samples i each sceario uder the Threshold estimator satisfies m = O log Note that Theorem 3 is ituitive give the first exit time iterpretatio of Figure 4. I particular, for large values of, the amout of time required for a radom walk startig at the origi with drift 0 to exit the iterval is approximately /. If the radom walk has zero drift, the exit time is approximately 2. I our case, the expected umber of samples m is averaged over various possibilities of drift give by L c/. The probability of this drift beig exactly zero is zero, by the secod coditio of Assumptio 2. However, arbitrarily small drifts are possible, ad thus m is slightly larger tha O. Although Theorem 3 provides a O log boud o the expected umber of ier stage samples per sceario, it might be the case that the realized umber of ier stage samples per sceario is larger. The followig theorem guaratees that, so log as the umber of scearios is sufficietly large, a O log boud cotiues to hold o the umber of realized samples per sceario with high probability. The proof ca be foud i the appedix. Theorem 4. Uder Assumptio 2, suppose that C 0 0 > 0 are costats so that, for all 0, m C 0 log. (Such costats are guarateed to exist by Theorem 3.) Furthermore, suppose the umber of scearios is chose as a fuctio of ad that there exist costats C 1 1 > 0, so that, for all 1, C 1 ; that is, asymptotically grows at least liearly i. The, for ay > 0, there exists 2 > 0 so that, for all 2, ( ) 1 P m i C 0 + log < 5.3. Optimal Nouiform Threshold Estimator Theorems 2 ad 3 ad Corollary 2 allow a compariso betwee the Uiform estimator ad the ouiform Threshold estimator. I particular, suppose ˆ m is the Uiform estimate with scearios ad m ier stage samples. As discussed i 3, whe m, this has asymptotic bias ad variace E ˆ m = c m + Om 3/2 (17) 1 Var ˆ m = + Om 1 1 O the other had, suppose that is the ouiform Threshold estimator with scearios ad a threshold of. By Theorem 3, this estimator will employ, o average, m m = O 1+ ier stage samples per sceario for ay positive. We ca express the asymptotic bias ad variace results of Theorem 2 ad Corollary 2 as a fuctio of ad m by Var = for all positive. E = O m O m 2+ 1 (18)

12 Maagemet Sciece 57(6), pp , 2011 INFORMS 1183 Comparig (17) ad (18), we see that, up to the domiat term, the two algorithms achieve the same asymptotic variace of order 1. This is cosistet with the discussio i 3, which suggests that the asymptotic variace is determied by the radomess i sceario geeratio. This is exactly the same i the two algorithms. The ier stage samplig is differet, however, ad this results i a differece i bias for the estimators. Specifically, as a fuctio of the average umber of ier stage samples per sceario, the bias of the ouiform Threshold estimator decays approximately as the square of the bias of the Uiform estimator. Give a total work budget of k (i.e., m k), we saw i 3 that the optimal Uiform estimator (i the sese of miimum MSE) would utilize a umber of scearios of order k 1/3, a umber of ier stage samples per sceario m of order k 2/3, ad result i a MSE of order k 2/3. For the ouiform Threshold estimator, from the results of 5.2, we ca boud the MSE by E C 4 for sufficietly large ad ad a appropriate choice of the costat C. We ca fid a ouiform Threshold estimator with low MSE by miimizig this upper boud over choices of, subject to a expected total work costrait; that is, we cosider optimizatio problem miimize subject to 1 m k 0 + C 4 (19) For ay positive ad give a work budget k, suppose we choose k 1/5 ad k 4/5. The, we have that m = Ok 1 log k = ok. Thus, for sufficietly large k, the expected total work will be less tha k. Ideed, because satisfy the coditios of Theorem 4, for sufficietly large k the realized total work will also be less tha k with high probability. This choice will result i a MSE of Ok 4/5+. Hece, the optimal ouiform Threshold estimator coverges at a faster rate tha ay uiform estimator. This is accomplished by geeratig more outer scearios (k 4/5 versus k 2/3 ) ad geeratig fewer ier stage samples o average i each sceario (k 1/5 versus k 1/3 ) tha is optimal i the uiform case. 6. Adaptive Allocatio Algorithm The ouiform Sequetial estimator provides a way to determie the placemet of ier stage samples across scearios. The decisio of how to allocate computatioal effort betwee geeratig more scearios (i.e., the choice of ) ad geeratig more ier samples across scearios (i.e., the choice of m) is uaddressed, however. The discussio i 5.3 suggests that, give a total work budget of k, oe should asymptotically approximately choose k 4/5 ad m k 1/5. However, the costats i these asymptotic expressios are uspecified. The choice of these costats may have a eormous impact o the practical performace of these algorithms. Note that the Uiform estimator faces the same problem ideed, the optimal allocatio (6) suggested by the aalysis of 3 requires kowledge of the costat c. It is ot clear, i geeral, how to determie this costat. I this sectio we will cosider a adaptive allocatio approach. This algorithm is a heuristic that estimates the optimal choice of m ad at each poit i time. It refies these estimates over the course of the simulatio. The mai idea of this approach is that, based o the results of 5, the variace is determied by the umber of scearios (), ad the bias squared is determied by the amout of ier samplig ( m). The adaptive algorithm estimates these quatities ad the either icreases the umber of scearios or icreases the umber of ier samples depedig o whether the MSE is domiated by the variace or the biased squared. Specifically, the Adaptive estimator of Algorithm 4 proceeds as follows: 1. The simulatio is iitialized (lies 2 7) by geeratig 0 scearios with m 0 ier samples for each sceario. 2. The work budget of the simulatio k is divided ito K k/ e itervals (or epochs) of legth e (ote that we assume for simplicity of expositio that K is itegral ad that the first epoch is oly of legth e 0 m 0 because of the iitializatio). 3. At the begiig of the lth epoch (lie 9), estimates are made for the bias squared ad variace of the loss probability estimate, give the scearios ad samples that have bee geerated thus far. Specifically, give the loss probability estimate ˆ = 1 ˆL i c the bias is approximated accordig to where E ˆ B ˆ (20) 1 ( ) mi ˆL i c This approximatio is based o a cetral limit theorem heuristic: i each sceario i, whe the umber of samples m i is large, each loss estimate ˆL i ca be approximated by a ormal distributio with mea i

13 1184 Maagemet Sciece 57(6), pp , 2011 INFORMS equal to L i ad with variace 2 i /m i. Hece, give a fixed set of scearios 1, oe might estimate the bias via E ˆ = 1 1 P ˆL i c Li c { ( mi L i c i ) } Li c Because each true loss L i is ukow i practice, we ca approximate this with its realized estimate ˆL i. This results i (20). By makig a similar heuristic approximatio for the variace, we arrive at the expressio Var ˆ V 1 (21) Note that the estimators (20) ad (21) are meat oly as heuristics. Better estimators may be possible, ad bias i particular is otoriously difficult to estimate. For our purposes, however, they oly eed to be accurate withi orders of magitude so as to allocate computatioal effort betwee ier samples ad outer scearios. We will see i the umerical results of 7 that, empirically, they suffice for this purpose. 4. Suppose there are outer scearios ad a average of m 1/ m i ier samples per sceario at the begiig of the lth epoch. From the results i 5, we expect the bias squared to decrease accordig to m 4+ ad the variace to decrease i proportio to 1. The, assume that the umber of scearios ad samples at the ed of the lth epoch is give by ad m. We ca estimate the bias squared at the ed of the lth epoch, as a fuctio of the bias estimate B at the begiig, by B 2 m/ m 4. Similarly, the variace at the ed of the lth epoch ca be estimated by V /. Thus, at the begiig of the lth epoch, we cosider the followig optimizatio problem: ( ) m 4 ( ) miimize B 2 + V m m subject to m = m + e + e m 0 (22) This problem seeks to make a choice of m that results i a miimal mea squared error at the ed of the lth epoch. The first costrait esures that the total umber of ier samples i the lth epoch will equal the epoch legth e. The secod costrait esures that the umber of scearios at the ed of the lth epoch is at least the umber of scearios at the begiig, ad icreases by at most the legth of the epoch. The solutio to (22) is give by { {( ) V 1/5 } = mi max 4 B 2 m m + e 4 } + e 4 m = m + e (23) After obtaiig the target umber of scearios (lie 10), additioal scearios are geerated. 5. Over the course of the lth epoch (lies 13 21), e ier samples are geerated. These are distributed to esure that every sceario has at least m 0 ier samples i total (ot per epoch). Oce that is the case, ier samples are allocated myopically accordig to miimum error margi as i the Sequetial estimator. Algorithm 4 (Estimate the probability of a large loss usig a adaptive ouiform ested simulatio. This estimator employs a sequetial algorithm to determie the placemet of ier stage samples across scearios ad adaptively decides the umber of scearios ad ier samples to add by estimatig the bias ad variace. The parameters 0 ad m 0 are the iitial umber of scearios ad ier samples per sceario, respectively. The parameter e is the epoch legth. The parameter k is the total umber of ier samples. Note that each stadard deviatio i ca be estimated i a olie fashio over the course of the simulatio, as is discussed i 7.4.) 1: procedure Adaptive(m 0 0 e k) 2: geerate scearios : 0 4: for i 1 to 0 do 5: coditioed o sceario i, geerate i.i.d. samples Z i 1 Z im 0 of portfolio losses 6: m i m 0 7: ed for 8: for l 1 to k/ e do 9: estimate the curret bias ad variace by B ad V from (20) (21) 10: determie a target umber of scearios by { mi max {( V 4 B 2 m 4 m+ e 4 11: geerate scearios +1, set m i 0 for i = : 13: while m i < l e do 14: if mi i m i < m 0 the 15: set i arg mi i m i 16: else 17: set i arg mi i m i ˆL i c/ i 18: ed if ) 1/5 } + e }