科技部補助專題研究計畫成果報告
期末報告
複雜衍生性商品信用價值調整 (CVA) 高效演算法研究
計 畫 類 別 ： 個別型計畫
計 畫 編 號 ： MOST
104-2410-H-004-036-執 行 期 間 ： 104年08月01日至105年12月31日
執 行 單 位 ： 國立政治大學風險管理與保險學系
計 畫 主 持 人 ： 謝明華
計畫參與人員： 碩士班研究生-兼任助理人員：劉羿圻
碩士班研究生-兼任助理人員：陳思雅
報 告 附 件 ： 出席國際學術會議心得報告
中 華 民 國 106 年 03 月 31 日
中 文 摘 要 ： 信用價值調整（CVA）自2008年的金融危機以來已經成為金融業的銀
行監管和會計準則的通用標準。因此，從業者和學者都非常關注
CVA的相關問題。巴塞爾協議III要求銀行針對交易對手信用風險
（CCR）計提資本要求。 CCR包括交易對手違約和CVA價值的變化。
CVA衡量CCR的評價部分，即 CVA根據交易對手信用狀態調整相互間
的衍生性商品契約的價值。 CVA的計算是一個複雜的任務，因為它
涉及複雜的衍生工具的違約選擇權的評價。對複雜衍生性商品契約
而言，情況更加困難。因為它們的定價, 蒙特卡羅模擬是唯一可行
的計算工具。因此，設計高效的演算法, 對於這些複雜的衍生性商
品契約的CVA計算就變得非常重要。針對此一議題，本計畫將一個重
要的複雜的衍生性商品契約：一籃子信用違約交換合約，這是流行
的對沖信貸組合
中 文 關 鍵 詞 ： 信用價值調整, 亞式選擇權, 一籃子信用違約交換合約, 蒙地卡羅
法,變異數縮減技術
英 文 摘 要 ： Credit Valuation Adjustments (CVA) has become a common
standard in bank regulation and accounting rules since the
2008 crisis in the financial industry. Therefore,
practitioners and academics draw a lot of attentions to CVA
related problems. Basel III requires banks to set a part of
their capital requirement for counterparty credit risk
(CCR). CCR includes counterparty default and CVA changes.
CVA measures the valuation component of this risk, i.e. the
adjustment to the price of a product due to this risk. The
computation of CVA is a complex task, because it involves
valuation of the default option on complex derivatives. The
situation is even worse for complex derivatives, because
Monte Carlo simulation is the only viable computation tool
for their pricing. Therefore, designing effective
algorithms for the CVA computation of these complex
derivatives becomes very important. We develop fast
valuation algorithm for basket credit default swaps, which
are popular for hedging credit portfolio.
英 文 關 鍵 詞 ： Credit value adjustment, Asian options, Basket credit
default swap, Monte Carlo, Variance reduction
Valuations of BDS with Counterparty Risk
March 31, 2017
Paper Outline I. Introduction
II. Valuation of kth-to-Default BDS with Counterparty Risk II-1: Characterization of Default Time Correlations
II-2: Valuation Problem Setting III. Proposed Algorithm
III-1: Model Implementation: naive MC simulation
III-2: Model Implementation: Simulation with Importance Sampling IV. Numerical Results
V. Conclusion
1
Introduction
The recent credit crisis has highlighted the importance of counterparty risk in connection with valuation and risk management of credit derivatives. Counterparty risk in general is the risk that the party to a financial contract may fail to make all the payments required by the con-tract, causing losses to the other party. Contracts privately negotiated between counterparties like over-the-counter (OTC) derivatives are most likely subject to counterparty risk. Modelling counterparty credit exposure for credit derivatives is more complicated than for other noncredit products, since the reference credit and counterparty may display some sort of default corre-lation. In the credit default swap (CDS) market, the increased correlation between reference entities and protection sellers of CDS has diminished the effectiveness of the clean transfer of risk. (to add some thing to emphasize the importance of CP)
A few studies have been made to analyze the valuation of counterparty risk within a CDS. For example, Jarrow and Yu (2001) [?] propose an intensity-based model to examine the impact of a default on a surviving firm. Hull and White (2001) [?] address the counterparty risk problem for CDS by resorting to default barrier correlated models. In contrast, little attention has been given to analyze the counterparty risk embedded in a basket default swap (BDS). Therefore, we take into account of counterparty risk for the valuation of a BDS in this paper, aiming to fill the gap in the literature on the analysis of counterparty risk within credit derivatives.
A BDS is like an insurance contract that offers protection against the event of the kth default on a basket of n (n ≥ k) underlying names. It works in a similar manner to a single-name CDS, with a crucial difference that the credit event to insure against is the event of the kth default.
Depending on the ranking of default protections, a basket credit default swap can be known as a 1st-to-default basket, a 2nd-to-default basket, or more generally, a kth-to-default basket. The valuation of basket credit default swaps requires a full specification for the joint distribution of default times. Li (2000) assumes that the dependence structure between default times of underlying obligors is captured by a Gaussian copula. The Li (2000) model, or commonly known as the Gaussian Copula approach, has become an industry standard for valuing basket default swaps for its ease of implementation via Monte Carlo simulations.
Though being conceptually simple and easy to implement, Monte Carlo simulations are seen to be unstable and slow in convergence when dealing with default events. The problem will be getting worse for pricing basket default swaps, where a default payoff is trigger only when a k-th default has taken place before the maturity date in a simulated path. Chiang et al. (2006) propose an effective importance sampling algorithm for the valuation of k-th to default basket default swaps.
(* What we did)
We introduce default dependence among obligors through the specification of a joint distri-bution for the default times using a copula. Under a factor form representation proposed by Laurent and Gregory (2005), we can explicitly establish the correlation between default of the protection seller and default of the BDS reference obligors.
We follow the literature and define the credit value adjustment (CVA) as the devaluation of a contract due to counterparty default. Therefore, CVA is the market value of counterparty credit risk. In this paper, we study the credit value adjustment (CVA) of BDS, and examine the impacts of the default correlations on CVAs.
(*) Organization of the Paper This paper is organized as follows. (*) Literature Review
The dependence between defaults caused by common factors has received a lot of attention in the credit risk literature, as it can and has been modelled in the standard reduced form credit risk models such as Lando (1998) [?] or Duffie and Singleton (1999) [?]; for empirical work on the specification of an appropriate factor structure see for instance Duffee (1999) [?] or Driessen (2005) [?]. In contrast, researchers became only recently interested in counterparty risk. This interest stems from at least two reasons: first, there is substantial empirical evidence for counterparty risk; for instance Lang and Stulz (1992) [?] have shown that bankruptcy filings do impact stock returns (and most likely also default probabilities) of non-defaulted companies. Moreover, as has been pointed out by Hull and White (2001) [?], the correlation between defaults obtainable in reduced form models are often quite low, so that these models may not be able to mimic the clustering of defaults around economic recessions observed in real data (see for instance Keenan (2000) [?]). Obviously, this calls for an incorporation of other sources of default dependence such as counterparty risk into the model.
The value of a CDS depends on the risk of default of both the reference entity and the counterparty, as well as on the dependence of these two risks on one another. Ignoring correlation among underlying and counterparty can be dangerous. This credit underlying case involves default correlation, that is perceived in the market to have impact in counterparty risk credit valuation adjustments.
Jarrow and Yu (2001) [?] are the first to propose an intensity-based model, which allows
for counterparty risk. In their framework the impact of defaults on the default intensities of surviving firms is explicitly modelled, which is a very intuitive parametrization of counterparty risk; see also Davis and Lo (2001) [?] for a related approach. The construction of default processes in Jarrow and Yu (2001) works only for a very special type of interaction between defaults, the so-called primary secondary framework, which excludes many interesting examples of cyclical default dependency. This and other mathematical aspects of the Jarrow-Yu model are discussed in Kusuoka (1999) , Bielecki and Rutkowski (2002), and Collin-Dufresne, Goldstein, and Hugonnier (2002) [?]. Yu (2002) has carried out an interesting simulation study. He analyzes the default correlations which can be obtained for different parametrizations of the standard reduced form models and of the Jarrow-Yu model.
(Reduced-form) Jarrow and Yildirim (2002) [?] proposed an intensity-based valuation model of CDS with correlated market and credit risk.
Hull and White (2001) [?]address the counterparty risk problem for CDS by resorting to default barrier correlated models, without considering explicitly credit spread volatility in the reference CDS.
Brigo and Chourdakis (2008) [?] consider counterparty risk for CDS in presence of correlation between default of the counterparty and default of the CDS reference credit. Besides default correlation, they also model credit spread volatility. Stochastic intensity models are adopted for the default events, and defaults are connected through a copula function. We find that both default correlation and credit spread volatility have a relevant impact on the positive counterparty-risk credit valuation adjustment to be subtracted from the counterparty-risk free price.
Counterparty risk is also present in the popular copula model (see for instance Li (2001) or Schonbucher and Schubert (2001)). Schonbucher and Schubert (2001) specify a model within it the default intensity of the surviving firms jumps at the default time of one obligor in the portfolio in the copula framework However, direction and size of this jump depend on higher order derivatives of the copula, which makes the copula parametrization of counterparty risk quite unintuitive.
Leung and Kwok (2005) [?], building on Collin-Dufresne et al. (2004) [?], model default intensities as deterministic constants with default indicators of other names as feeds. The exponential triggers of the default times are taken to be independent and default correlation results from the cross feeds.
(Structural-form) Kim and Kim (2003) [?] suggests a methodology for valuing CDS that takes account of counterparty default risk as well as correlated market and credit risk. It incorporates market risk into determining default correlation between multiple firms using the fist-passage time approach.
2
Valuation of kth-to-Default BDS with Counterparty Risk
2.1 Characterization of Default Time Correlations
(*) how to model the dependence structure between protection seller and underlying obligors
Up to now the industry standard for the joint default probability of many underlying obligors has relied on a model of joint default-times. The copula approach for specifying the dependence structure among default-times was developed by Li (2000), and later extended to many obligors by Laurent and Gregory (2005) where the correlation structure is represented in a factor form, common known as the factor-copula approach. The factor-copula approach conceptually coin-cides with the conditional independence assumption among default events, i.e. conditional on the common factor, default events are independent.
In the following we give a brief account for the joint default-time model that our approach is based upon. Let τi denote the default time for an underlying obligor i, where i = 1, · · · , n.
τi is a positive random variable and its distribution is characterized in terms of a hazard rate
function hi(·):
prob (τi > t) = e−
Rt
0hi(u)du,
and let Fi(t) denote the cumulative default probability before time t for an obligor i (the
marginal distribution of the time-until-default for obligor i), Fi(t) = prob (τi ≤ t) = 1 − e−
Rt
0hi(u)du
Let Si(t) be the survival function of obligor i, Si(t) can then be expressed as Si(t) = 1 − Fi(t) .
The marginal distribution of of the default time for each obligor i is typically extracted from the quoted market prices of CDSs; these market prices are used to construct a hazard rate func-tion hi(·) from which we get the distribution Fi(t). However, the cash-flows of portfolio credit
derivatives are functions of a whole sequence of random default times (τ1, · · · , τn) . Therefore,
in order to evaluate multi-name credit derivatives, the modelling challenge is to characterize the dependence structure for the default times, τi.
By sampling a set of correlated uniform variates (U1, U2, · · · , Un), one can then specify a
copula function C (u1, u2, · · · , un) , which defines the dependence structure among default times
to link univariate marginals into their full multivariate joint distribution, i.e.: C (u1, u2, · · · un, ρ) = prob (U1 ≤ u1, U2 ≤ u2, · · · , Un≤ un)
The copula-based approach hence involves conducting the following steps: First of all, one generates correlated random numbers Xi, where i = 1, · · · , n; Secondly, uniformly distributed
random variates, Ui = Φ (Xi), are obtained from the cumulative normal distribution function,
Φ (·); The final step involves the computation of default times for each individual obligor via an inverse mapping of their marginal distributions, τi= Fi−1(Φ (Xi)).
In this article, we extend a single-factor form representation of Laurent and Gregory (2005) for the Gaussian copula, and specify a two-factor formalism for an underlying entity i, i = 1, . . . , n, as
Xi = φiM + ρiDu+
q
1 − φ2i − ρ2
iZi (1)
where M is the common factor upon which default events of all underlying obligors are de-pendent; Du represents the industry-specific risk factors, with u ∈ {1, · · · , m}, m is the total
number of industries; and Zi represents the firm-specific risk factors. All M , Du and Zi are
in-dependent standard normal variables. The parameter φi captures how strongly Xi is correlated
to the evolution of the common factor M , and the parameter ρi determines how strongly Xi
is correlated to the evolution of the industry-specific factor Du. The correlation between two
entities i and j is Corr (Xi, Xj) = φiφj + ρiρj. The default time for each underlying entity i is
then computed as τi = Fi−1(Φ (Xi)).
Based on the two-factor Gaussian copula model of equation (??), we can also write down the factor-form representation for the protection seller as
Xs= φsM + ρsDs+
p 1 − φ2
s− ρ2sZs
where Ds represents the firm-specific risk factors for a protection seller, and the the default
time of the protection seller is computed via an inverse mapping of its marginal distributions, τs= Fs−1(Φ (Xs)).
2.2 Valuation Problem Setting
In this section, we briefly review the valuation procedure of a generic kth-to-default credit default swap relative to a portfolio of n (n ≥ k) reference risky obligors. Throughout the section we shall adapt the following notations : tj denotes the time for the jth premium payment to take
place; δj−1,j is the time increment between premium payments at the (j − 1)th and the jthtime
points in units of years; B (0, ti) = e−rti is the discount factor for one dollar received at time
ti, r is the constant short rate; Ri denotes the recovery rate for the ith obligor when default
happens, and we assume that Ri is equal to a constant R for all i; and Rs is the recovery rate
for protection seller, Ai is the notional amount for credit i, and we assume Ai to be equal to
a constant amount A for all i. We denote τu as the k-th default time among the underlying obligors; and τs as the default time of protection seller. The time-to-maturity of the basket default swap is set to be T , where T = tN; Fi(tj) is the probability that an underlying credit i
defaults before or at time tj, hence by definition, Fi(tj) = Pr (τi≤ tj). S (t) denotes the survival
function of the k-th default time, S (t) = Pr (τ > t); the distribution function of τ is therefore given by: F (t) = Pr (τ ≤ t) = 1 − S (t); Finally, Q denotes the risk-neutral probability measure; SCR denotes the credit spread of a BDS contract that is subject to counterparty risk, and I{·}
is the indicator function.
τiu : default time of underlying obligor i
τu : the k-th default time of underlying obligors, i.e. τu is the k-th order statistics of (τu
1, . . . , τnu)
τs : default time of protection seller, i.e. counterparty
ΠC(t, T ) : the sum of all payoff terms between time t and T subject to counterparty risk. Π (t, T ) : the sum of all payoff terms between time t and T without the consideration of counterparty risk.
We define PVpremiumCR to be the present value of the premium leg of the k-th to default basket swap that is subject to counterparty risk, then
P VCRpremium= E " N X i=1 B (0, ti) × SCR× I{min(τs,τu)>t i}× A # (2)
where the expectation is taken under the risk-neutral pricing measure. If accrued premiums are
considered, then the present value of accrued premium PVAccruedPremiumCR can be computed as P VCRAccruedP remium= E " N X i=1 B (0, ti) × SCR× min (τs, τu) − t i−1 ti− ti−1 × I{min(τs,τu)>t i}× A #
On the other hand, we define PVdefaultCR to be the present value of the default leg of the k-th to default basket swap subject to counterparty risk, and
P VCRdef ault = (1 − R) EB (0, τu+ 4) × I{τu≤T }× I{τs>(τu+4)}× A
+RsEtB (0, τu+ 4) × (1 − R) × I{t<τs≤τu≤T }× A
(3) where 4 is the length of the settlement period, (τu+ 4) represents the settlement date.
We can therefore derive the fair spread of the k-th to default basket swap that is subject to counterparty risk as follows
SCR = (1 − R) EtB (0, τu+ 4) × I{τu≤T }× I{τs>(τu+4)} E h PN i=1B (0, ti) × I{min(τs,τu)>t i} i +R sE tB (0, τu+ 4) × (1 − R) × I{t<τs≤τu≤T } EhPN i=1B (0, ti) × I{min(τs,τu)>t i} i (4)
When τs ≤ τu, the protection seller defaults prior to the default of reference entity. If we
assume that Rs= 0, protection seller would not pay anything to protection buyer.
When τs < T , the protection seller defaults before contract maturity T and cannot fulfill his obligations. At default time τs, we calculate the net present value (NPV) of the residual payoff until maturity and denote it as follows
N P V (τs, T ) = Eτs[Π (τs, T )]
For a protection buyer, if N P V (τs, T ) > 0, only a recovery fraction of the NPV is received by the protection buyer due to the default of the protection seller. Therefore, from the veiwpoint of a protection buyer, the value of the expected payoff of a BDS contract subject to counterparty risk is
EtΠC(t, T ) = Et[Π (t, T )] − (1 − Rs) EtI{t<τs≤T }D (t, τs) (N P V (τs, T ))+
(5) It is clear that the value of a counterparty defaultable claim is the value of the correspond-ing default-free claim minus an option part; a call option (with zero strike) on the residual NPV giving nonzero contribution only in scenarios where τs ≤ T . This adjustment, including the recovery factor Rs, is called counterparty-risk credit valuation adjustment. Equation (??) demonstrates that counterparty risk adds an optionality level to the original payoff.
From the above analysis, we can write
EtΠC(t, T ) = Et[Π (t, T )] − CV A
where CV A is the credit value adjustment for a protection buyer that is vulnerable to the default risk of a protection seller. For further analysis, we can write CV A as
CV A = P Vdef ault− P Vpremium−P VCRdef ault− P VCRpremium
= P Vdef ault− P VCRdef ault+P VCRpremium− P Vpremium
= (1 − R) EB(0, τu)I {τu≤T } − (1 − R) E B (0, τu+ 4) I{τu≤T }I{τs>(τu+4)} +E " N X i=1 δi−1,iB (0, ti) SCRI{min(τs,τu)>t i} # − E "N X i=1 δi−1,iB(0, ti)SI{τu>t i} # = (1 − R)E B(0, τu) × I {τu≤T } − E B (0, τu+ 4) × I{τu≤T }× I{τs>(τu+4)} +δE " N X i=1 B(0, ti)SCR× I{min(τs,τu)>t i}− S × I{τu>ti} # (6)
where S is the credit spread of a BDS without taking into account of counterparty risk,
P Vdef ault( P Vpremium) is the present value of the default (premium) leg of the k-th to default
basket swap without taking into consideration of counterparty risk. We assume the notional amount A = 1, the recovery rate Rs= 0, and the tenor period is constant, i.e. δi−1,i= δ.
3
Proposed Algorithm
3.1 Model Implementation: naive MC simulation
If we assume the recovery rate of protection seller is zero (Rs = 0) and a unit of notional amount (A = 1), the present value of default leg in equation (??) can be simplified as
P VCRdef ault = (1 − R) EB (0, τu+ 4) × I{τu≤T }× I{τs>(τu+4)}
(7) The crude Monte Carlo procedure for computing the value of the default leg (hereby abbre-viated as DL) can be described as follows:
1. Generate independent samples of standard normal variates M, Du, Ds, Z1, . . . , Zn and
Zs
2. Generate correlated normal variates by setting Xi = φiM + ρiDu+ q 1 − φ2i − ρ2 iZi, 1 ≤ i ≤ n and Xs= φsM + ρsDs+ p 1 − φ2 s− ρ2sZs 3. Set Ui= Φ(Xi), 1 ≤ i ≤ n; and Us= Φ(Xs) 4. Set τiu= Fi−1(Ui), 1 ≤ i ≤ n; and τs = Fs−1(Us)
5. Set τu= the k-th order statistic of (τ1u, . . . , τnu)
6. Set the discounted payoff = (1 − R)I{τu≤T,τs>(τu+4)}B(0, τu)
Repeat step 1 to 6 q times; the confidence interval of the DL can then be constructed by the q copies of the discounted payoff.
The above procedure provides us with a point estimate for the DL, denoted by α, as
α = 1 q
q
X
i=1
(1 − R)I(τu(i)≤ T, τs(i)>τu(i)+ 4)B0, τu(i), (8) where τu(i) and τs(i)are the i-th independent samples of τu and τs respectively.
3.2 Proposed IS Algorithm
Case 1: Du and Ds are different
1. Generate independent samples of standard normal variates Du and Z1, . . . , Zn
2. Set hi = Φ−1(Fi(T )) − ρiDu− q 1 − φ2i − ρ2 iZi φi , 1 ≤ i ≤ n
3. Let h = (n − k + 1)-th order statistic of (h1, . . . , hn), and L1 = Φ (h)
4. Generate common factor M according to the formula Φ−1(L1U1), where U1is a uniform(0, 1)
random variate 5. Set Xi = φiM + ρiDu+ q 1 − φ2i − ρ2 iZi; 1 ≤ i ≤ n 6. Set τiu= Fi−1(Φ(Xi)), 1 ≤ i ≤ n
7. Set τu= the k-th order statistic of (τ1u, . . . , τnu) 8. Set hs= Φ−1(Fs(τu+ ∆)) − φsM p1 − φ2 s 9. Set L2= Φ (−hs)
10. Set the discounted payoff = (1 − R)B(0, τ )L1L2
11. Repeat step 1 to 10 q times; confidence interval of DL can then be constructed by the q copies of the discounted payoff
The above algorithm guarantees the event {τu ≤ T, τs> (τu+ 4)} happen and the
likeli-hood ratio L = L1L2 < 1.
Case 2: Du and Ds are the same
1. Generate independent samples of standard normal variates Du = Ds and Z1, . . . , Zn
2. Set hi = Φ−1(Fi(T )) − ρiDu− q 1 − φ2 i − ρ2iZi φi , 1 ≤ i ≤ n
3. Let h = (n − k + 1)-th order statistic of (h1, . . . , hn), and L1 = Φ (h)
4. Generate common factor M according to the formula Φ−1(L1U1), where U1is a uniform(0, 1) random variate 5. Set Xi = φiM + ρiDu+ q 1 − φ2 i − ρ2iZi; 1 ≤ i ≤ n 6. Set τiu= Fi−1(Φ(Xi)), 1 ≤ i ≤ n
7. Set τu= the k-th order statistic of (τ1u, . . . , τnu) 8. Set hs= Φ−1(Fs(τu+ ∆)) − φsM − ρsDs p1 − φ2 s− ρ2s 9. Set L2= Φ (−hs)
10. Set the discounted payoff = (1 − R)B(0, τ )L1L2
11. Repeat step 1 to 10 q times; confidence interval of DL can then be constructed by the q copies of the discounted payoff
The above algorithm guarantees the event {τu ≤ T, τs> (τu+ 4)} happen and the
likeli-hood ratio L = L1L2 < 1.
3.3 Algorithm Analysis
Let us consider the simple event {τi≤ T } first.
{τi≤ T } ≡ {Fi(τi) ≤ Fi(T )} ≡ {Xi ≤ Φ−1(Fi(T ))} ≡ {φiM + ρiDu+ q 1 − φ2 i − ρ2iZi ≤ Φ−1(Fi(T ))} ≡ {M ≤ Φ −1(F i(T )) − ρiDu+ q 1 − φ2i − ρ2 iZi φi } Set hi = (Φ−1(Fi(T )) − ρiDu− q 1 − φ2i − ρ2 iZi)/φi, then we have {τi ≤ T } ≡ {M ≤ hi}.
Next, we consider the event {τu ≤ T }. Notice that I(τu≤ T ) = 1 ⇔ n X i=1 I(τi ≤ T ) ≥ k ⇔ n X i=1 I(M ≤ hi) ≥ k
Set h = (n − k + 1)-th order statistic of (h1, . . . , hn). It is straightforward to show that n
X
i=1
I(M ≤ hi) ≥ k ⇔ I(M ≤ h) = 1
Now, we can consider the expectation of interest EB (0, τu) I {τu≤T }I{τs>(τu+4)}. Note that EB (0, τu) I {τu≤T }I{τs>(τu+4)} = E{EB (0, τu) I {τu≤T }I{τs>(τu+4)} M, Du, Z1, . . . , Zn]} = E{B (0, τu) I{τu≤T }EI{τs>(τu+4)} M, Du, Z1, . . . , Zn]} = E[B (0, τu) I{τu≤T }Φ(−hs)] = E[B (0, τu) I{τu≤T }L2] Moreover, E[B (0, τu) I{τu≤T }L2] = E[E[B (0, τu) I{τu≤T }L2|Du, Z1, . . . , Zn]] = E[E[B (0, τu) I{M ≤h}L2|Du, Z1, . . . , Zn]]
Given Du, Z1, . . . , Zn, both B(0, τu) and L2 are functions of M . If we sample M as a truncated
normal random variable with truncated region (h, ∞), it is easy to see that ˜
E[B(0, τu)Φ(h)L2|Du, Z1, . . . , Zn] = ˜E[B(0, τu)L1L2|Du, Z1, . . . , Zn]
has the same expectation of
E[B (0, τu) I{M ≤h}L2|Du, Z1, . . . , Zn]
Here, we use ˜E to denote the measure of truncated normal M . From above analysis, it guarantees that the resulted estimator has smaller variance than the estimator based on crude Monte Carlo.
