一籃子信用違約交換之評價---考量交易對手違約風險

行政院國家科學委員會專題研究計畫 成果報告

一籃子信用違約交換之評價-考量交易對手違約風險

研究成果報告(精簡版)

計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 99-2410-H-004-078- 執 行 期 間 ： 99 年 08 月 01 日至 101 年 07 月 31 日 執 行 單 位 ： 國立政治大學財務管理學系 計 畫 主 持 人 ： 岳夢蘭 計畫參與人員： 博士班研究生-兼任助理人員：謝依婷 博士班研究生-兼任助理人員：邱信瑜 博士班研究生-兼任助理人員：傅信豪 報 告 附 件 ： 出席國際會議研究心得報告及發表論文 公 開 資 訊 ： 本計畫可公開查詢

中 華 民 國 101 年 07 月 31 日

中 文 摘 要 ： 交易對手風險指的是衍生性商品契約交易的一方不願意或無 法履行契約的義務，因而導致與其交易的另一方產生損失的 風險。自從 2008 年全球最大的保險集團 American

International Group Inc. (AIG-US) 因使用信用違約交換 （Credit Default Swaps）而引發鉅額虧損瀕臨破產後，如 何衡量信用衍生性商品的交易對手的違約風險便成為一個重 要的研究課題。

本研究計畫的目的是在考慮交易對手有可能違約的情況 下，探討一籃子信用違約交換（basket default swaps）的 評價問題。信用違約交換是一種簡單的信用衍生性商品，契 約中的違約保護買方（protection buyer）因持有具風險性 的標的資產，而希望將此資產的信用風險移轉給違約的保護 賣方（protection seller），因而定期支付固定費用給違約 的保護賣方以獲得違約風險的保護。當標的資產是由多個風 險性資產購成的投資組合時，此契約便稱為一籃子信用違約 交換。若投資組合中出現第一個信用違約事件時，違約保護 賣方就要賠償違約保護買方的損失時，此契約稱為首次違約 的信用違約交換（first-to default basket default swaps）。若直到投資組合中出現第 K 個信用違約事件時，違 約保護賣方才會賠償違約保護買方的損失時，此契約便稱為 第 K 個違約的信用違約交換（Kth-to default basket default swaps）。一籃子信用違約交換契約的分析與評價， 除了需考慮標的投資組合內個別資產的違約機率及其違約損 失值外，還需同時考量此一籃子內各風險性資產的關聯係 數，以正確衡量標的投資組合的損失分配，因此在評價一籃 子信用違約交換上已較單一資產的 CDS 評價問題複雜。若此 時再加入交易對手違約風險的考量，則整個評價問題將更趨 困難，因此截至目前為止，文獻中尚未有考量交易對手違約 風險的一籃子信用違約交換定價之相關研究。

本研究計畫首先利用 Laurent and Gregory（2005）提 出的因子連繫結構模型（factor model）來描述一籃子中個 別標的資產間的相關性。接著將擴展此模型至多因子的形 式，以藉此描述個別標的資產和違約保護賣方間的相關性。 在此多因子連繫結構模型的架構下，我們將發展減小變異數 （variance reduction）的有效模擬方法來評價考量了交易 對手違約風險的一籃子信用違約交換。在定義信用價值調整 項（credit value adjustment）為一籃子信用違約交換契約 因其交易對手有可能違約而造成一籃子信用違約交換契約價 值減損的部分後，本計劃也將進一步分析標的資產間的相關 性，以及標的資產和違約保護賣方之間的相關性，將如何影 響此一信用價值調整項。最後本計劃將探討模型中的各參數

將如何影響評價方式的效率性。

中文關鍵詞： 一籃子信用違約交換，交易對手違約風險，因子連繫結構模 型，減小變異數模擬法，信用價值調整

英 文 摘 要 ： 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 contract, causing losses to the other party. Contracts privately negotiated between counterparties like over-the-counter (OTC) derivatives are most likely subject to counterparty risk. The value of a credit derivative should depend on not only the risk of defaults of both the

reference entity and the counterparty, but also the dependence of these two risks on one another.

Ignoring correlations between underlying entities and the counterparty cannot reflect the true value of a credit derivative.

This research takes into account of counterparty risk for the valuation of a basket default swap (BDS). In this research, we will 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. We will first study the CVA of a BDS. We will then explore the impact of model parameters on the efficiency of the proposed pricing method.

英文關鍵詞： basket default swap, counterparty risk, factor-copula model, variance-reduction, credit value adjustment

An Efficient Algorithm for Basket Default Swap

Valuation

An Efficient Algorithm for Basket Default Swap Valuation

Abstract

The recent credit crisis has highlighted the importance of counterparty risk in connection with valuation and risk management of credit derivatives. Counter-party risk in general is the risk that the Counter-party to a financial contract may fail to make all the payments required by the contract, causing losses to the other party. Contracts privately negotiated between counterparties like over-the-counter (OTC) derivatives are most likely subject to counterparty risk. The value of a credit derivative should depend on not only the risk of defaults of both the reference entity and the counterparty, but also the dependence of these two risks on one another. Ignoring correlations between underlying entities and the counterparty cannot reflect the true value of a credit derivative.

This paper takes into account of counterparty risk for the valuation of a bas-ket default swap (BDS), aiming to fill the gap in the literature on the analysis of counterparty risk within credit derivatives.

We propose an effective IS algorithm for the valuation of BDS with counter-party risk. The algorithm is simple to implement and guarantees variance reduc-tion. We have established a way of ensuring that for every simulation path gener-ated, the desired default events always take place, and this is achieved under an appropriate choice of IS distribution among the set of possible measures that can force the required events of interest to happen. Our numerical results show that the proposed algorithm is considerably more efficient than the crude simulation method.

1

Introduction

The recent credit crisis has highlighted the importance of counterparty risk in connec-tion with valuaconnec-tion 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 pay-ments required by the contract, 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 correlation. 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. A few studies have been made to analyze the valuation of counterparty risk within a CDS. For example, Jarrow and Yu (2001) Jarrow and Yu [2001] propose an intensity-based model to examine the impact of a default on a surviving firm. Hull and White (2001) Hull and White [2001] address the counterparty risk problem for CDS by re-sorting 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, aim-ing 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 de-fault swaps requires a full specification for the joint distribution of dede-fault 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 simula-tions 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 matu-rity 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.

To model default dependence among obligors and between obligors and a protec-tion seller, we specify a joint distribuprotec-tion for the default times using a copula. Under a factor form representation proposed by Laurent and Gregory (2005), we can explic-itly 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 adjust-ment (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 also will study the credit value adjustment (CVA) of BDS, and examine the impacts of the default correlations on CVAs.

This paper is organized as follows. Section 2 begins with a brief review on credit risk modelling and ends with a problem formulation for the valuation of kth-to-default basket default swaps with counterparty risk being considered. Section 3 proposes an IS algorithm that is simple to implement and guarantees variance reduction. In Sec-tion 4, we present numerical results for the variance reducSec-tion effect for kth-to-default BDS. Section 5 concludes this paper.

2

Valuation of kth-to-Default BDS with Counterparty Risk

2.1

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 stan-dard reduced form credit risk models such as Lando (1998) Lando [1998] or Duffie and Singleton (1999) Duffie and Singleton [1999]; for empirical work on the specifica-tion of an appropriate factor structure see for instance Duffee (1999) Duffee [1999] or Driessen (2005) Driessen [2005]. In contrast, researchers became only recently inter-ested 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) Lang and Stulz [1992] have shown that bankruptcy filings do impact stock re-turns (and most likely also default probabilities) of non-defaulted companies. More-over, as has been pointed out by Hull and White (2001) 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 eco-nomic recessions observed in real data (see for instance Keenan (2000) 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. Ig-noring 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) Jar-row 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 intensi-ties of surviving firms is explicitly modelled, which is a very intuitive parametriza-tion of counterparty risk; see also Davis and Lo (2001) Davis and LO [2001] for a re-lated 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 sec-ondary framework, which excludes many interesting examples of cyclical default de-pendency. 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) Collin-Dufresne et al. [2002]. Yu (2002) has carried out an inter-esting 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. Other scholars also provide different frameworks to take into account of the counterparty risk when valuing credit derivatives. Jarrow and Yildirim (2002) Jarrow and Yildirim [2002] propose an intensity-based valuation model of CDS with correlated market and credit risk. Hull and White (2001) Hull and White [2001]ad-dress 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) 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. They 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) Le-ung and Kwok [2005], building on Collin-Dufresne et al. (2004) 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. Kim and Kim (2003) valuing CDS that takes account of counterparty default risk as well as correlated mar-ket and credit risk. It incorporates marmar-ket risk into determining default correlation between multiple firms using the first-passage time approach.

2.2

Characterization of Default Time Correlations

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 speci-fying the dependence structure among default-times was developed by Li (2000), and later extended to many obligors by Laurent and Gregory (2005) where the correla-tion structure is represented in a factor form, common known as the factor-copula approach. The factor-copula approach conceptually coincides with the conditional

in-dependence 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. τ_i is a positive random variable and its distribution is

character-ized in terms of a hazard rate function h_i(·):

prob(τi >t) = e−

Rt

0hi(u)du_,

and let F_i(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 S_i(t)be the survival function of obligor i, S_i(t) can then be expressed as S_i(t) = 1−F_i(t).

The marginal distribution of of the default time for each obligor i is typically ex-tracted from the quoted market prices of CDSs; these market prices are used to

con-struct a hazard rate function h_i(·) from which we get the distribution F_i(t).

How-ever, 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 dis-tribution, 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 X_i, where i = 1,· · · , n; Secondly,

uniformly distributed random variates, U_i = Φ_{(Xi), are obtained from the}

cumula-tive 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 =F_i−1(Φ(Xi)).

In this article, we extend a single-factor form representation of Laurent and Gre-gory (2005) for the Gaussian copula, and specify a two-factor formalism for an under-lying entity i, i =1, . . . , n, as

X_i =φiM+ρiDu+

q

1−φ2_i −ρ2_iZ_i (1)

are dependent; Durepresents the industry-specific risk factors, with u∈ {1,· · · , m}, m

is the total number of industries; and Z_irepresents the firm-specific risk factors. All M,

Duand Ziare independent standard normal variables. The parameter φicaptures how

strongly Xiis 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 τ_i =F_i−1(Φ_(Xi)).

Based on the two-factor Gaussian copula model of equation (1), we can also write down the factor-form representation for the protection seller as

Xs =φsM+ρsDs+

q

1−φ2_s −ρ2_sZs

where Ds represents the industry-specific risk factors for a protection seller, and the

default time of the protection seller is computed via an inverse mapping of its marginal distributions, τs = Fs−1(Φ(Xs)).

2.3

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.

Through-out the section we shall adapt the following notations : t_j denotes the time for the jth

premium payment to take place; δj−1,jis the time increment between premium

pay-ments at the (j−1)th and the jth time points in units of years; B(0, t_i) = e−rti _{is the}

discount factor for one dollar received at time ti, r is the constant short rate; Ridenotes

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, Aiis the

notional amount for credit i, and we assume A_ito be equal to a constant amount A for

all i. We denote τ_iu as the default time of underlying obligor i, τs as the default time

of protection seller, and τu as the k-th default time among the underlying obligors,

i.e. τu is the k-th order statistics of τ₁u, . . . , τ_nu
. The 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

func-tion of τ is therefore given by: F(t) = Pr(τ ≤t) = 1−S(t); Finally, Q denotes the

risk-neutral probability measure; b_CR (b)denotes the credit spread of a BDS contract

that is (not) subject to counterparty risk, and I_{·} is the indicator function. To study

the credit value adjustment, we further define ΠC(t, T) as the sum of all payoff terms

between time t and T subject to counterparty risk, and Π(t, T)as the sum of all payoff

terms between time t and T without the consideration of counterparty risk. We define PVpremium_CR to be the present value of the premium leg of the k-th to default basket swap

that is subject to counterparty risk, then PV_CRpremium =E " N

∑

i=1 B(0, ti) ×bCR×I{min(τs_,τu_)>t i}×A # (2) where the expectation is taken under the risk-neutral pricing measure. If accrued

pre-miums are considered, then the present value of accrued premium PVAccruedPremium_CR

can be computed as PV_CRAccruedPremium =E " N

∑

i=1 B(0, ti) ×bCR× min(τ s_{, τ}u_{) −t} i−1 t_i−t_i−1  ×I_{min(τs_,τu_)>t i}×A #

On the other hand, we define PVdefault_CR to be the present value of the default leg of

the k-th to default basket swap subject to counterparty risk, and

PV_CRde f ault = (1−R)EhB(0, τu+ △) ×I_{τu_≤T}×I_{τs>(_τu+△)}×A

i

+RsEhB(0, τu+ △) × (1−R) ×I_{t<_τs≤_τu≤_T}×A

i

(3)

where △ is the length of the settlement period, (τu+ △) 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

b_CR = (1−R)Et h B(0, τu_{+ △) ×I} {τu_≤T}×I_{τs>_(τu_+△)} i Eh∑N_i=1B(0, t_i) ×I_{min(τs_,τu_)>ti} i +R s_E t h B(0, τu_{+ △) × (1−R) ×I} {t<_τs≤_τu≤_T} i Eh∑_iN₌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:

NPV(τs, T) = Eτs[Π(τs, T)]

For a protection buyer, if NPV(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 Et h ΠC_{(t, T)}i_{= Et[}Π_{(t, T)] − (1−R}s_)Eth_I{t<τs_≤T}D(t, τs) (NPV(τs, T))+ i (5) It is clear that the value of a counterparty defaultable claim is the value of the corre-sponding 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

valua-tion adjustment. Equavalua-tion (5) demonstrates that counterparty risk adds an opvalua-tionality level to the original payoff.

From the above analysis, we can write

Et

h

ΠC_{(t, T)}i _=Et[Π_{(t, T)] −CVA}

where CVA 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 CVA as

CVA = PVde f ault−PVpremium−PV_CRde f ault−PV_CRpremium

= PVde f ault−PV_CRde f ault+PV_CRpremium−PVpremium

= (1−R)EhB(0, τu)I_{τu_≤T} i − (1−R)EhB(0, τu+ △)I_{τu_≤T}I_{τs>(_τu+△)} i +E " N

∑

i=1 δ_i−1,iB(0, ti)SCRI{min(τs_,τu_)>ti} # −E " N

∑

i=1 δ_i−1,iB(0, ti)SI{τu>t_i} # = (1−R)nEhB(0, τu) ×I_{τu_≤T} i −EhB(0, τu+ △) ×I_{τu_≤T}×I_{τs>(_τu+△)} io +δE " N

∑

i=1 B(0, t_i)nb_CR×I_{min(τs_,τu_)>ti}−b×I_{τu>t_i} o # (6)

where PVde f ault( PVpremium) is the present value of the default (premium) leg of the

k-th to default basket swap wik-thout taking into consideration of counterparty risk. We

assume the notional amount A = 1, the recovery rate Rs = 0, and the tenor period is

constant as δ_i−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

sim-plified as

PV_CRde f ault= (1−R)EhB(0, τu+ △) ×I_{τu_≤T}×I_{τs>(_τu+△)}

i

(7) The crude Monte Carlo procedure for computing the value of the default leg (hereby abbreviated 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

X_i =φiM+ρiDu+ q 1−φ2_i −ρ2_iZ_i, 1≤i ≤n and Xs =φsM+ρsDs+ q 1−φ2 s −ρ2sZs 3. Set U_i =Φ(X_i), 1≤i ≤n; and Us =Φ(Xs) 4. Set τ_iu =F_i−1(U_i), 1≤i ≤n; and τs = F_s−1(Us)

5. Set τu =the k-th order statistic of(τ₁u, . . . , τ_nu)

6. Set the discounted payoff= (1−R)I_{τu_≤T,τs>(_τu+△)}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

∑

j=1 (1−R)I(τu(j) ≤T, τs(j) >  τu(j)+ △)B0, τu(j), (8)

where τu(j)and τs(j)are the j-th independent samples of τu and τs respectively.

3.2

Proposed IS Algorithm

We propose an algorithm which guarantees variance reduction when running simu-lations to value a BDS subject to counterparty risk in the following two cases:

Case 1: Du and Dsare different

1. Generate independent samples of standard normal variates Duand Z1, . . . , Zn

2. Set h_i = 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 U₁ is a

uniform(0, 1)random variate

5. Set X_i =φiM+ρiDu+

q

6. Set τ_iu =F_i−1(Φ_{(Xi)), 1≤i≤n}

7. Set τu =the k-th order statistic of(τ₁u, . . . , τ_nu) 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+ △)} happen and

the likelihood ratio L =L₁L2<1.

Case 2: Du and Dsare the same

1. Generate independent samples of standard normal variates Du = Dsand Z1, . . . , Zn

2. Set h_i = Φ−1(F_i(T)) −ρiDu− q 1−φ2_i −ρ2_iZ_i 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 U1 is a

uniform(0, 1)random variate

5. Set Xi =φiM+ρiDu+

q

1−φ2_i −ρ2_iZ_i; 1 ≤i ≤n

6. Set τ_iu =F_i−1(Φ_{(Xi)), 1≤i≤n}

7. Set τu =the k-th order statistic of(τ₁u, . . . , τ_nu) 8. Set hs = Φ−1_(Fs(τu_{+∆)) −φ} sM−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+ △)} happen and

the likelihood ratio L =L₁L₂<1.

3.3

Algorithm Analysis

The efficiency of Monte Carlo simulation is measured by the variance of the estimator and the effort to generate a replication (Glynn and Whitt (1992)). Our goal is twofold: First, we establish a way of selecting an appropriate IS distribution the set of possible

measures that ensures the k default events of interest to take place on every generated path. Second, we need to ensure that the gain in variance reduction of the proposed algorithm always outweighs a crude Monte Carlo algorithm.

Let f be the density function of τ and suppose g(x) is another density function

such that, g(x) > 0 for 0 ≤ x ≤ T. Let L(x) = f(x)/g(x), which is known as the

likelihood ratio, and τ(j) are independent copies sampled from g(x). Then, it is clear

that the point estimator

β= 1 m m

∑

j=1 (1−R)I(τ(j) ≤T)P0, τ(j)L(τ(j)) (9)

is an alternative estimator for the DL. The point estimator β is called IS estimator

(Glynn and Iglehart Glasserman the choice of IS density, g(x). In order to select an

ap-propriate g(x), we provide a simple alternative characterization for the default event

of the basket,{τ ≤T}.

Proposition 1. The default event of the basket {τ ≤ T} is equivalent to the event {M≤h} ifhis the (n−k+1)-th order statistic of (h₁, . . . , hn), wherehi = (Φ−1(Fi(T)) −

ρiDu−

q

1−φ_i2−ρ2_iZi)/φi. That is, conditional onZ1, . . . , Zn, the event of interest can

be determined solely by the common factorM.

Proof. Since {τi ≤T} ≡ {Fi(τi) ≤ Fi(T)} ≡ {Xi ≤Φ−1(Fi(T))} ≡ {φiM+ρiDu+ q 1−φ2_i −ρ2_iZ_i ≤Φ−1(F_i(T))} ≡ {M ≤ Φ−1₍F_i(T)) −ρiDu− q 1−φ2_i −ρ2_iZ_i φi } ≡ {M ≤h_i}

we therefore arrive at an alternative characterization for the default event {τi ≤T},

i.e. the event{τi ≤T}is equivalent to{M ≤hi}.

Next, we consider the event{τu ≤T}. Notice that

I(τu ≤ T) = 1 ⇔ n

∑

i=1 I(τ_i ≤T) ≥ k ⇔ n

∑

i=1 I(M ≤h_i) ≥ k

show that

n

∑

i=1

I(M≤h_i) ≥k ⇔ I(M ≤h) = 1

Thus{τu ≤T}if and only if{M ≤h}.

Now, we can consider the expectation of interest EhB(0, τu)I_{τu_≤T}I_{τs>(_τu+△)}

i . Note that EhB(0, τu)I_{τu_≤T}I_{τs>(_τu+△)} i = E{EhB(0, τu)I_{τu_≤T}I_{τs>(_τu+△)}    M, Du, Z1, . . . , Zn]} = E{B(0, τu)I_{τu_≤T}E h I_{τs>_(τu_+△)}    M, Du, Z1, . . . , Zn]} = E[B(0, τu)I_{τu_≤T}Φ(−h_s)] = 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|D_u, Z₁, . . . , Z_n]] = 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.

4

Numerical Results

Based on the IS algorithm proposed in the previous section, in this section we present the numerical results for the estimation of fair spreads for kth-to-default basket fault swaps. The market quotes for standardized five-year kth-to-default basket de-fault swaps are obtained from J. P. Morgan, which are defined using five underlying reference entities. The risk-neutral default probabilities of the reference entities dur-ing the five years are extracted directly from the market quoted credit default swap

spreads. All five underlying references are of equal unit nominal and assumed with a recovery rate of 40%.

We study the kth-to-default basket default swaps with auto sector. Our focus is placed on the variation reduction of the estimation of the expected values of default

leg, which are vital inputs for the calculation of the fair spreads b_CR for the

kth-to-default basket kth-to-default swaps subject to counterparty risk. In our simulations, the val-ues for k are chosen to be 1 and 3, and we take the risk free rate r = 5%. To see how correlations/factor loadings affect the variance reduction efficiency of our importance sampling algorithm, we repeat the same calculation for different constant factor load-ings.

Case 1. auto

First-to-Default Basket Default Swap (k =1)

variance φi ρi φs ρs reduction 0.5 0.7 0.2 0.5 1.8465 0.5 0.7 0.5 0.5 1.7944 0.5 0.7 0.8 0.5 1.7271 0.8 0.5 0.2 0.5 4.3752 0.8 0.5 0.5 0.5 4.1659 0.8 0.5 0.8 0.5 3.8533 0.95 0.2 0.2 0.5 16.6187 0.95 0.2 0.5 0.5 15.6960 0.95 0.2 0.8 0.5 14.4875 0.99 0.1 0.2 0.5 76.1741 0.99 0.1 0.5 0.5 72.8592 0.99 0.1 0.8 0.5 68.7843

Third-to-Default Basket Default Swap (k=3)

variance φi ρi φs ρs reduction 0.5 0.7 0.2 0.5 2.4250 0.5 0.7 0.5 0.5 2.2369 0.5 0.7 0.8 0.5 2.0164 0.8 0.5 0.2 0.5 7.9163 0.8 0.5 0.5 0.5 6.9064 0.8 0.5 0.8 0.5 5.0983 0.95 0.2 0.2 0.5 46.5202 0.95 0.2 0.5 0.5 39.4247 0.95 0.2 0.8 0.5 24.2808 0.99 0.1 0.2 0.5 191.3913 0.99 0.1 0.5 0.5 163.6716 0.99 0.1 0.8 0.5 98.8807

5

Conclusion

In this paper, we have proposed an importance sampling algorithm for the valuation of basket default swaps that is subject to counterparty risk. The algorithm is simple to implement and guarantees variance reduction via an appropriate choice of the impor-tance sampling distribution.

In Proposition 1, we have established a way of ensuring that for every path gen-erated, k default events always take place. In addition, we have established an addi-tional guideline for selecting an appropriate IS distribution among the set of possible measures that can force the required events of interest to take place. While IS can actually be prone to ineffective variance reduction, or at cases increase variance, the strength of our proposed algorithm ensures variance reduction efficiency.

Our numerical results confirm that the gain in variance reduction efficiency of our proposed algorithm considerably outweighs a crude simulation estimator. The effi-ciency gain is more significant for highly correlated underlying credits and especially for basket default swaps with higher values of k. Our proposed algorithm thus pro-vides a sound basis for the estimation of kth-to-default BDS that is subject to counter-party risk.

REFERENCES

D. Brigo and K. Chourdakis. Counterparty Risk for Credit Default Swaps: Impact of Spread Volatility and Default Correlation. Working Paper, 2008.

P. Collin-Dufresne, R. Goldstein, and J. Hugonnier. A General Formula for Valuing Defaultable Securities. Working Paper, 2002.

P. Collin-Dufresne, R. Goldstein, and J. Hugonnier. A General Formula for Valuing

Defaultable Securities. Econometric, 72:1377–1407, 2004.

M. Davis and V. LO. Infectious Defaults. Quantitative Finance, 1:382–387, 2001.

J. Driessen. Is Default Event Risk Priced in Corporate Bonds? The Review of Financial

Studies, 18:165–195, 2005.

G. Duffee. Estimating the Price of Default Risk. Review of Financial Studies, 12:197–

226, 1999.

D. Duffie and K. Singleton. Modeling Term Structure Models of Defaultable Bonds. Review of Financial Studies, 12:687–720, 1999.

J. Hull and A. White. Valuing Credit Default Swaps II: Modelling Default Correlations. Journal of Derivatives, 8(3):12–22, 2001.

R. Jarrow and Y. Yildirim. Valuing Default Swaps Under Market and Credit Risk

Correlation. Journal of Fixed Income, 11:7–19, 2002.

R. Jarrow and F. Yu. Counterparty Risk and the Pricing of Defaultable Securities. Journal of Finance, 53:2225–2243, 2001.

S. Keenan. Historical Default Rates of Corporate Bond Issuers 1920-1999. Moody’s Investors Services, 2000.

D. Lando. Cox Processes and Credit Risky Securities.Review of Derivatives Research,

2:99–120, 1998.

L. Lang and R. Stulz. Contagion and Competitive Intra-industry Effects of Bankruptcy

Announcements. Journal of Financial Economics, 32:45–60, 1992.

S. Y. Leung and Y. K. Kwok. Credit Default Swap Valuation with Counterparty Risk. Kyoto Economic Review, 74:25–45, 2005.

國科會補助專題研究計畫項下出席國際學術會議心得報告

日期：2012 年 7 月 10 日

計畫編號

NSC99－2410－H－004－078

計畫名稱

一籃子信用違約交換之評價-考量交易對手違約風險

出國人員

姓名

岳夢蘭

服務機構

及職稱

政治大學財務管理學系

副教授

會議時間

2012 年 7 月 2 日

至

2012 年 7 月 4 日

會議地點

日本筑波

會議名稱

(中文)第二屆數學統計學會環太平洋區岸研討會

(英文) IMS-APRM2012

發表論文

題目

(中文) 固定期間信用違約交換之評價

(英文) Valuation of Constant Maturity Credit Default

Swaps

一、 參加會議經過

數學統計學會環太平洋區岸研討會原訂於去年 2011 年舉行，但因去年日

本發生福島核災，因此研討會主辦單位決定將會議延至今年 2012 年舉行。

第二屆數學統計學會環太平洋區岸研討會(IMS-APRM2012)乃是環太平洋

區岸之大型數學統計研討會，此次在日本筑波的國際會議中心舉行。研討

會共計三日，時間為 7 月 2 日至 7 月 4 日，共計約有 104 個 concurrent

sessions，論文發表內容涵蓋統計各領域，共有 507 與會者參與此次會議。

會議中除了安排各個不同的 session 外，也規劃了 poster session，提

供論文無法被接受發表者另一研究交流的機會。

二、 與會心得

此次本人參與會議並發表論文，論文名稱為“Valuation of Constant

Maturity Swaps“。此篇論文主要應用隨機微積分以及統計的方法，評價

信用違約交換的衍生性商品。此商品在實務上具有良好的避險以及投機功

能，因此在理論以及實務上有相當的重要性。

本人參加的 session 為“Financial Risk Managements”，該 session 共

有五篇論文報告，各篇論文的主旨都和當今財務領域最重要的議題風險管

理有關。其中來自日本央行的論文發表者，更和大家分享了日本央行如何

因應 Basel III 的實施，使用非線性的模型，估計所需提列的資本準備

金。此外論文發表者也分享了歐債危機對日本央行的衝擊。另外，我也和

某些學者討論未來合作的可能性。有些研究資料，本校因經費短缺而無法

購置資料庫時，因此也和部分學者討論，是否能籍由雙方合作的關係，讓

我們能使用到對方的資料庫。

由於風險管理的議題在近來相當的被重視，藉由參與此次會議，使我能在

和風險管理相關的研究議題上，有更深的體悟。藉由和與會者的討論與心

得交換，對未來的研究也有了新的想法。也希望回國後，仍有機會繼續雙

邊的合作計劃。

總之，此次參加第二屆數學統計學會環太平洋區岸研討會(IMS-APRM2012)，

在學術交流與實務經歷方面的收穫都相當豐富。

三、考察參觀活動

無考察參觀活動。

四、建議

無。

五、攜回資料名稱及內容

1. 研討會議程等相關資料

Chapter 1

Methodology and Theory for Partial Least Squares Applied to Functional Data. Peter Hall* (University of Melbourne, Australia)

Chair. Byeong U. Park (Seoul National University, Korea)

Large Deviations - Some Unusual Examples. S.R.S. Varadhan• (New York University, USA)

Chair: Tadahisa Funald (University of Tokyo, Japan}

Sponsor. Korea/KSS

Multiscale Modeling.

Organizer: Hee-Seok Oh {Seoul National University, Korea) Chair: Kyusang Yu (Konkuk University, Korea}

Speakers:

2,July (Mon)

PL-1 [Main Convention Hall]

PL-2 [Convention Hall 200]

IP-12 [lOll

1. Time--Threshold Maps: Using Information from Wavelet ReeonGtn.lctions with All Threshold Values Simultaneously. Piotr Fryzlewfcz• (London School of Economics, UK) 2. A Tour for Empirical Mode Dacompcsltlon. Hee-Seok Oh• (Seoul National University, Korea), Donghoh Kim (Sejong University, Korea)

3. Generalized Fiducial inference for Modem Statistical Problema. Jan Hanning* (University of North Carolina at Chapel Hill, USA), Thomas C.M. Lee {The University of California Davis, USA)

4. Testing the Equality of Regression Curves In Multiple Scale. Cheolwoo Park (University of Georgia, USA), Jan Hannig (University of North Carolina at Chapel Hill, USA), Kee-Hoon Kang' {Hankuk U. Foreign Studies, Korea)

Sponsor. Korea/KSS IP-11 [102]

Statistical Tools for Genome-wide Associstion Study.

Co-Organizers: Jaewon Lee (Korea University, Korea), Sungho Won (Chung-Ang University, Korea) Chair: Makoto Aoshima (University of Tsukuba, Japan)

Speakers:

1. Detecting Rare Variants in Admixed Populations. Xlaofeng Zhu'" {Case We..stem Reserve Univf".rslty, USA), Huizhen Qin (Case Western Reserve University, USA) 2. Emclent Strategy to Detect Gene x Gene Interaction and Its Application to Schizophrenia. Sungho Won* (Chung-Ang University, Korea), Christoph Lange (Harvard

School of Public Health, USA)

3. Homozygosity Disequilibrium and Its Applications. H.sin-Chou Yang* (Academia Sinica, Taiwan)

4. Assoclatlon Analyals of Recurrent Gap Time Data via Multivariate Kendall's Tau. Shu-Hui Chang• {National Taiwan University, Taiwan), Tsung-Chiang Fu {National Taiwan University, 'Taiwan)

Sponsor. IMS

Statistical Inference with Large Covariance Matrices.

Chair: Runze Li (Penn State University, USA)

Adaptive Estimation of Large Covariance Matrices. Dl speaker: Tony Cai* (University of Pennsylvania, USA) Invited speakers:

1, Minimax Lower Bounds In Covariance Matrices Estimation. Harrison Zhou* (Yale University, USA)

2. High Dimensional Covariance Matrix Estimation with Group Structures. Ming Yuan• (Georgia Institute of Technology, USA}

DL- 03 [Convention Hall 200]

- - -

---Sponsor: Bernoulli Society [p- 30 [201 A]

Interface of Probability and Mathematical Ststistics I.

Organizer: Edward Waymire (Oregan State University, USA) Chair: S.RS. Varadhan (New York University, USA) Speakers:

1, Cramer Type Moderate Deviations for Self-normalized Processes. Qi-Man Shao• (Hong Kong University of Science and Technology, Hong Kong), Wenxin Zhou (Hong Kong t;niversity of Sdence and Technolgy, Hong Kong)

2, Modelling Genetic Variationa Using Fragmentation-coagulation Proceues. Yee Whye Teh* (University College London, UK), Charles Blundell (University College London, L'X), Uoyrl T. Elliott (University College London, UK)

3. Structured Selection In Partial Llkellhccd. Jelena Bradic* (University of California, San Diego, USA), Ruf Song {Colorado State University, USA)

Sponsor. Japan/JSS IP- 16 [201 B)

Stochastic Analysis on Large Scale Interacting Systems.

Organizer/Chair: Tadahisa Funaki (University of Tokyo, Japan) Speakers:

1. Glnlbre Random Point Field. lfuofumi Osada* {Kyushu University, Japan)

2. Scaling Limit for Multi-species Exclusion Process. Yuki.o Nagahata* (Osaka University, Japan)

3. A Singular 1-0 Hamllton.Jacobl Equation, with Application to Large Deviation of Diffusions. Xiaoxue Deng (Tsinghua University, China), Jin Feng (Kansas University, USA), Yong lJu"' (Peking University, China)

Sponsor: lMS

Recent Advance in Multiple Testing, Regularization Method and Network.

Organizer: Runze li (Penn State University, USA} Chair: Ying Wei (Columbia University, USA) Speakers:

IP - 03 (202A]

1. Modeling MultJple Relationship& In ScelaiNetwork: Data. Tyler McCormick* (University of Washington, USA), Natesh S. Pillai (Harvard University, USA)

2. A Penalized EM Algorithm for Multivariate Gaussian Parameter EatJmatlon wHi1 Non~lgnorable Mining Data. Lin Chen* (University of Chicago, USA), Ross L. Prentice (Fred Hutchinson Cancer Research Center, USA), Pci Wang (Fred Hutchinson Cancer Research Center, USA)

3. Robust High Dimensional Statistical Inference. Yunda Zhong (University of Chicago, USA), Hongyuan Cao* (University of Chicago, USA), Wei-Biao Wu (University of Chicago, USA)

4, Joint Linear Trend Recovery Using L1 Regularization. Xiaoli Gao• (Oakland University,USA), Yuan Wu (University of California at San Dle.go, USA), S. &iaz Ahmed (University of Windsor and Brock University, USA)

Chapter 1

2,July (Mon)

Sponsor. IMS DL- 04 [Convention Hall 300]

Functional Data Analysis.

Chair. Ming-Yen Cheng (National Taiwan University, Taiwan) Dimenaion Reduc:tion for Func:tional Data.

Dl speaker: Jane-Ung Wang• (University of California at Davis, USA) Invited speakers:

I. Smooth Backflttlng: Additive Modala for Longitudinal Data. Xiaoke Zhang (University of California, USA), Jane-Ling Wang (University of California, USA), Byeong u. Park* (Seoul National University, Korea)

2. Robuat Eatim.tora under a Func:tional Common Principal Component& Model. Lucas Bali (Universidad de Buenos Aires and CONICET, Argentina), Graciela Boente• (University of Bueno Aires, Argentina)

Sponsor. India IP-21 [303]

Design of Experiments.

Organizer/Chair. Aloke Dey (Indian Statistical Institute, India) Speakers:

I. Combln.torlca on Reaolvablllly of lncompiN Block Dealgna. Sanpei Kageyama• (Hiroshima Institute of Teclmology, Japan)

2. Efficient Row-Column Dealgns for 2-colour Single Factor Mlcroarrey Exparlmanta. Rajender Parsad• (Indian Agricultural Statistics Research Institute, India), Sukanta Dash (IASRI, library Avenue, India), VK Gupta (IASRI, library Avenue, India)

3. OpOmal Supersaturated Designs. Ashish Das• (Indian Institute of Technology, India), Chung-Yi Suen (Cleveland State University, USA), Kashinath Chatterjee (Visva Bharatl University, India), Feng-Shun Chai (Institute of Statistical Science Academia Sinica, Taiwan)

4. Discunant's Remarka in the Sesaion on Deaign of Experiments. Discussant: Aloke Dey" (Indian Statistical Institute, India)

Recent Development in Joint Models and Their Applications.

Co-organizers: Yangxin Huang {University of South Florida, USA), Lang Wu (University of British Columbia, UK) Chair. lang Wu {University of British Columbia, UK)

Speakers:

I. Joint Spatial Modeling of Longitudinal and Survival Data. Farouk Nathoo• (University of Victoria, Canada) 2. Joint Modeling of Survival and Functional Data Aaaociation. Jirnln Ding" (Washington University in St. Louis, USA)

TCP- 03 [401]

3. Time-varying Func:tlonal Regreaalon for Predlc:Ung Rlak of Event Ualng Put Hl.tory of Longitudinal Covariate TraJec:torlea. Wen Ye• (University of Michigan, USA), Jeremy Taylor (University of Michigan, USA), Lu Wang (University of Michigan, USA)

Financial Risk Managements. TCP- 21 [402] Organizer/Chair. Ryozo Miura (Hitotsubashi University, Japan)

Speakers:

I. Default Timing and Recovery Rata. Yuki Itoh" (Yokohama National University, Japan)

2. Valuation ofConatant Maturity Credit Default Swapa. Hidetoshi Nakagawa (Hitotsubashi University, Japan), Meng-Lan Yueh* (National Chengchi University, Taiwan), Ming-Hua Hsieh (National Chengchi University, Taiwan)

3. Calibrating the L.eval of Capital: The Way We Sea lt. Ryo Kato (Bank of Japan, Japan), Shun Kobayashi* (Bank of Japan, Japan), Yumi Salta (Tohmatsu, Japan)

4. A statistical Model for Hedge Fund Returns. Daisuke Yokouchi" (Hitotsubashi University, Japan), Yoshimitsu Aoki (The Graduate University for Advanced Studies, Japan), Takeshi Kato (Sophia University, Japan), Ryozo Miura (Hitotsubashi University,Japan)

5. On Portfolio Optimization with Levy Proceaaea. Olivier Le Courtois" (Ecole de Management de Lyon, France)

Sponsor. Bernoulli Society

Biostatistics.

Organizer: Choongrak Kim (Pusan National University, Korea) Chair. Hwan Chung (Korea University, Korea)

Speakers:

I. lsauea In Temperatura-mortality Studlea. Ho Kim• (Seoul National University, Korea)

2. Analyzing the Clustered and lntarval-c:ansorad Data Baaed on the Semlparametrlc Frailty Modal. Jlnhewn Kim" (University of Suwon, Korea)

TCP- 30 [403]

3. Detection of Gena11ane lntaractlonsln Family-baaed Genetic Asaoclatlon Study. Hyojung Lee (Korea University, Korea), Seohoon Jin (Korea University, Korea), Wonseok Woo (Korea University, Korea), Mira Park" (Eulji University, Korea)

4. A Logistic Regreaaion Method in Eatimating Median Survival Time in lnterval-cenaored Data. Choongrak Kim• (Pusan National University, Korea), Eunyoung Yoon

(Pusan National University, Korea)

Survival Analysis. CP-03 [404]

Chair. Katsuto Tanaka (Hitotsubashi University, Japan) Speakers:

I. Uae ofAlternative Time 8calea in Cox Proportional Hazard Models. Beth Ann Griffin• (RAND Corporation, USA), Garnet Anderson (Fred Hutchinson Cancer Research Center, USA), Regina Shih (RAND Corporation, USA), Eric Whitsel (University of North Carolina at Chapel Hill, USA)

2. Maximum Penalized Ukellhood Estimation to Baaellne Hazard and Regrasalon Coal'llclenta In Proportional Hazard Modala. Jun Ma* (Macquarie University, Australia), Stephane Heritier (George Institute, The University of Sydney, Australia), Serigne Lo (George Institute, The University of Sydney, Australia)

3. Marginal Additive Hazarda Modal for Case-cohort Studies with Multiple Dl•-• Outcornea. Sangwook Kang* (University of Connecticut, USA), Jianwen Cal (University of North Carolina, USA), Uoyd Chambless (University of North Carolina, USA)

4. Eatlmatlon of Kendall's Tau for Blvartata Survival Date with Truncation. Hong Zhu* (The Ohio State University, USA)

5. Conaiatent Eatimationa in the Accelerated Failure Time Model When Covariate& are Subjac:t to Measurement Errons. Yih-Huei Huang* (Tamkang University, Taiwan) 6. Explicit Solution for Ruin Probabilitiea in the Ren-al Riak Modal with Conatent Interest Force. K.K. Thampi• (Mahatma Gandhi University, India)

Sponsor: IMS IP - 07 [405]

Time Series Analysis.

Organizer: Qiwei Yao (london School of Economics, UK) Chair: Wai Keung li {University of Hong Kong, Hong Kong) Speakers:

I. Poluon Thl'llllhold Autoragl'llllalon. Abdullah Almarashi (University of Strathclyde, UK), Jiazhu Pan" (University of Strathclyde, UK)

2. Asymptotic Properties ofTime Sarles Non-life Insurance Modal. Kentaro Kobayashi (Niigata University, Japan), Junlchi Hirukawa* (Niigata University, Japan) 3. Least Absolute Deviation Estimation for NonstaUonary Vector Autoregl'llllslva Time Sarles Models with Pure Unit Roota. Guodong u• (Hong Kong University, Hong

Kong), Jianhong Wu (Zhejiang Gongshang University, China), Wai Keung U (University of Hong Kong, Hong Kong)

4. On the Quasi-maximum Likelihood Eatimation of a Threahold Double AR Model. U Dong (University of Iowa, USA), Shiqing Ling* (Hong Kong University of Science and Technology, Hong Kong)

Chapter 1

U"t;!odfAtJ

Sponsor: n5A

Some Recent Advances in Nonparametric Statistics.

Organizer: Bodhisattva Sen (Columbia University, USA) Chair: Johan Um (Seoul National University, Korea) Speakers:

I. Properti. .crf the Adjueted Empirical Likelihood. Jiahua Chen• (University of British Columbia, Canada)

2,July (Mon)

IP-23 [101]

2. Log-concave Denaltlea and Bl·log-concave Distribution Functions. Lutz Dflmbgen• (University of Bern, Switzerland), Petro Kolesnyk (University of Bern, Switzerland), Kaspar Ru:!ibach (University of Zilrich, Switzerland), Dominic Schulunacher (University of Bern, Switzerland)

3. Nonparametrlc Leut Squares Estimation crf a Multivariate Convex Regression Function. Emilio Seijo• (Columbia University, USA)

Sponsor: China

Biostatistics and Related Topics.

Organizers: Dayue Chen (Peking University, China) Chair: Geng Zhi (Peking University, China) Speakers:

IP-38 [102]

I. Construction of Orthogonal Latin Hypereubes of Ordera One and Two. Mingyao Ai• (Peking University, China), Yuanzhen He (Peking University, China), Senmao Uu (Peking University, China)

2. Conetruction crf N. .ted Orthogonal Latin Hypereube Deaigna. Jlnyu Yang (Nankai University, China), Min-Qian Uu• (Nankai University, China), Dennis K. J. I1n ('The Pennsylvania State University, USA)

3. Asymptotic Behavior of the Laval Set Eatlmate for Censored Data. Yangfeng Wang (Tsinghua University, China), Y"mg Yang• (Tsinghua University, China)

4. Teat Conditional independence Baaed on a Likelihood Ratio Proce... Xiaogang Duan (Beijing Normal University, China), Qihua Wang• (Chinese Academy of Sciences, China), Jing Qin (National Institute of Allergy and Infectious Diseases, NIH, USA), Bingyi Jing (Hong Kong University of Science and Technology, China)

Sponsor: Singapore

Recent Advances on High-<iimensional Data Analysis.

Chair: Jin-Ting Zhang (National University of Singapore, Singapore) A ROAD to Nonparametric Cluaification.

Dl speaker: Jianqing Fan* {Princeton University, USA)

Yang Feng (Columbia University, USA), Xin Tong (Princeton University, USA) Invited Speakers:

DL • 12 [Convention Hall 200]

1. Consnlned L.uao Optimization. Gareth James• (University of South California, USA), Paat Rusmevichientong (University of Southern California, USA), Courtney Paulson (University of Southern California, USA)

Sponsor: IMS

Applications of Mathematical Statistics in the Behavioral Sciences.

Organizer: Linda Collins (Pennsylvania State University, USA) Chair: John W. Graham {Penn State University, USA) Speakers:

IP- 01 [201 A]

I. Fractional Factorial Experimente and Cluater Randomization: An Integration of Engineering Rea"rch Methoda and Educational Ruearch Methoda. John J. Dziak {The Pennsylvania State University, USA), lnbal Nahum-Shani (University of Michigan, USA), linda M. Collins" (Pennsylvania State University, USA)

2. Statlatlcal Modela for Longitudinal Zero-lnflatad Count Data with Application• to the Substance Abuse Field. Arme Buu• (University of Michigan, USA), RWlZe U (Pennsylvania State University, USA), Xiaruning Tan (Pennsylvania State University, USA), Robert A. Zucker (University of Michigan, USA)

3. A Dynamical Syatema Approach for Adaptive Behavioral intervention Development. Jessica Trail• (Pellllllylvania State University, USA), Linda M. Collim ('The Pennsylvania State University, USA), Daniel E. Rivera (Arizona State University, USA), Megan E. Piper (University of Wisconsin, USA)

4. A Slnlctural Model for Examining Tim-.rying Effect Modendion: Estimating the Effect ofAdditional Substance Uae Treatment aa a Function ofTime-varying Severity. Daniel Almira!!• (University of Michigan. USA), Daniel F. McCalfrey (RAND, USA), Beth Ann Griffin (RAND, USA), Rajeev Ramchand (RAND, USA), Susan A. Murphy {University of Michigan, USA)

Sponsor: IMS IP - 43 [201 Bl

Spatial Statistics.

Organizer: Marc G. Genton (Texas A&M University, USA) Chair: Hidetoshi Shimodaira (Tokyo lnstirute of Technology, Japan) Speakers:

I. Nonstatlonary Random Flelda and the Gravitational Lenalng ofthe Coamlc Mlcrowavt1 Background. Ethan Anderes* (University of California at Davis, USA) 2. Hierarchical Spatial Model for Precipitation Data from MuHiple Satellltea. Avishek Chakraborty (Texas A&M University, USA), Huiyan Sang• (Texas A&M University,

USA), Bani Mallick (Texas A&M University, USA), Kenneth Bowman (Texas A&M University, USA)

3. Geoatatistical Regrvaion Model Selection. Chih-Hao Chang (National Chiao Tung University, Taiwan), Hsin-Cheng Huang* (Academia Slnica, Taiwan), Ching-Kang log

(Academia Sinica, Taiwan)

Sponsor: Japan/JSS

Inference for Stochastic Processes from High-Frequency Data.

Organizer/Chair: Takaki Hayashi (Keio University, Japan) Speakers:

I. Limit Theorems and Estimation for Dlffualona. Nakahiro Yoshida• {University of Tokyo, Japan)

2. Adaptive Eatlmatlon for Dlacretely Obaervecl Ergodic Dlffualon Procue... Masayuki Uchida• (Osaka University, Japan)

IP- 13 [202A]

3. Nonparametric Estimation for Quadratic Covartatlon of Two-dlmenalonal Diffusion under Nonaychronoua High Frequency Obaervallona. Keiji Nagai* (Yokoharna National University, Japan), Yoshihiko Nishiyama (Kyoto University, Japan), Kotaro Hitomi (Kyoto Institute of Technology, Japan)

4. Empirical Speclnd Measure Approach to Econometric Analyala of High Frequency Financial Data. Shinsuke Ikeda* (National Graduate Institute for Policy Studies, Japan)

Sponsor: n5A

Bayesian Nonparametrics.

Chair: Lanslot James (Hong Kong University of Science and Technology, Hong Kong)

Bayeaian Nonparametrica ·A Survey of N- Priora, Conaiatency luu" in Eatimation and Testing, Application• to Clustering. Dl speaker: J. K. Ghosh* (Purdue University, USA)

Invited speakers:

I. On Some Asymptotic Features ofSemi-Parametric Baynlan Modela. Judith Rousseau• (Universi~ Paris Dauphine, France)

DL-15 [Convention Hall 300]

2. Poaterlor Consistency ofSpeclea Sampling Priors. Gun Ho Jang (University of Pennsylvania, USA), Jaeyong Lee• (Seoul National University, Korea), Sangyeol Lee (Seoul National University, Korea)