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Interacting Particle Systems for the

Computation of CDO Tranche Spreads

with Rare Defaults ∗

Douglas Vestal†Ren´e Carmona‡& Jean-Pierre Fouque §

January 24, 2008

Abstract

We propose an Interacting Particle System method to accurately cal-

culate the distribution of the losses in a highly dimensional portfolio by

using a selection and mutation algorithm. We demonstrate the eﬃciency

of this method for computing rare default probabilities on a toy model

for which we have explicit formulas. This method has the advantage of

accurately computing small probabilities without requiring the user to

compute a change of measure as in the Importance Sampling method.

This method will be useful for computing the senior tranche spreads in

Collateralized Debt Obligations (CDOs).

1 Introduction

The past few years have seen an explosion in the credit risk market. At the

same time, the ﬁeld of credit risk and credit derivatives research has substan-

tially increased. As the credit derivative products have grown in complexity, so

has the need for fast and accurate numerical methods to accurately price such

derivatives.

In this paper, we consider the pricing of CDOs under the ﬁrst passage model.

A CDO is a credit derivative that pools together many diﬀerent ﬁrms (125 is a

typical contract) and sells exposure to diﬀerent default levels of the portfolio;

the so-called tranches. This segmentation of the risk in a portfolio enables

the buyer to purchase only the tranches that they deem appropriate for their

∗Work supported by NSF-FRG-DMS-0455982

†Department of Statistics and Applied Probability, University of California, Santa Barbara,

CA 93106-3110 vestal@pstat.ucsb.edu.

‡Department of Operations Research & Financial Engineering, Princeton University, E-

Quad, Princeton, NJ 08544 rcarmona@princeton.edu.

§Department of Statistics and Applied Probability, University of California, Santa Barbara,

CA 93106-3110 fouque@pstat.ucsb.edu.

1

hedging positions. Since the tranche levels are fairly standardized, there are

also new products called bespoke CDOs that sell customized default levels.

The main diﬃculty in pricing CDOs is the high-dimensional nature of the

problem. To accurately price the CDO, the distribution of the joint default for

many names is needed. Even if the distribution of joint defaults is found explic-

itly, there is no guarantee that the expectations that are then needed to compute

tranche spreads can be found analytically. Therefore, the user has to rely on

numerical schemes to compute the CDO prices. Due to the high-dimensional

nature of the problem, PDE based methods are ruled out and Monte Carlo

(MC) methods are heavily relied upon. While MC methods are easy to imple-

ment in high-dimensional problems, they do suﬀer from slow convergence. In

addition, due to the rare nature of joint defaults, the computational problem

is exacerbated since many MC simulations are needed to observe the joint de-

fault of many names. Therefore, variance reduction and eﬃciency become very

important for these MC methods.

The main variance reduction technique used is importance sampling. There

have been many successful applications of importance sampling towards credit

risk [1, 12]. However, most authors have concentrated on multifactor Gaussian

copula models or the reduced-form model for credit risk. We have not found any

reference in the literature that applies importance sampling to the ﬁrst passage

model. In addition, the diﬃculty with implementing importance sampling is

computing the change of measure under which one simulates the random vari-

ables. More information about importance sampling and the diﬃculty required

can be found in [11] and the references therein.

In this paper we are concerned with ﬁrst passage models where it is imprac-

tical, if not impossible, to compute the importance sampling change of measure

explicitly. Examples we have in mind are ﬁrst passage models with stochastic

volatility (see for instance [10]), and/or regime switching models with many

factors.

In this situation, our solution is to use an Interacting Particle System (IPS)

technique that can brieﬂy be loosely described as applying importance sampling

techniques in path space. That is, we never compute a change of measure ex-

plicitly but rather decide on a judicious choice of a potential function and a

selection/mutation algorithm under the original measure such that the measure

in path space converges to the desired “twisted” measure. This “twisted” mea-

sure will be exactly the change of measure that one would use in an importance

sampling scheme. The advantage is that the random processes are always simu-

lated under the original measure, but through the selection/mutation algorithm

converge to the desired importance sampling measure. The IPS techniques and

theoretical support we use to design our algorithm can be found in the book [5]

where Pierre Del Moral developed the theory of IPS, and in [6] which provides

applications as well as a toy model very similar to the one used in this paper in

the one-dimensional case.

In this paper, we are interested in the intersection of two events; a compli-

cated model that doesn’t lend itself to importance sampling, and the computa-

tion of rare probabilities under such a model. The rest of the paper is organized

2

as follows. Section 2 discusses the ﬁrst passage model that we will be using as

a toy model. Section 3 gives an overview of Feynman-Kac measures and the

associated IPS interpretation. Section 4 provides the algorithm we propose and

outlines its implementation on our toy model. Section 5 discusses the numerical

results of using IPS on our toy model and the comparison with traditional MC.

2 Problem Formulation

In contrast to intensity-based approaches to credit risk where default is given

by an exonously deﬁned process, default in the ﬁrm value model has a very nice

economical appeal. The ﬁrm value approach, or structural model, models the

total value of the ﬁrm that has issued the defaultable bonds. Typically, the

value of the ﬁrm includes the value of equity (shares) and the debt (bonds) of

the ﬁrm [15]. There are two main approaches to modeling default in the ﬁrm

value approach: one is that default can only happen at the maturity of the

bond, and the second is that default can occur any time before maturity. The

latter is referred to as the ﬁrst-passage model and is the one we consider in this

paper.

2.1 Review of the First Passage Model

We follow both [14, 2] and assume that the value of the ﬁrm, St, follows geomet-

ric Brownian motion. We also assume that interest rates are constant. Under

the risk-neutral probability measure Pwe have,

dS(t) = rS(t)dt +σS(t)dW (t),(1)

where ris the risk-free interest rate, and σis constant volatility. At any time

t≤T, the price of the unit-notional nondefaultable bond is Γ(t, T ) = e−r(T−t).

We also assume that at time 0, the ﬁrm issued non-coupon corporate bonds

expiring at time T. The price of the defaultable bond at time t≤Tis denoted

by ¯

Γ(t, T ).

In [14] default was assumed to only occur at the expiration date T. Fur-

thermore, default at time Tis triggered if the value of the ﬁrm was below some

default threshold B; that is if ST≤B. Therefore, assuming zero recovery, the

price of the defaultable bond satisﬁes,

¯

Γ(t, T ) = Ehe−r(T−t)1ST>B |Sti

= Γ(t, T )P(ST> B|St)

= Γ(t, T )N(d2),

where N(·) is the standard cumulative normal distribution function and,

d2=ln St

B+ (r−1

2σ2)(T−t)

σ√T−t.

3

The next model, the Black-Cox model, is also known as the ﬁrst passage ap-

proach. Developed in [2], default can occur anytime before the expiration of

the bond and the barrier level, B(t), is some deterministic function of time. In

[2], they assume that the default barrier is given by the function B(t) = Keηt

(exponentially increasing barrier) with K > 0 and η≥0. The default time τis

deﬁned by

τ= inf {t:St≤B(t)}.

That is, default happens the ﬁrst time the value of the ﬁrm passes below the

default barrier. This has the very appealing economical intuition that default

occurs when the value of the ﬁrm falls below its debt level, thereby not allowing

the ﬁrm to pay oﬀ its debt.

Assuming zero recovery, we can price the defaultable bond by pricing a

barrier option. Therefore,

¯

Γ(t, T ) = 1τ >tEhe−r(T−t)1τ >T |Sti

=1τ >tΓ(t, T )P(τ > T |St)

=1τ >tΓ(t, T )N(d+

2)−St

B(t)p

N(d−

2),

where

d±

2=±ln St

B(t)+ (r−η−1

2σ2)(T−t)

σ√T−t,

p= 1 −2(r−η)

σ2.

In addition, we denote the probability of default for the ﬁrm between time t

and time Tby P(t, T ). Hence,

P(t, T ) = 1 −N(d+

2)−St

B(t)p

N(d−

2).(2)

The yield spread of the defaultable bond is deﬁned as

Y(t, T ) = −1

T−tln ¯

Γ(t, T )

Γ(t, T ).

We remark that in the ﬁrst passage model above, and all ﬁrm value models,

the yield spread for very short maturities goes to 0 in contrast to the empirical

evidence found in [8]. However, by incorporating a fast mean-reverting stochas-

tic volatility σt, the authors in [9] were able to raise the yield spread for short

maturities.

2.2 Multiname Model

For our purposes, we consider a CDO written on Nﬁrms under the ﬁrst passage

model. That is, we assume that the ﬁrm values for the Nnames have the

4

following dynamics,

dSi(t) = rSi(t)dt +σiSi(t)dWi(t), i = 1,...,N (3)

where ris the risk-free interest rate, σiis constant volatility, and the driving

innovations dWi(t) are inﬁnitesimal increments of Wiener processes Wiwith

correlation

dhWi, Wjit=ρijdt.

Each ﬁrm iis also assumed to have a deterministic boundary process Bi(t) and

default for ﬁrm iis given by

τi= inf {t:Si(t)≤Bi(t)}.(4)

We deﬁne the loss function as

L(T) =

N

X

i=1

1{τi≤T}.(5)

That is, L(T) counts the number of ﬁrms among the Nﬁrms that have defaulted

before time T. We remark that in the independent homogeneous portfolio case,

the distribution of L(T) is Binomial(N,P(0, T )) where P(0, T ) is deﬁned in

(2).

It is well known (see for instance [7]) that the spread on a single tranche can

be computed from the knowledge of expectations of the form,

E{(L(Ti)−Kj)+},

where Tiis any of the coupon payment dates, and Kjis proportional to the

attachment points of the tranche.

The most interesting and challenging computational problem is when all of

the names in the portfolio are correlated. In [18], the distribution of losses

for N= 2 is found by ﬁnding the distribution of the hitting times of a pair

of correlated Brownian motions. However, the distribution is given in terms

of modiﬁed Bessel functions and a tractable general result for N > 2 is not

available. Since the distribution of L(T) is not known in the dependent case,

for N > 2, Monte Carlo methods are generally used to calculate the spread on

the tranches. Since Nis typically very large (125 names is a standard contract),

PDE based methods are ruled out and one has to use Monte Carlo.

Instead of computing the spread on the tranches numerically, our goal is to

calculate the probability mass function for L(T), that is calculate

P(L(T) = i) = pi(T), i = 0,...,N. (6)

In this manner, we will then be able to calculate all expectations that are a

function of L(T), not just the spreads. In addition, as the reader will see, our

method is dynamically consistent in time so that we can actually calculate, for

all coupon dates Tj≤T,

P(L(Tj) = i) = pi(Tj), i = 0,...,N,

5

with one Monte Carlo run. This is in contrast to a lot of importance sampling

techniques where the change of measure has a dependence on the ﬁnal time

through the Girsanov transformation thereby requiring a diﬀerent MC run for

each coupon date.

3 Feynman-Kac Path Measures and IPS

Feynman-Kac path measures, and their subsequent interacting particle system

interpretation, are closely related to the stochastic ﬁltering techniques used in

mathematical ﬁnance. In this paper, we adapt an original interacting particle

system developed in [6] to the computation of the probability mass function

(6). In [6], the authors develop a general interacting particle system method

for calculating the probabilities of rare events. For the sake of completeness,

we provide a fairly thorough overview of IPS as developed in [6] and the main

results that will be the foundation of this paper. Brieﬂy, the method can be

described as formulating a Markov process and conducting a mutation-selection

algorithm so that the chain is “forced” into the rare event regime.

We suppose that we have a continuous time non-homogeneous Markov chain,

(˜

Xt)t∈[0,T ]. However, for the variance analysis and to foreshadow the method

ahead, we only consider the chain (Xp)0≤p≤n= ( ˜

XpT/n)0≤p≤n, where nis ﬁxed.

The chain Xntakes values in some measurable state space (En,En) with Markov

transitions Kn(xn−1, dxn). We denote by Ynthe historical process of Xn, that

is

Yndef.

= (X0,...,Xn)∈Fndef.

= (E0× · ·· × En).

Let Mn(yn−1, dyn) be the Markov transitions associated with the chain Yn. Let

Bb(E) denote the space of bounded, measurable functions with the uniform

norm on some measurable space (E , E). Then, given any fn∈Bb(Fn), and the

pair of potentials/transitions (Gn, Mn), we have the following Feynman-Kac

measure deﬁned by

γn(fn) = E

f(Yn)Y

1≤k<n

Gk(Yk)

.(7)

We denote by ηn(·) the normalized measure deﬁned as

ηn(fn) =

Ehf(Yn)Q1≤k<n Gk(Yk)i

EhQ1≤k<n Gk(Yk)i=γn(fn)/γn(1).(8)

In addition, in [6] they assume that the potential functions are chosen such

that

sup

(yn,¯yn)∈F2

n

Gn(yn)/Gn(¯yn)<∞.

However, the authors note that this condition can be relaxed by considering tra-

ditional cut-oﬀ techniques, among other techniques (see [6, 5] for more details).

6

A very important observation is that

γn+1(1) = γn(Gn) = ηn(Gn)γn(1) =

n

Y

p=1

ηp(Gp).

Therefore, given any bounded measurable function fn, we have

γn(fn) = ηn(fn)Y

1≤p<n

ηp(Gp).

The above relationship is crucial because it allows us to relate the un-normalized

measure in terms of only the normalized “twisted” measures. In our study, we

will also make use of the distribution ﬂow (γ−

n, η−

n) deﬁned exactly the same

way as (γn, ηn) except we replace Gpby its inverse

G−

p= 1/Gp.

Then, using the deﬁnition for γnand ηnit is easy to see that E[fn(Yn)] admits

the following representation,

E[fn(Yn)] = E

fn(Yn)Y

1≤p<n

G−

p(Yp)×Y

1≤p<n

Gp(Yp)

=γn

fnY

1≤p<n

G−

p

=ηn

fnY

1≤p<n

G−

p

Y

1≤p<n

ηp(Gp).

Finally, it can be checked that the measures (ηn)n≥1satisfy the nonlinear re-

cursive equation

ηn= Φn(ηn−1)def.

=ZFn−1

ηn−1(dyn−1)Gn−1(yn−1)Mn(yn−1,·)/ηn−1(Gn−1),

starting from η1=M1(x0,·).

3.1 IPS Interpretation and General Algorithm

The above deﬁnitions and results lend themselves to a very natural interacting

path-particle interpretation. We denote the Markov chain taking values in the

product space FM

nwith transformation Φnby ξn= (ξi

n)1≤i≤M, for each time

n≥1. One constructs a numerical algorithm so that each path-particle

ξi

n= (ξi

0,n, ξ i

1,n,...,ξi

n,n)∈Fn= (E0× · · · × En),

is sampled almost according to the twisted measure ηn.

7

We start with an initial conﬁguration ξ1= (ξi

1)1≤i≤Mthat consists of M

independent and identically distributed random variables with distribution,

η1(d(y0, y1)) = M1(x0, d(y0, y1)) = δx0(dy0)K1(y0, dy1),

i.e., ξi

1

def.

= (ξi

0,1, ξi

1,1) = (x0, ξi

1,1)∈F1= (E0×E1). Then, the elementary

transitions ξn−1→ξnfrom FM

n−1into FM

nare deﬁned by

P(ξn∈d(y1

n,...,yM

n)|ξn−1) =

M

Y

j=1

Φn(m(ξn−1))(dyi

n),(9)

where m(ξn−1)def.

=1

MPM

i=1 δξi

n−1, and d(y1

n,...,yM

n) is an inﬁnitesimal neigh-

borhood of the point (y1

n,...,yM

n)∈Fm

n. From the deﬁnition of Φn, one can

see that (9) is the overlapping of a simple selection and mutation transition,

ξn−1∈FM

n−1

selection

−→ ˆ

ξn−1∈FM

n−1

mutation

−→ ξn∈FM

n.

The selection stage is performed by choosing randomly and independently M

path-particles

ˆ

ξi

n−1= (ˆ

ξi

0,n−1,ˆ

ξi

1,n−1,...,ˆ

ξi

n−1,n−1)∈Fn−1,

according to the Boltzmann-Gibbs particle measure

M

X

j=1

Gn−1(ξj

0,n−1,...,ξj

n−1,n−1)

PM

i=1 Gn−1(ξi

0,n−1,...,ξi

n−1,n−1)δ(ξj

0,n−1,...,ξj

n−1,n−1).

Then, for the mutation stage, each selected path-particle ˆ

ξi

n−1is extended by

ξi

n= ((ξi

0,n,...,ξi

n−1,n), ξi

n,n)

= ((ˆ

ξi

0,n,..., ˆ

ξi

n−1,n), ξi

n,n)∈Fn=Fn−1×En,

where ξi

n,n is a random variable with distribution Kn(ˆ

ξi

n−1,n−1,·). In other

words, the transition is made by applying the original kernel Kn. All of the

mutations are performed independently. We just quote the results from [5, 6]

in stating the weak convergence result:

ηM

n

def.

=1

M

M

X

i=1

δ(ξi

0,n,ξi

1,n,...,ξi

n,n)

N→∞

−→ ηn.

Furthermore, there are several propagation of chaos estimates that ensure that

(ξi

0,n, ξ i

1,n,...,ξi

n,n) are asymptotically independent and identically distributed

with distribution ηn[5]. Therefore, we can form the particle approximation γM

n

deﬁned as

γM

n(fn) = ηM

n(fn)Y

1≤p<n

ηM

p(Gp).

8

Lemma 1 ([6]).γM

nis an unbiased estimator for γn, in the sense that for any

p≥1and fn∈Bb(Fn)with ||fn|| ≤ 1, we have

E(γM

n(fn)) = γn(fn),

and in addition

sup

M≥1

√ME[|γM

n(fn)−γn(fn)|p]1/p ≤cp(n),

for some constant cp(n)<∞whose value does not depend on the function fn.

Proof. Refer to [6].

4 Pricing CDOs using IPS

In this section, we present our adaptation of the interacting particle system

approach to computing rare event probabilities in credit risk with a structural

based approach by applying it to the following model.

Our Markov process is the 3 ×Ndimensional process ( ˜

Xt)t∈[0,T ]deﬁned as:

˜

Xt=S1(t),min

u≤tS1(u),1τ1≤t, S2(t),min

u≤tS2(u),1τ2≤t,···

···, SN(t),min

u≤tSN(t),1τN≤t,

where the dynamics of Si(t) are given in equation (3). We assume a constant

barrier Bifor each ﬁrm 1 ≤i≤N. While it is redundant to also include

1τi≤tin the above expression since we also know minu≤tSi(u), we keep track

of it because it will tell us the default time of the ﬁrm when we implement the

algorithm numerically. We divide the time interval [0, T ] into n equal intervals

[ti−1, ti], i= 1,2,...,n. These are the times we stop and perform the selection

step. We introduce the chain (Xp)0≤p≤n= ( ˜

XpT/n)0≤p≤nand the whole history

of the chain is denoted by Yp= (X0,...,Xp).

Since it is not possible to sample directly from the distribution of (Xp)0≤p≤n

for N > 2, we will have to apply an Euler scheme during the mutation stage;

we let △tdenote the suﬃciently small time step used. In general △twill be

chosen so that △t << T /n.

Our general strategy is to ﬁnd a potential function that increases the likeli-

hood of default among the ﬁrms. In the IPS algorithm, given a particular choice

of weight function G(·), particles with low scores are replaced by particles with

high scores. Therefore, we would like to select a potential function G(·) that

places a higher score on ﬁrms which have reduced their distance to default dur-

ing the previous mutation step. Since the rare event in this case is that the

minimum of the ﬁrm value falls below a certain level, we would like to put more

emphasis on particles whose ﬁrm minimums are decreasing during a mutation

step. Therefore, we ﬁx some parameter α < 0, and deﬁne the potential function,

Gα(Yp) = exp[α(V(Xp)−V(Xp−1))],(10)

9

where

V(Xp) =

N

X

i=1

log(min

u≤tp

Si(u)).

The choice of α < 0 may seem peculiar initially, but it is chosen to be

negative because the potential function Gα(Yp) can be written in the form,

Gα(Yp) = exp[α(V(Xp)−V(Xp−1))]

= exp "N

X

i=1

αlog min

u≤tp

(Si(u)/min

u≤tp−1

Si(u)#,

where,

log min

u≤tp

(Si(u)/min

u≤tp−1

Si(u)≤0.

Therefore, to place more weight on the ﬁrms whose minimum has decreased, we

must multiply by α < 0. In addition, by choosing the weight function as we did,

if the minimum value did not decrease during the last mutation, then that ﬁrm

has a small relative contribution to the total empirical measure. Therefore, we

will be putting more weight onto path-particles whose minimum has decreased

the most between two mutation times.

In addition, there are several computational advantages for choosing the

weight function above. Chieﬂy among them are:

1. Our choice of weight function, while not unique in this regard, will only

require us to keep track of (Xp−1, Xp) instead of the full history Yp=

(X0, X1,···, Xp) thereby minimizing the increased dimensionality of using

an IPS scheme.

2. In addition, our weight function has the added advantage of having the

property that Q1≤k<p G(Yk) = exp[α(V(Xp−1)−V(X0))] thereby ensur-

ing that the Feynman-Kac measures deﬁned in equations (7) and (8) are

simpler to analyze.

4.1 Detailed IPS Algorithm

Our algorithm is built with the weight function deﬁned in equation (10).

Initialization. We start with Midentical copies, ˆ

X(i)

0, 1 ≤i≤M, of the

initial condition X0. That is,

ˆ

X(i)

0= (S1(0), S1(0),0, S2(0), S2(0),0,···, SN(0), SN(0),0),1≤i≤M.

We also have a set of “parents”, ˆ

W(i)

0, deﬁned by ˆ

W(i)

0=ˆ

X(i)

0. We denote

V0

def.

=V(ˆ

W(i)

0). This forms a set of Mparticles ( ˆ

W(i)

0,ˆ

X(i)

0), 1 ≤i≤M.

Now suppose that at time p, we have the set of Mparticles ( ˆ

W(i)

p,ˆ

X(i)

p),

1≤i≤M.

10

Selection Stage

We ﬁrst compute the normalizing constant,

ˆηM

p=1

M

M

X

i=1

exp hαV(ˆ

X(i)

p)−V(ˆ

W(i)

p)i.(11)

Then, we choose independently Mparticles according to the empirical dis-

tribution,

ηM

p(dˇ

W , d ˇ

X) = 1

MˆηM

p

M

X

i=1

exp hαV(ˆ

X(i)

p)−V(ˆ

W(i)

p)i×δ(ˆ

W(i)

p,ˆ

X(i)

p)(dˇ

W , d ˇ

X).

(12)

The particles that are selected are denoted ( ˇ

W(i)

p,ˇ

X(i)

p).

Mutation Stage

For each of the selected particles, ( ˇ

W(i)

p,ˇ

X(i)

p), we apply an Euler scheme

from time tpto time tp+1 with step size △tfor each ˇ

X(i)

pso that ˇ

X(i)

pbecomes

ˆ

X(i)

p+1. We then set ˆ

W(i)

p+1 =ˇ

X(i)

pIt should be noted, that each of the particles

are evolved independently and that the true dynamics (given in equation (3))

of Xpis applied rather than some other measure. It is this fact that separates

IPS from IS (Importance Sampling).

Then let,

f(ˆ

X(i)

n) =

N

X

j=1

1{minu≤TS(i)

j(u)≤Bj}

denote the number of ﬁrms that have defaulted by time Tfor the ith particle.

Then, the estimator for P(L(T) = k) = pk(T) is given by

PM

k(T) = "1

M

M

X

i=1

1f(ˆ

X(i)

n)=kexp(−α(V(ˆ

W(i)

n)−V0))#×"n−1

Y

p=0

ˆηM

p#.(13)

This estimator is unbiased in the sense that E[PM

k(T)] = pk(T). The unbiased-

ness follows directly from Lemma 1.

4.2 Single-Name Case: Variance Analysis

We analyze the variance of the estimator in equation (13) for the single name

case. Therefore, we take N= 1, with constant barrier Band we are interested

in computing, using IPS, the probability of default before maturity T. That is,

we compute

PB(0, T ) = P(min

u≤TS(u)≤B) = E[1minu≤TS(u)≤B].

Of course, we have an explicit formula for PB(0, T ) given by (2), and this case

will be precisely our toy model used to compare the variance for IPS and pure

11

MC. In the more general case, where Nis large and the names are correlated,

we will provide an empirical comparison. It should be noted that we are only

interested in values of Bthat make the above event rare.

We remark that it is a standard result that the variance associated with

the traditional Monte Carlo method for computing PB(0, T ) is PB(0, T )(1 −

PB(0, T )). We also remark that for a single name the Markov chain (Xp)0≤p≤n

deﬁned in Section 4 simpliﬁes to

Xp= (S(tp),min

u≤tp

S(u),1τ≤tp).

Then, following the setup described in Section 3, we see that the rare event

probability PB(0, T ) has the following Feynman-Kac representation:

PB(0, T ) = γn(L(B)

n(1)),

where L(B)

n(1) is given by the weighted indicator function deﬁned for any path

yn= (x0,...,xn)∈Fnby

L(B)

n(1)(yn) = L(B)

n(1)(x0,...,xn) = 1{minu≤TS(u)≤B}Y

1≤p<n

G−

p(x0,...,xp)

=1{minu≤TS(u)≤B}e−α(V(xn−1)−V(x0))

=1{minu≤TS(u)≤B}e−α(log(minu≤tn−1S(u)/S0)).

Also, notice that ||L(B)

n(1)(yn)|| ≤ 1 since log(minu≤tn−1S(u)/S0)≤0 and

−α > 0 by assumption.

Next, following the IPS selection-mutation algorithm outlined in Section 4.1,

we form the estimator

PB

M(0, T ) = γM

n(L(B)

n(1)) = ηM

n(L(B)

n(1)) Y

1≤p<n

ηM

p(Gp).(14)

By Lemma 1, PB

M(0, T ) is an unbiased consistent estimator of PB(0, T ). While

many estimators are unbiased, the key to determining the eﬃciency of our es-

timator is to look at its variance. As such, we have the following central limit

theorem for our estimator.

Theorem 1. The estimator PM

B(0, T )given in equation (14) is unbiased, and

it satisﬁes the central limit theorem

√ME[PB

M(0, T )−PB(0, T )] M→∞

−→ N(0, σB

n(α)2),

with the asymptotic variance

σB

n(α)2

=

n

X

p=1 hEneαlog(minu≤tp−1S(u))o×EnP2

B,p,n e−αlog(minu≤tp−1S(u))o−PB(0, T )2i,

(15)

12

where PB,p,n is the collection of functions deﬁned by

PB,p,n(x) = E1mint≤TS(t)≤B|Xp=x,

and PB(0, T )is given by (2).

Proof. The proof follows directly by applying Theorem 2.3 in [6] with the weight

function that we have deﬁned in (10).

In the constant volatility single-name case, the asymptotic variance σB

n(α)2

can be obtained explicitly in terms of double and triple integrals with respect

to explicit densities. This will be used in our comparison of variances for IPS

and pure MC in Section 5.1. The details of these explicit formulas are given in

the Appendix A.

As shown numerically in the next section the variance for IPS is of order

p2with p=PB(0, T ) (small in the regime of interest), in contrast to being of

order pfor the direct MC simulation. This is indeed a very signiﬁcant variance

reduction in the regime psmall, as already observed in [6], in a diﬀerent context.

5 Numerical Results

In this section we investigate numerically the results of implementing the IPS

procedure for estimating the probability mass function of the loss function for

single names and multinames.

5.1 Single-Name Case

For the single-name case, we compute the probability of default for diﬀerent

values of the barrier using IPS and traditional Monte Carlo. In addition, for each

method, we implemented the continuity correction for the barrier level described

in [4] to account for the fact that we are using a discrete approximation to the

continuous barrier for both IPS and MC. For the diﬀerent values of the barrier

we use, we can calculate the exact probability of default from equation (2).

The following are the parameters we used for both IPS and MC.

rσ S0△tT n (# of mutations in IPS) M

.06 .25 80 .001 1 20 20000

/Users/dougvestal/Desktop/IPS Paper/VCF-paper.tex The number of sim-

ulations Mis the same for IPS and MC, and from an empirical investigation,

we chose α=−18.5 in the IPS method. The results are shown in Figure 1.

Indeed probabilities of order 10−14 will be irrelevant in the context of default

probabilities but the user can see that IPS is capturing the rare events proba-

bilities for the single name case whereas traditional Monte Carlo is not able to

capture these values below 10−4.

In Figure 2 we show how the variance decreases with the barrier level, and

therefore with the default probability, for MC and IPS. In the IPS case the

13

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Barrier/S0

Probabilitiy of Default

MC vs IPS

IPS

MC

True Value

Figure 1: Default probabilities for diﬀerent barrier levels for IPS and MC

variance is obtained empirically and using the integral formulas derived in the

Appendix. We deduce that the variance for IPS decreases as p2(pis the default

probability), as opposed to pin the case of MC simulation.

Each MC and IPS simulation gives an estimate of the probability of default

(whose theoretical value does not depend on the method) as well as an estimate

of the standard deviation of the estimator (whose theoretical value does depend

on the method). Therefore, it is instructive from a practical point of view to

compare the two methods by comparing the empirical ratios of their standard

deviation to the probability of default for each method. If p(B) is the probability

of default for a certain barrier level B, then the standard deviation, p2(B), for

traditional MC is given by,

pMC

2(B) = pp(B)×p(1 −p(B)),

and the theoretical ratio for MC is given by

pMC

2(B)

p(B)=p(1 −p(B))

pp(B),

which can computed using (2).

For IPS, the corresponding ratio is

pIPS

2(B)

p(B)=σB

n(α)

p(B),

14

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

10−30

10−25

10−20

10−15

10−10

10−5

100

105

B/S0

Variance

Comparison of MC with IPS

IPS Empirical Variance

IPS Theoretical Variance

MC Theoretical Variance

MC prob. default squared

Figure 2: Variances for diﬀerent barrier levels for IPS and MC

where σB

n(α) is given in Theorem 1. It is computed using the formula given in

Corollary 1 in the Appendix.

In Figure 3 one sees that there are speciﬁc regimes where it is more eﬃcient

to use IPS as opposed to traditional MC for certain values of the barrier level

(below .60 ×S0). This is to be expected since IPS is well suited to rare event

probabilities whereas MC is not.

5.2 Multiname Case

For the multiname case, we tested using 25 ﬁrms (N= 25). In addition, we took

all of the ﬁrms to be homogeneous, meaning that they have the same parame-

ters, starting value, and default barrier in (3). The following are the parameters

that we used:

r σiSi(0) Bi△tT n M

.06 .3 90 36 .001 1 20 10000

In addition, we took the correlation between the driving Brownian motions

to be ρij =.4 for i6=j. The above parameters give, in the independent case,

a probability of default of .0018, a realistic default probability for highly rated

ﬁrms. For the IPS simulation we used α=−18.5/25 so that it is consistent

with the value used in the single-name case.

The following picture illustrates the diﬀerence between using MC and IPS to

estimate the Loss probability mass function. That is, we calculate numerically

15

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

100

101

102

103

104

105

106

107

B/S0

p2(x)/p(x)

Standard Deviation to Probability Ratio for MC and IPS

IPS Theoretical

IPS Empirical

MC Theoretical

MC Emprical

Figure 3: Standard deviation-to-probability ratio for MC and IPS

P(L(T) = k) where L(T) is the number of ﬁrms that have defaulted before time

Tas deﬁned in (5). In Figure 4 we plot the pmf on a log scale more adapted

to our range of values. One can see that the IPS method is picking up more of

the tail events for the pmf of the loss function and the distinction between IPS

and MC becomes clear. in this regime, MC in only good for estimating the pmf

for K= 0,1,2,3,4, but IPS is good for estimating the pmf for K= 0,1,...,10.

Considering the fact that most contracts are written only on the ﬁrst 40% of

losses and 10 = .4×25, we see that IPS does a good job of describing the rare,

but economically (in the sense of contracts) signiﬁcant events.

6 Conclusion

In this paper, we adapted an original IPS approach to the computation of rare

probabilities [6] to the ﬁeld of credit risk under the ﬁrst passage model. We

showed that with our choice of weight function, IPS performs better than tra-

ditional MC methods for computing the probability of tail events in credit risk

under a simple toy model. We also derived an explicit formula (up to double

and triple integrals) for the asymptotic variance of the IPS estimator in the

single-name case. In addition, we also showed that there are speciﬁc regimes

where IPS is better suited to credit risk than traditional MC methods. For

16

0 2 4 6 8 10

10−10

10−8

10−6

10−4

10−2

100

Number of Defaults (K)

P(Loss = K)

Density function for Loss given by different numerical procedures

MC

IPS

Figure 4: Probability mass function of the Loss shown in log-scale, with N= 25.

practical purposes our algorithm can be adapted to more complicated models,

for instance, with stochastic volatility [10].

Appendix

A Formulas for the Variance in the Single-Name

Constant Volatility Case

The asymptotic variance σB

n(α)2of the IPS estimator is given by (15) in The-

orem 1. As a Corollary, we deduce in this Appendix the explicit formulas used

in Section 5.1.

Corollary 1. Given the choice of weight function in (10) with α < 0, and

17

constant barrier B, we have that

σB

n(α)2

=

n

X

p=1 "f(α, tp−1)e−1

2θ2tp Z(1

σ) ln(B/S0)

−∞ Z0

yZ∞

y

e−ασx+θz Ψ(tp−1,tp)(x, y, z )dzdxdy

+Z(1

σ) ln(B/S0)

−∞ Z∞

x

e−ασx+θz Υ(tp−1,tp)(x, z )dzdx

+Z0

(1

σ) ln(B/S0)Z0

yZ∞

y

h(tp, z)2e−ασx+θ z Ψ(tp−1,tp)(x, y, z )dzdxdy

+Z0

(1

σ) ln(B/S0)Z∞

x

h(tp, z)2e−ασx+θ z Υ(tp−1,tp)(x, z)dzdx!

− 1−N(d+

2(0,0)) + S0

B1−2r

σ2

N(d−

2(0,0))!2#,(16)

where,

f(α, tp−1) = ασ +θ

ασ + 2θeασ(ασ+2θ)tp−1/2Erfc (ασ +θ)√tp−1

√2+θ

ασ + 2θErfc −θ√tp−1

√2,

Erfc(x) = 2

√πZ∞

x

e−v2dv,

θ=r−1

2σ2

σ,

Ψ(tp−1,tp)(x, y, z)

=2√2e−(2x−2y+z)2/2tp

qπtpt2

p−1(tp−tp−1)2×"σ(tp−1,tp)(µ(1)

(tp−1,tp)(x, y, z)−x+z−2y)

√2πe−(x−µ(1)

(tp−1,tp)(x,y,y))2/2σ2

(tp−1,tp)

+σ2

(tp−1,tp)+ (µ(1)

(tp−1,tp)(x, y, z))2

+µ(1)

(tp−1,tp)(x, y, z)(z−2y−2x)−2x(z−2y)

1−Φ

x−µ(1)

(tp−1,tp)(x, y, z)

σ(tp−1,tp)

#,

18

Υ(tp−1,tp)(x, z)

=s2

πtptp−12×"e−(2x−z)2/2tp σ(tp−1,tp)

√2πe−(x−µ(2)

(tp−1,tp)(x,z))2/2σ2

(tp−1,tp)

+(µ(2)

(tp−1,tp)(x, z)−2x)1−Φx−µ(2)

(tp−1,tp)(x, z)

σ(tp−1,tp)!

−e−z2/2tp σ(tp−1,tp)

√2πe−(x−µ(3)

(tp−1,tp)(x,z))2/2σ2

(tp−1,tp)

+(µ(3)

(tp−1,tp)(x, z)−2x)1−Φx−µ(3)

(tp−1,tp)(x, z)

σ(tp−1,tp)!#,

Φ(x) = Zx

−∞

1

√2πe−y2/2dy,

µ(1)

(tp−1,tp)(x, y, z) = 2x(tp−tp−1) + (2y−z)tp−1

tp

,

µ(2)

(tp−1,tp)(x, z) = 2x(tp−tp−1) + ztp−1

tp

,

µ(3)

(tp−1,tp)(x, z) = 2xtp−ztp−1

tp

,

σ2

(tp−1,tp)=tp−1

tp

(tp−tp−1),

h(tp, z) = 1−N(d+

2(tp, z)) + S0eσz

B1−2r

σ2

N(d−

2(tp, z))!,

d±

2(tp, z) = ±(ln(S0) + σz −ln(B)) + (r−1

2σ2)(T−tp)

σpT−tp

.

Proof. As can be seen in (15), we need to compute the joint distribution of

min

u≤tp−1

S(u),min

u≤tp

S(u), S(tp).

First, recall that since S(u) = S0e(r−1

2σ2)u+σWu,σ > 0 by assumption, and

log is an increasing function we have:

Eneαlog(minu≤tp−1S(u))o=Eeαlog“minu≤tp−1S0e(r−1

2σ2)u+σWu)”

=Sα

0Eneα(minu≤tp−1(r−1

2)u+σWu)o

=Sα

0Eneασ(minu≤tp−1c

Wu)o,

19

where c

Wu=θu +Wuis a Brownian motion with deterministic drift θ=

r−1

2σ2

σ. Therefore, the above computation simpliﬁes to computing the mo-

ment generating function of the running minimum of Brownian motion with

drift. This formula is well known (see for instance [3]) and so we have,

Eneαlog(minu≤tp−1S(u))o

=Sα

0ασ +θ

ασ + 2θeασ(ασ+2θ)tp−1/2Erfc (ασ +θ)√tp−1

√2+θ

ασ + 2θErfc −θ√tp−1

√2

:= Sα

0f(α, tp−1),(17)

where

Erfc(x) = 2

√πZ∞

x

e−v2dv.

Now, we compute EnP2

B,p,ne−αlog(minu≤tp−1S(u))o. The general goal will be to

write everything in terms of expectations of functionals of c

Wu. First, we note

that

PB,p,n(x) = E1minu≤TS(u)≤B|Xp=x

=E1minu≤TS(u)≤B|Xp= (min

u≤tp

S(u), S(tp),1τ≤tp)

=1minu≤tpS(u)≤B+1minu≤tpS(u)>BP( min

tp≤u≤TS(u)≤B|S(tp))

=1minu≤tpS(u)≤B+1minu≤tpS(u)>B 1−N(d+

2) + S(tp)

B1−2r

σ2

N(d−

2)!,

where

d±

2=±ln S(tp)

B+ (r−1

2σ2)(T−tp)

σpT−tp

.

In the formula for d±

2we will ﬁnd it useful to substitute the formula S(tp) =

S0eσc

Wtpand write the dependence on tpand c

Wtpexplicitly as,

d±

2(tp,c

Wtp) = ±ln S0eσc

Wtp

B+ (r−1

2σ2)(T−tp)

σpT−tp

=±ln(S0) + σc

Wtp−ln(B)+ (r−1

2σ2)(T−tp)

σpT−tp

,

In addition, we also substitute S(u) = S0eσc

Wuinto the expression for PB,p,n

20

and rearrange to get,

PB,p,n(x) = 1minu≤tpc

Wu≤(1

σ) ln(B/S0)

+1minu≤tpc

Wu>(1

σ) ln(B/S0)

1−N(d+

2(tp,c

Wtp)) + S0eσc

Wtp

B!1−2r

σ2

N(d−

2(tp,c

Wtp))

Hence,

PB,p,n(x)2=1minu≤tpc

Wu≤(1

σ) ln(B/S0)+1minu≤tpc

Wu>(1

σ) ln(B/S0)h(tp,c

Wtp)2,

where

h(tp,c

Wtp) =

1−N(d+

2(tp,c

Wtp)) + S0eσc

Wtp

B!1−2r

σ2

N(d−

2(tp,c

Wtp))

.

Hence, plugging in the expression for P2

B,p,n into EnP2

B,p,ne−αlog(minu≤tp−1S(u))o

we have

EnP2

B,p,ne−αlog(minu≤tp−1S(u))o

=S−α

0En1minu≤tpc

Wu≤(1

σ) ln(B/S0)e−ασ minu≤tp−1c

Wuo

+S−α

0E(1minu≤tpc

Wu>(1

σ) ln(B/S0)h(tp,c

Wtp)2e−ασ minu≤tp−1c

Wu),(18)

where the expectation above is taken with respect to the measure Pfor which

c

Wuis a Brownian motion with drift. Recall that under P,

dc

Wt=θdt +dWt

where Wtis a Pstandard Brownian motion and θ=r−1

2σ2

σ. Using Girsanov’s

theorem (see [13] or [16]), c

Wtis a standard Brownian motion under b

Pand the

Radon-Nikodym density, Z(t), is given by

Z(t) = exp −Zt

0

θdWu−1

2Zt

0

θ2du

= exp −Zt

0

θ(dc

Wu−θu)−1

2θ2t

= exp −θc

Wt+1

2θ2t.

21

Therefore, we rewrite (18) as an expectation under b

Pto get

EnP2

B,p,ne−αlog(minu≤tp−1S(u))o

=S−α

0b

En1minu≤tpc

Wu≤(1

σ) ln(B/S0)e−ασ minu≤tp−1c

WuZ(tp)−1o

+S−α

0b

E(1minu≤tpc

Wu>(1

σ) ln(B/S0)h(tp,c

Wtp)2e−ασ minu≤tp−1c

WuZ(tp)−1)

=S−α

0b

En1minu≤tpc

Wu≤(1

σ) ln(B/S0)e−ασ minu≤tp−1c

Wueθc

Wtp−1

2θ2tpo

+S−α

0b

E(1minu≤tpc

Wu>(1

σ) ln(B/S0)h(tp,c

Wtp)2e−ασ minu≤tp−1c

Wueθc

Wtp−1

2θ2tp)

=S−α

0e−1

2θ2tp ZZZ1y≤(1

σ) ln(B/S0)e−ασx+θzΓ(tp−1,tp)(dx, dy , dz)

+Z Z Z 1y> 1

σln(B/S0)h(tp, z )2e−ασx+θz Γ(tp−1,tp)(dx, dy, dz)!,(19)

where,

Γ(tp−1,tp)(dx, dy, dz) = b

P( min

u≤tp−1c

Wu∈dx, min

u≤tpc

Wu∈dy, c

Wtp∈dz),(20)

and as stated before, c

Wuis a standard Brownian motion under b

P.

The formula for Γ(tp−1,tp)(dx, dy, dz) is given by

Γ(tp−1,tp)(dx, dy, dz) = Ψ(tp−1,tp)(x, y, z )1y<x,y≤z,x≤0dzdydx

+Υ(tp−1,tp)(x, z)1x≤z,x≤0δx(dy)dzdx,

where the functions Ψ(tp−1,tp)and Υ(tp−1,tp)are given in Corollary 1.

The derivation of these formulas is obtained by:

1. Introducing the value of c

Wuat the intermediate time tp−1in (20).

2. Using the Markov property at present time tp−1.

3. Using the classical joint distribution of a Brownian motion and its running

minimum.

4. Re-integrating with respect to c

Wtp−1.

The details of this derivation are in [17]. Substituting the formulas for Ψ(tp−1,tp)

and Υ(tp−1,tp)into (19) ends the proof of Corollary 1.

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23