Let represent the time asset value of the firm on a per share basis which solves the

next stochastic differential equation

where is the asset volatility and is the firm value at time 0 at which we stand. We assume that the

asset value processes have correlations, i.e., , where is the quadratic

covariation. Filtrations generated by observed asset values are denoted by. There is a

random default threshold such that firm default as soon as the asset value falls to the level, where denotes the recovery rate at default and is a positive constant representing debt per share, which may given by accounting reports. Then the default time of the firm is a random variable given by

Here random variables are mutually independent of the. More complicated stochastic processes for such as stochastic volatility may be possible, however, we shed

lights to the multi-name setting and model the so-called default contagion. Let be a right-continuous

process which indicate the default status of the firm at time and we denote by the associated filtration.

With the view to analyzing how the period of past default memories affect succeeding defaults, we consider the incomplete information framework which is known to represent default contagion. In order to depict the incomplete information structure more concretely, in addition to the assumption of the randomness of the default threshold, we postulate the following assumptions.

Assumption 1 Public investors can observe firm values and default events although they can not directly

observe the firm’s default thresholds except for the default time.

Define the set of survived firms and the set of defaulted firms

at the time. We write, the number of elements in the set. Since

investors have knowledge that the surviving firms have lower default barrier than their running minimum of the firm value processes, it is natural to suppose the next assumption.

Assumption 2 At time, we assume every firm in the portfolio are surviving, i.e., and then the

inequality holds for all under the condition.

Let be normally distributed random variable with mean vector

and variance-covariance matrix. Here, , are

some constants. And we assume that be the truncation of above

. We denote.

Remark 1 The definition of the mean vector is given along the line of

original CreditGrades model. Finger, Finkelstein, Pan, Lardy and Ta [12] proposed that the random recovery

rate is modeled as with, where.

Assumption 3 There is a consensus on the prior joint distribution of firm’s default thresholds among the public investors. More concretely, investor’s uncertainty about the default thresholds is expressed by

where is a -dimensional Normal distribution.

Assumption 4 For each default time, public investors update their belief on the joint distribution function of surviving firm’s default thresholds based on the Assumption 3 and newly arrived information, i.e., the realized recovery rate.

Remark 2 Since public investors observe all the history of the firm value, they know that the unobservable threshold should be located below the running minimum of the firm value. Despite these knowledge, we assume that public investors treat the logarithm of the recovery rate as normally distributed random variable.

Assumption 1, Assumption 3 and Assumption 4 provide the default contagion mechanism; The default of a firm reveals information about the default threshold and then public investors update their beliefs on surviving firm’s joint distribution of thresholds. From public investors’ perspective, this naturally causes the sudden change of default probabilities of survived firms, which is just what we wanted to model. The situation of contagious defaults can be translated to the recession, however, it will not continue forever. In our model, we further assume that public investors view the crisis will return to normal condition after some finite time interval.

Assumption 5 The covariance parameter jumps from to at time for some

constants and. This can be captured by introducing time-depending covariance parameters defined as

and then assume that the elements of the variance-covariance matrix are given by (0.4). We call the memory period of after.

Thus the mean vector and the variance-covariance matrix at time can be defined as

Assumption 6 for.

Define the set at time and let be the number of

elements in the set. Rearrange the order of firm identity numbers in such a way that the elements of

come after the elements of and the elements of are located the last. Let be a

submatrix formed by selecting the rows and columns from the subset and let and be corres-

ponding -dimensional vectors respectively.

Assumption 6 implies that during the memory period, public investors remember the firm values at which the defaults occurred. We note that

By virtue of Assumption 3, we can deduce the conditional joint distribution of the default thresholds as follows. Here we don’t eliminate the possibility of simultaneous defaults, i.e., we don’t need to assume.

Proposition 1 Let be a -dimensional vector consists of the logarithm of the realized recovery rate at time. Partition the vector, and the matrix into

where and are dimensional vectors, and are dimensional vectors, is a

matrix, and is a matrix. Then -conditional joint density of is given by

where

Proof 1 From the continuity of the asset process and Equation (0.5), public investors know that

for all defaulted firms and for all survived firms.

Here, whenever defaults occur, let the order of the firms be rearranged in such a way that the elements of come after the elements of. Define the set

with the special case

to be the possible range of the recovery rate vector under the condition of. In particular,

for, takes value. Let. As in the proof of the Lemma 4.1 of [5],

from Bayes’ Theorem,

The last equality holds because is independent of. Hence the joint distribution of the surviving firm’s

logarithm of recovery rates are given by the conditional distribution of given at which the realization

hold for all. Conditional distributions of the multivariate normal distribution are

well known. See for example [13] for details. However, from Assumption 1, public investors have already know the following inequalities hold.

Therefore the conditional distribution should be truncated above given by (0.7).

Remark 3 In the case for all, the problem become quite easy because first-passage time of 1-dimensional Geometric Brownian motion is well known. In fact, in such a case, Giesecke and Goldberg [6]

showed that the counting process has intensity process and they proposed the simulation method

based on the total hazard with the case are not observable.

Let be an ordered default times of. Figure 1 illustrate an example of sequence of

defaults with and the corresponding memory periods. At time 0, since all the firms are active and then

the unconditional joint density of is given by

For example, square bracket bottom of the figure indicate that the random vector should be sampled under the condition at time. At the first default time, updated default threshold is sampled under the condition and this

condition remains effective until. This is shown by a square bracket which

indicate that the random vector should be sampled under the condition

at time. If the second default occurred at time, then the updated default threshold at

should be sampled under the condition. However, at the

third default time, investor’s interest have changed from the first default to the second default

completely, i.e., the memory period of 5 after have finished. Therefore updated default threshold should be

sampled under the condition. Notice also that

In this subsection we see how the conditional distribution of the default threshold change at the default time.

In this subsection we see how the conditional distribution of the default threshold change at the default time.

Suppose that the first default occurred at time. Let denote the unconditional density of

and let denote the conditional density of given. The distributions of

, change at from to then the default probabilities

, which is restricted before, change from

to

Figure 2 and Figure 3 show the conditional distribution of at and with

. Distributions are truncated above the running minimum of the firm value

0.95. We see that the control the contagion impact effectively.

Remark 4 Giesecke [5] showed that does not admit the default intensity. Define the supermartingale with respect to as

where then can be expressed as

where is cumulative distribution function of. Define the nondecreasing process by the Stiel- tjes integral

where is the compensator of. Therefore

Consequently, by introducing the density function of, one can obtain

which implies that is singular because is concentrated on the set, which has

Lebesgue masure 0. Therefore does not admit the representation. This is the reason why

we argue the default probabilities instead of the default intensities in our model.

Let be the time horizon. Partition the interval into subintervals of length. Let be

the discrete time Markov chain given by where, in our model, continuous time

Markov chain is given by

which will be discussed later. In general, the random element takes values in some measurable state space

that can change with. We denote by the transition kernels of the Markov chain

at time, and we denote by the historical process of defined by

Next, we let denote the transition kernels of the inhomogeneous Markov chain. We

finally let be the space of all bounded measurable functions on some measurable space, and we equip with the uniform norm.

The Interacting Particle System consists of a set of path-particles evolving from time

to. The initial generation at time is a set of independent copies of and the system evolves as if strong animals produce many offsprings, however the rest die.

For each, we consider non-negative measurable functions defined on equipped with the product -algebra, and we call these functions as potential functions. We associate to the pair of potentials and

transitions the Feynman-Kac path measure defined for any test function by the

formula

We also introduce the corresponding normalized measure

Note that

Therefore, for any given bounded measurable function, we have

The above relationship has the merit of relating the un-normalized expectations in the left hand side to nor-

malized expectations in the right hand side. Furthermore, for any path, we define the

weighted indicator function

Then we see that

This formula can be applied to the situation that the are the rare events and it can be

computed via computation of normalized measures. It is known (see Del Moral and Garnier [15] ) that the computation of the sequence of normalized measures is achieved by the following non-linear recursive equation

starting from. Equation (0.15) has the differential form

which can be easily seen by substituting

and

into the integrand of the right hand side of (0.15).

For the purpose of numerical computations of the rare events of the form (0.14), we introduce an interacting particle system. We choose a large integer which we shall interpret as the number of particles. We construct

a Markov chain whose state at time can be interpreted as a set of samples of

particles with respect to the measure,

We start with an initial configuration that consists of independent and identically distri-

buted random samples from the distribution

i.e., where the are drawn independently of each other from the

distribution. The transitions of particles from into are defined by

where is the empirical measure defined by

and represents an infinitesimal neighborhood of the point. By the definition

of, one sees that (0.16) is the superposition of two identifiable transitions, a selection followed by a mutation as shown below.

The selection stage is performed by choosing randomly and independently particles

according to the Boltzmann-Gibbs measure

During the mutation stage, each selected particle is extended in time by an elementary -transition.

In other words, we set

where is a random variable with distribution. The mutations are performed independently.

A result of [17] reproduced in [15] states that for each fixed time, the empirical measure converges in distribution, as, toward the normalized Feynman-Kac measure, i.e.,

Mimicking (0.13), unbiased particle approximation measures of the un-normalized measure are defined as

and then, by (0.14), we can get the particle approximation of the rare event probabilities. More precisely, if we let

then is an unbiased estimator of the rare event probabilities such that

We refer to Del Moral [17] and Del Moral and Garnier [15] for the details of the convergence results. Their complete proofs rely on a precise propagation of chaos type analysis and they can be found in Section 7.4 on Pages 239-241 and Theorem 7.4.1 on Page 232 in Del Moral [17] .

We introduce special notations and which indicate that these are the input to the selection stage, while another notation and indicate that these are the input to the mutation stage of the IPS algorithm. Here, as we will describe later, indicate the parent of.

According to Del Moral and Garnier [15], one of the recommended potential functions are of the form

for some and suitable function so as to satisfying

This regularity condition ensures that the normalizing constants and the measure are bounded and positive. Thanks to the form of this potential function, we note that we need only to keep track of and, and then the selection would be implemented with those two particles. The form of the distribution (0.17) shows that in order to have more simulation paths realizing great many defaults, it is important to choose a potential function becoming larger as the likelihood of default increases. To meet our purpose, we choose the function as follows.

Then our potential function is given by

The first term of (0.21) reflects the fact that the high weights are assigned to the particle which had renewed the running minimum during the period. When is not random, i.e., the default barrier is

observable, it is known that the IPS is effective to simulate the counting process with reasonable

accuracy by Carmona, Fouque and Vestal [14] . We borrowed the form of potential function from Carmona, Fouque and Vestal [14] . Detailed IPS algorithm is summarized as follows.

Algorithm 2 Assume that we have a set of particles at time denoted by. We

define the Markov process

where, and define the discrete time Markov process.

To generate an estimate of for, perform the following:

Step 0. Initialize the particles and indicators. Choose as a discretized time step for the

firm value processes, to be some small value. We start with a set of i.i.d. initial conditions,

chosen according to the initial distribution of.

Set (initial value of) for all and then form a set of particles

Step 1. For each step, repeat the following steps.

• Selection.

Compute the normalizing constant

and choose independently particles according to the empirical distribution

The new particles are denoted by.

• Mutation.

For every, using the Algorithm 3.1, the particle is transformed into

independently by

and set.

Step 2. The estimator of the probability is given by

It is known that this estimator is unbiased in the sense that and satisfies the central limit

theorem. (Refer to [15] [17] )

Instead of explicit calculation of the asymptotic variance, we notice that the approximate variance

defined by

can be easily calculated within the above IPS algorithm. This provides criteria to choose the parameter to be a suitable level.