Conclusiones del estudio de mercado

In this section we detail the dynamic copula modelling framework first developed by Schu- bert and Schönbucher (2001).

Assume we have

n

obligors and recall in Chapter 4 (Note 3) we demonstrated how a con- ditionally independent framework could be constructed. For obligors

i

∈ {1, . . . , n}

we determined that default times could be constructed as:

τ

i

= inf{t∈R

+

:

Z

t

0

˜

where

ζ

i are random variables with uniform distribution on

[0,1]

. In Chapter 4 (Note 3) we

assumed these random variables were mutually independent, here we relax this assump- tion. Moreover in the models subsequently considered in Chapter 4 we assumed that

(˜λ

i

t

)

t≥0

were deterministic, in this chapter we take

(˜λ

i_t

)

t≥0 to be

F−adapted

continuous stochastic

processes.

We will use the notation

λ

and

˜λ

to denote intensities in the setting where we have full information on the default times of all obligors and the setting where we only have informa- tion on a specific obligor respectively. The process

(˜λ

i

t

)

t≥0 will be shown to be the intensity

of obligor

i

in the setting where we only have information about the occurrence (or not) of default of obligor

i. For this reason Schubert and Schönbucher (2001) call

(˜λ

i

t

)

t≥0 the

pseudo intensity of obligor

i. We will subsequently denote the intensity of obligors defined

on the filtration that contains the default information of all obligors by

(λ

i

t

)

t≥0. We call this

intensity thefull intensity.

In this chapter the setup of Note 3 (Chapter 4) will enable us to generate spread dynamics using stochastic intensities. Although it is feasible to correlate the processes

(˜λ

i

t

)

t≥0 in

order to induce some level of default correlation, we will not do this. Default dependence is generated by the joint distribution of

ζ

i

∀i∈ {1, . . . , n}

using a copula function.

This setup works well because a dependence structure is completely characterised by a cop- ula function. By Sklar’s theorem (Theorem 15 in Chapter 4), the use of a copula function does not affect the marginal distribution of default times. Hence in this setup the calibra- tion of single obligor intensities are not affected by the default dependence relationship between obligors as it is given by a copula. This is important as it allows for efficient multi- obligor modelling. In contrast, in Chapter 4 (Section 4.2), we showed that when one tries to describe the default dependence relationship between obligors by correlating intensities the calibration of the volatility parameter is affected.

5.1.1

The full intensity of an obligor

Recall the definition of the enlarged filtration:

G=F∨D,

where

D= (D

t

)

t≥0,

D

t

=∨

n_i=1

D

tiand

D

itis the smallest sigma-algebra containing information

about the default of an obligor

i∈ {1, . . . , n}

at time

t≥0

.

G

contains information about the

default of all obligors. We can also define a smaller filtration:

˜

where

D

i

= (D

ti

)

t≥0.

G˜

contains information about the default of only obligor

i.

• The filtration

G˜

is the one we used in Chapter 3, where only one obligor was consid-

ered.

• The filtration

G

is the one used in Chapter 4 and includes information on the default

times of all obligors. We call this filtration thefull filtration.

Under

G˜

we have established in Chapter 3 (Section 3.4.1) that the intensity of obligor

i

is

the

F−adapted

process

(˜λ

it

)

t≥0 in Note 3 (Chapter 4).

We now consider the default intensity of obligor

i

under the full filtration,

G

, where we have

default information about all obligors.

Assumption 5.1. LetU

={ζ

1

, . . . , ζ

n

}

. Under

G

,Uis distributed with an

n−dimensional

copula which is twice differentiable. We call the copula function forUthe threshold copula, denoted by

C

T.

RecallUis independent of

F

.

Theorem 18. Let

(λ

i_t

)

t≥0 be the intensity of obligor

i

under

G

, then

∀t≥0

, If no defaults

have occured, we have:

• The intensity of obligor

i

is:

λ

i_t

= ˜λ

i_te− Rt 0˜λisds ∂ ∂xi

C

T

_(x

1

, . . . , x

n

)

C

T

_(x

1

, . . . , x

n

)

,

(5.1.2) where

x

i

=

e− Rt 0λ˜ i sds

_{∀i∈ {1, . . . , n}}

_.

• The dynamics of the intensity of obligor

i

is:

dλ

i_t

λ

i_t

=

dλ˜

i_t

˜

λ

i t

+

λ

i_t

1−

C

xixi

C

C

2 xi

−˜λ

i_t

dt−dN

i

+

n

X

j=1,j6=i

_C

xixj

C

C

xi

C

xj

−1

dN

j

−λ

j_t

dt

,

(5.1.3) where

C

=C

T

(x

1

, . . . , x

n

)

,

C

xi denotes the partial derivative of

C

with respect to the

i

th _argument,

_C

xixj is the second order partial derivative with respect to the

i

th _and

j

thargument,

N

i is the point process of obligor

i

.

Proof. See propositions 4.3 and 4.7 in Schubert and Schönbucher (2001).

Similar results can be found for the case where defaults have occurred, see Schubert and Schönbucher (2001). A few important points arise from Theorem 18:

• Under

G

the intensity of obligor

i

still depends on

(˜λ

it

)

t≥0. However, now, the addi-

tional information from

ζ

j

∀j∈ {1, . . . , n}&j6=i

is incorporated.

• Consider the case of two obligors. Let us assume

ζ

1and

ζ

2(as given in Note 3 (Chapter

4)) have perfect positive dependence. If obligor 1 has not defaulted, then by Note 3 this implies that the realisation of

ζ

1 was not high, as no default has been triggered

e.g. e−R0tλ˜

i

sds

_{> ζ}

1 holds. As a result of perfect positive correlation, we have that

ζ

1

=

ζ

2. Hence

ζ

2 is also not high, meaning it is less likely obligor 2 has defaulted.

This is the effect that comes through when the intensity of obligor

i

is considered in the full filtration: the conditional value of

ζ

1 influences directly the full intensity of

obligor 2. • Let:

π

ij

=

_C

xixj

C

C

xi

C

xj

−1

.

Schönbucher (2003a) shows that

π

ij is positive if and only if there is locally positive

dependence between obligors

j

and

i. Hence in the case of positive dependence

between obligors we have

π

ij

>0∀i, j

∈ {1, . . . , n}

. From Theorem 18 we can consider

the influence of obligor

j

on obligor

i, withj6=i:

– If obligor

j

has not defaulted then

dN

j

= 0

and the only influence from obligor

j

is

−π

ij

λ

jt

<0

. Hence, in the case of no default of obligor

j

the intensity of obligor

i

reduces. This is what we expect.

– If obligor

j

has defaulted then

dN

j

= 1

and the influence from obligor

j

is to add

π

ij

>0

to the intensity of obligor

j. Again a desired feature.

– Therefore default contagion feeds naturally into this model setup. When there is positive correlation in the quantities

ζ

i

∀i∈ {1, . . . , n}

, then upon a default of an

obligor, all other obligors’ full intensities increase leading to greater chances of further defaults.

5.1.2

The survival and threshold copula

In this sub-section we make a connection between the threshold copula and a copula called the survival copula. Recall that the threshold copula was the copula that described the relationship between quantities

ζ

i

∀i∈ {1, . . . , n}

and was denoted by

C

T.

Now consider the conditional survival functions,

Q

i

(t, t

i

)∀i∈ {1, . . . , n}

, defined in Chapter

τ

1

, . . . , τ

ncan be denoted by:

S(t

1

, . . . , t

n

) =Q(τ

1

> t

1

, . . . , τ

n

> t

n

|F

t

).

(5.1.4)

By Sklar’s theorem (Theorem 15) we have:

S(t

1

, . . . , t

n

) =C

S

(Q

1

(0, t

1

), . . . , Q

n

(0, t

n

)).

(5.1.5)

C

Sis called the survival copula. Jouanin et al. (2001) provide the following important result:

Proposition 19. The relationship between

C

T and

C

S is given,

∀t≥0

, by:

C

S

(Q

1(

t, t

1)

, . . . , Q

n

(t, t

n

)) =E[C

T

(

e− Rt₁ t ˜λ1sds

_{, . . . ,}

_e− Rtn t ˜λ n s ds

_)|F

t

],

(5.1.6)

where the notation

(˜λ

i_t

)

t≥0refers to the intensity of obligor

i

with respect to the filtration

G˜

that contains default information only about obligor

i

.

Proof. See Jouanin et al. (2001).

We note that

C

T and

C

S are equivalent when intensities are deterministic, which is the case we considered in Chapter 4. The theorem tells us that the survival copula, which is the natural default time copula (see Andersen (2006)), can be generated as the conditional expectation of the threshold copula.

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