Unexpected Default in an Information based model

Unexpected Default in an Information based model
Dr. Matteo L. BEDINI
Intesa Sanpaolo - DRFM, Derivatives
Milano, 27 January 2016

Summary
The work provides suﬃcient conditions for a default time τ for being
totally inaccessible in a framework where market information is modelled
explicitly through a Brownian bridge between 0 and 0 on the random time
interval [0, τ].
This talk is based on a joint work with Prof. Dr. Rainer Buckdahn and
Prof. Dr. Hans-Jürgen Engelbert:
MLB, R. Buckdahn, H.-J. Engelbert, Unexpected Default in an
Information Based model, Preprint, 2016 (Submitted). Available at
https://arxiv.org/abs/1611.02952.
Disclaimer
The opinions expressed in these slides are solely of the author and do not
necessarily represent those of the present or past employers.
Work partially supported by the European Community’s FP 7 Programme
under contract PITN-GA-2008-213841, Marie Curie ITN "Controlled
Systems".

Outline
1 Objective and Motivation
2 The Information process
3 Main result and its proof
4 Further developments and Bibliography

Objective and Motivation
Outline
M. L. Bedini (ISP - DRFM) Unexpected Default & Information QFW, UniMiB, 27/01/2017 4 / 23

Objective and Motivation The flow of information on a default: reduced-form models
In most of the credit-risk models used by practitioner, the information on a
default time τ is modelled by H = (Ht)t≥0, the smallest filtration making
τ a stopping time, which is generated by the single-jump process occurring
at τ, meaning that market agents just know if the default has occurred or
not.
Figure: The minimal filtration making τ a stopping time.

Objective and Motivation Explicitly modelling the information on a default ([BBE])
Financial reality can be more complex, there are periods where the default
is more likely to happen than in others. For this reason in the
information-based approach the flow of market information on the default
is modelled by Fβ = Fβ
t
t≥0
, the filtration generated by β = (βt, t ≥ 0),
a Brownian bridge between 0 and 0 on the random time interval [0, τ].
Figure: The filtration generated by the information process β.

Objective and Motivation Earlier works
Key question
Is the default time a predictable, accessible or totally inaccessible stopping
time?
Structural approach to credit risk (see, e.g., [M74]). Default time
is predictable (as any stopping time in a Brownian filtration) .
Reduced-form models (see, e.g., [DSS] or [EJY]). Key result of
Dellacherie and Meyer ([DM]): if the law of τ is diffuse, then τ is a
totally inaccessible stopping time with respect to H.
The fact that financial markets cannot foresee the time of default of a
company (non-negligible credit-spread even for short maturities) makes the
reduced-form models well accepted by practitioners. In this sense, totally
inaccessible default times seem to be the best candidates for modelling
times of bankruptcy.
See, e.g. Jarrow and Protter [JP] and Giesecke [G] on the relations
between financial information and the properties of the default time.

Objective and Motivation Main result
Our focus is on the classification of the default time with respect to the filtration
Fβ
generated by the information process and our main result is the following: if
the distribution of the default time τ admits a density f with respect to the
Lebesgue measure, then τ is a totally inaccessible stopping time and its
compensator K = (Kt, t ≥ 0) is given by
Kt =
t∧τˆ
0
f (s)
´
(s,+∞)
v
2πs(v−s) f (v) dv
dLβ
(s, 0) ,
where Lβ
(t, 0) is the local time of the information process β at level 0 up to time
t.
Main features
Common assumption that τ admits a continuous density with respect to the
Lebesgue measure.
The default time is a totally-inaccessible stopping time.
The model for the flow of market information on the default is more
sophisticated than the standard approach.

The Information process
Outline

The Information process Brownian bridges on random intervals
(Ω, F, P) complete probability space, N collection of (P, F)-null sets,
W = (Wt, t ≥ 0) a B.m., τ : Ω → (0, +∞) a r.v. independent of W .
Given r ∈ (0, +∞), a standard Brownian bridge βr = (βr
t , t ≥ 0) between
0 and 0 on [0, r] is given by:
βr
t := Wt −
t
r ∨ t
Wr∨t, t ≥ 0.
(see, e.g., [KS] Section 5.6.B).
Deﬁnition (see [BBE], Def. 3.1)
The process β = (βt, t ≥ 0) is called information process:
βt := Wt −
t
τ ∨ t
Wτ∨t, t ≥ 0.
Fβ = Fβ
t := σ (βs, 0 ≤ s ≤ t) ∨ N
t≥0
(right-continuous and complete,
see [BBE] Cor. 6.1).

The Information process Conditional law
The law of βt, conditional to τ = r ∈ (0, +∞), is the same of a standard
Brownian bridge βr between 0 and 0 on the deterministic time interval
[0, r] ([BBE], Lem. 2.4, Cor. 2.2):
P (βt = ·|τ = r) = N 0,
t (r − t)
r
.
Denote by p (, t, ·, y), the Gaussian density with mean y and variance t:
p (t, x, y) :=
1
√
2πt
exp −
(x − y)2
2t
, x ∈ R. (1)
The conditional density of βt, knowing τ = r, is equal to the density
ϕt (r, x) of a standard Brownian bridge βr given by
ϕt (r, x) :=



p t(r−t)
r , x, 0 , 0 < t < r, x ∈ R,
0, r ≤ t, x ∈ R.
(2)

The Information process Main properties
For all t > 0, {βt = 0} = {τ ≤ t} , P-a.s ([BBE], Prop. 3.1).
τ is an Fβ-stopping time ([BBE], Cor. 3.1).
β is an Fβ-Markov process ([BBE], Theo. 6.1).
Deﬁne the a-posteriori density function of τ as
φt (r, x) :=
ϕt (r, x)
ˆ
(t,+∞)
ϕt (v, x) dF (v)
, (r, t) ∈ (0, +∞) × R+, x ∈ R , (3)
Let t > 0, g : R+ → R Borel function s.t. E [|g (τ)|] < +∞. Then
E g (τ) I{t<τ}|Fβ
t =
ˆ
(t,+∞)
g (r) φt (r, βt) dF (r) I{t<τ}, P-a.s.
(4)
([BBE], Theo. 4.1, Cor. 4.1 and Cor. 6.1 ).

The Information process Semimartingale decomposition & local time
Deﬁne
u (s, x) := E
βs
τ − s
I{s<τ}|βs = x , s ∈ R+, x ∈ R. (5)
Theorem ([BBE], Theo. 7.1).
The process b = (bt, t ≥ 0) given by
bt := βt +
tˆ
0
u (s, βs) ds,
is an Fβ
-Brownian motion stopped at τ and β is an Fβ
-semimartingale (loc.
mart. + BV).
Being β a semimartingale, its (right) local time Lβ
(t, x) at level x up to time t is
deﬁned through Tanaka’s formula (see, e.g., [RY], Theo VI.(1.2)):
Lβ
(t, x) = |βt − x| − |β0 − x| −
tˆ
0
sign (βs − x) dβs, t ≥ 0,
where sign (x) := 1 if x > 0 and sign (x) := −1 if x ≤ 0.

Main result and its proof
Outline

Main result and its proof Statement of the main result
Let H = Ht := I{t≤τ} be the single-jump process occurring at τ.
Theorem
Suppose that the distribution function F of τ admits a continuous density
f with respect to the Lebesgue measure. Then τ is an Fβ-totally
inaccessible stopping time and the process K = (Kt, t ≥ 0) deﬁned by
Kt :=
t∧τˆ
0
f (s)
´
(s,+∞)
v
2πs(v−s) f (v) dv
dLβ
(s, 0) , (6)
is the compensator of the Fβ-submartingale H1.
1
The Fβ
-compensator of H is its Fβ
-dual predictable projection, i.e. the unique
Fβ
-predictable increasing càdlàg process K with K0− = 0 and s.t. H − K is an
Fβ
-martingale.

Main result and its proof Key properties of the Local time
Well known: There exists a modiﬁcation of Lβ (t, x) , t ≥ 0, x ∈ R
s.t. (t, x) → Lβ (t, x) is continuous in t, càdlàg in x. We prove joint
continuity in t, x.
In particular: Lβ (t, 0) , t ≥ 0 is a continuous increasing process,
hence, the compensator K given by (6) is continuous, which is
equivalent to say that τ is a totally inacessible stopping time with
respect to Fβ (see, e.g., [K], Cor. 25.18).
The occupation time formula (see, e.g., [RW], Theo. IV.(45.4)), in
our framework, takes the following form:
t∧τˆ
0
h (s, βs) ds =
tˆ
0
h (s, βs) d β, β s =
ˆ
R



tˆ
0
h (s, x) dLβ
(s, x)


 dx,
for all t ≥ 0 and all non-negative Borel functions h on R+ × R, P-a.s.
The function x → Lβ (t, x) is bounded, for all t ∈ R+, P-a.s. (and
the bound may depend on t and ω).
Outside a negligible set, the sequence Lβ (·, xn) weakly converges to
Lβ (·, x) as xn → x ∈ R.

Main result and its proof Laplacian approach
Recall
Let (C, F) be an integrable increasing càd process, L the càd modiﬁcation of
of Lt = E [C∞|Ft] , t ≥ 0.
Potential generated by C: Xt := Lt − Ct, t ≥ 0. Suppose that X ∈ (D).
Notation: For h > 0:
phX = (phXt, t ≥ 0) is the càd modiﬁcation of the supermartingale
phXt = E [Xt+h|Ft] , t, h ≥ 0;
Ah
t := 1
h
´ t
0
(Xs − phXs) ds, s ≥ 0 (integrable increasing process).
Theorem (P.-A. Meyer [M66])
There exists a unique integrable F-predictable increasing process A generating the
potential X. For every stopping time η it holds
Ah
η
σ(L1
,L∞
)
−−−−−−→
h↓0
Aη.
In our setting: F = Fβ
, Ct = Ht = I{τ≤t}, t ≥ 0, C∞ = H∞ = 1; potential
generated by Xt := 1 − Ht = I{τ>t}, t ≥ 0.

Main result and its proof Proof (1/4)
For every h > 0 deﬁne the process Kh
= Kh
t , t ≥ 0 as
Kh
t :=
1
h
tˆ
0
I{s<τ} − E I{s+h<τ}|Fβ
s ds
=
tˆ
0
1
h
P s < τ < s + h|Fβ
s ds, P-a.s.
The proof is then made by two main parts:
1st part: Prove that Kt − Kt0
is the P-a.s. limit of Kh
t − Kh
t0
as h ↓ 0, for
every 0 < t0 < t.
2nd part: Prove that K is indistinguishable from the compensator of H.
Compatness Criterion of Dunford-Pettis: Khn
t − Khn
t0
n∈N
relatively
compact in the weak topology σ L1
, L∞
⇒ it is uniformly integrable.
Thus, P-a.s. convergence of Khn
t − Khn
t0
n∈N
⇒ L1
-convergence ⇒
convergence in σ L1
, L∞
to Kt − Kt0 .
The result follows by uniqueness of the limit in the weak topology
σ L1
, L∞
.

Let us focus on the ﬁrst part of the proof:
lim
h↓0
Kh
t − Kh
t0
= lim
h↓0
t∧τˆ
t0∧τ
1
h
´ s+h
s
ϕs (r, βs) f (r) dr
´ +∞
s
ϕs (v, βs) f (v) dv
ds
= lim
h↓0
t∧τˆ
t0∧τ
1
h
´ s+h
s
ϕs (r, βs) dr
´ +∞
s
f (s) ds (7)
+ lim
h↓0
t∧τˆ
t0∧τ
1
h
´ s+h
s
ϕs (r, βs) [f (r) − f (s)] dr
´ +∞
s
ds. (8)
With a procedure analogous to that used in the computation of the limit (7), one
can prove that the limit (8) is equal to 0 P-a.s. by using the uniform continuity of
f on [t0, t + 1].

It remains to compute:
lim
h↓0
t∧τˆ
t0∧τ
1
h
´ s+h
s
ϕs (r, βs) dr
´ +∞
s
f (s) ds
= lim
h↓0
t∧τˆ
t0∧τ
1
h
´ h
0
ϕs (u, βs) du
´ +∞
s
f (s) ds
= lim
h↓0
t∧τˆ
t0∧τ
1
h
hˆ
0
p
su
s + u
, βs, 0 du g (s, βs)f (s) ds
= lim
h↓0
t∧τˆ
t0∧τ
1
h
hˆ
0
p (u, βs, 0) du g (s, βs)f (s) ds
P-a.s., where the last equality is a consequence of the following (rather technical)
result:
lim
h↓0
t∧τˆ
t0∧τ
1
h
hˆ
0
p
su
s + u
, βs, 0 − p (u, βs, 0) du g (s, βs) f (s) ds = 0, P-a.s.

In the last step, by the occupation time formula:
lim
h↓0
t∧τˆ
t0∧τ
1
h
hˆ
0
p (u, βs, 0) du g (s, βs) f (s) ds = lim
h↓0
t∧τˆ
t0∧τ
q (h, βs) g (s, βs) f (s) ds
= lim
h↓0
+∞ˆ
−∞


tˆ
t0
g (s, x) f (s) dLβ
(s, x)

 q (h, x) dx, P-a.s.
For every h > 0, q (h, ·) is the probability density function of a probability
measure Qh that converges weakly to the Dirac measure δ0 at 0 as h ↓ 0. Since
the integrand is continuous and bouned, one can pass to the limit and using the
deﬁnition of g (s, x) we obtain:
lim
h↓0
+∞ˆ
−∞


tˆ
t0
g (s, x) f (s) dLβ
(s, x)

 q (h, x) dx =
tˆ
t0
g (s, 0) f (s) dLβ
(s, 0)
= Kt − Kt0
, P-a.s.

Further developments and Bibliography
Outline

Further developments and Bibliography Predictable default time and Enlargment of Filtrations
Non-trivial and sufficient conditions for making the default time a
predictable stopping time are considered in another paper, [BH].
Other topics related with Brownian bridges on stochastic intervals are
concerned with:
the problem of studying the progressive enlargement of a reference
filtration F by the filtration Fβ
generated by the information process,
further applications to Mathematical Finance.

Bibliography
[BBE] M. L. Bedini, R. Buckdahn, H.-J. Engelbert. Brownian Bridges on
Random Intervals. Teor. Veroyatnost. i Primenen., 61:1, 129–157,
2016.
[BH] M. L. Bedini, M. Hinz. Credit Defalt Prediction and Parabolic
Potential Theory. Statistics and Probability Letters (accepted),
2017.
[DM] C. Dellacherie, P.-A. Meyer. Probabilities and Potential.
North-Holland, 1978
[DSS] D. Duﬃe, M. Schroder, C. Skiadas. Recursive valuation of
defaultable securities and the timing of resolution of uncertainty.
Annals of Applied Probability, 6: 1075-1090, 1996.
[EJY] R.J. Elliott, M. Jeanblanc and M. Yor. On models of default risk.
Mathematical Finance, 10:179-196, 2000.
[G] K. Giesecke. Default and information. Journal of Economic
Dynamics and Control, 30:2281-2303, 2006.

Bibliography
[JP] R. Jarrow and P. Protter. Structural versus Reduced Form Models:
A New Information Based Perspective. Journal of Investment
Management, 2004.
[K] O. Kallenberg. Foundation of Modern Probability. Springer- Verlag,
New-York, Second edition, 2002.
[KS] I. Karatzas and S. Shreve. Brownian Motion and Stochastic
Calculus. Springer- Verlag, Berlin, Second edition, 1991.
[M74] R. Merton. On the pricing of Corporate Debt: The Risk Structure
of Interest Rates. Journal of Finance, 3:449-470, 1974.
[M66] P.-A. Meyer. Probability and Potentials. Blaisdail Publishing
Company, London 1966.
[RY] D. Revuz, M. Yor. Continuous Martingales and Brownian Motion.
Springer-Verlag, Berlin, Third edition, 1999.

Bibliography
[RW] L. C. G. Rogers, D. Williams. Diﬀusions, Markov Processes and
Martingales. Vol. 2: Itô Calculus. Cambridge University Press,
Second edition, 2000.

Unexpected Default in an Information based model

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