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

SummaryThe work provides sufficient conditions for a default time τ for beingtotally inaccessible in a framework where market information is modelledexplicitly through a Brownian bridge between 0 and 0 on the random timeinterval [0, τ ].

This talk is based on a joint work with Prof. Dr. Rainer Buckdahn andProf. Dr. Hans-Jürgen Engelbert:

MLB, R. Buckdahn, H.-J. Engelbert, Unexpected Default in anInformation Based model, Preprint, 2016 (Submitted). Available athttps://arxiv.org/abs/1611.02952.

DisclaimerThe opinions expressed in these slides are solely of the author and do notnecessarily represent those of the present or past employers.

Work partially supported by the European Community’s FP 7 Programmeunder contract PITN-GA-2008-213841, Marie Curie ITN "ControlledSystems".

https://arxiv.org/abs/1611.02952

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 adefault time τ is modelled by H = (Ht)t≥0, the smallest filtration makingτ a stopping time, which is generated by the single-jump process occurringat τ , meaning that market agents just know if the default has occurred ornot.

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 defaultis more likely to happen than in others. For this reason in theinformation-based approach the flow of market information on the defaultis modelled by Fβ =

(Fβt)

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.M. L. Bedini (ISP - DRFM) Unexpected Default & Information QFW, UniMiB, 27/01/2017 6 / 23

Objective and Motivation Earlier works

Key questionIs the default time a predictable, accessible or totally inaccessible stoppingtime?

Structural approach to credit risk (see, e.g., [M74]). Default timeis predictable (as any stopping time in a Brownian filtration) .Reduced-form models (see, e.g., [DSS] or [EJY]). Key result ofDellacherie and Meyer ([DM]): if the law of τ is diffuse, then τ is atotally inaccessible stopping time with respect to H.

The fact that financial markets cannot foresee the time of default of acompany (non-negligible credit-spread even for short maturities) makes thereduced-form models well accepted by practitioners. In this sense, totallyinaccessible default times seem to be the best candidates for modellingtimes of bankruptcy.See, e.g. Jarrow and Protter [JP] and Giesecke [G] on the relationsbetween 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 filtrationFβ generated by the information process and our main result is the following: ifthe distribution of the default time τ admits a density f with respect to theLebesgue measure, then τ is a totally inaccessible stopping time and itscompensator K = (Kt , t ≥ 0) is given by

Kt =t∧τˆ

0

f (s)´(s,+∞)

√v

2πs(v−s) f (v) dvdLβ (s, 0) ,

where Lβ (t, 0) is the local time of the information process β at level 0 up to timet.

Main featuresCommon assumption that τ admits a continuous density with respect to theLebesgue measure.The default time is a totally-inaccessible stopping time.The model for the flow of market information on the default is moresophisticated 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) between0 and 0 on [0, r ] is given by:

βrt := Wt −

tr ∨ tWr∨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 ∨ tWτ∨t , t ≥ 0.

Fβ =(Fβt := 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 standardBrownian 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 densityϕt (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) It<τ|Fβt

]=ˆ

(t,+∞)g (r)φt (r , βt) dF (r) It<τ, P-a.s.

(4)([BBE], Theo. 4.1, Cor. 4.1 and Cor. 6.1 ).


The Information process Semimartingale decomposition & local time

Defineu (s, x) := E

[βsτ − s Is<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 isdefined 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.M. L. Bedini (ISP - DRFM) Unexpected Default & Information QFW, UniMiB, 27/01/2017 13 / 23

Main result and its proof

Outline






Main result and its proof Statement of the main result

Let H =(Ht := It≤τ

)be the single-jump process occurring at τ .

TheoremSuppose that the distribution function F of τ admits a continuous densityf with respect to the Lebesgue measure. Then τ is an Fβ-totallyinaccessible stopping time and the process K = (Kt , t ≥ 0) defined by

Kt :=t∧τˆ

0

f (s)´(s,+∞)

√v

2πs(v−s) f (v) dvdLβ (s, 0) , (6)

is the compensator of the Fβ-submartingale H1.

1The Fβ-compensator of H is its Fβ-dual predictable projection, i.e. the uniqueFβ-predictable increasing càdlàg process K with K0− = 0 and s.t. H − K is anFβ-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 ∈ Rs.t. (t, x)→ Lβ (t, x) is continuous in t, càdlàg in x . We prove jointcontinuity 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 isequivalent to say that τ is a totally inacessible stopping time withrespect to Fβ (see, e.g., [K], Cor. 25.18).The occupation time formula (see, e.g., [RW], Theo. IV.(45.4)), inour 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 7→ Lβ (t, x) is bounded, for all t ∈ R+, P-a.s. (andthe bound may depend on t and ω).Outside a negligible set, the sequence Lβ (·, xn) weakly converges toLβ (·, 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 ofof Lt = E [C∞|Ft ] , t ≥ 0.Potential generated by C : Xt := Lt − Ct , t ≥ 0. Suppose that X ∈ (D).Notation: For h > 0:

I phX = (phXt , t ≥ 0) is the càd modification of the supermartingalephXt = E [Xt+h|Ft ] , t, h ≥ 0;

I Aht := 1

h´ t0 (Xs − phXs) ds, s ≥ 0 (integrable increasing process).

Theorem (P.-A. Meyer [M66])There exists a unique integrable F-predictable increasing process A generating thepotential X . For every stopping time η it holds

Ahη

σ(L1,L∞)−−−−−−→

h↓0Aη.

In our setting: F = Fβ , Ct = Ht = Iτ≤t, t ≥ 0, C∞ = H∞ = 1; potentialgenerated by Xt := 1− Ht = Iτ>t, t ≥ 0.


Main result and its proof Proof (1/4)

For every h > 0 define the process K h =(K h

t , t ≥ 0)as

K ht := 1

h

tˆ

0

(Is<τ − E

[Is+h<τ|Fβs

])ds

=tˆ

0

1hP(s < τ < s + h|Fβs

)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 K h

t − K ht0 as h ↓ 0, for

every 0 < t0 < t.2nd part: Prove that K is indistinguishable from the compensator of H.

I Compatness Criterion of Dunford-Pettis:(K hn

t − K hnt0

)n∈N

relativelycompact in the weak topology σ

(L1, L∞

)⇒ it is uniformly integrable.

I Thus, P-a.s. convergence of(K hn

t − K hnt0

)n∈N⇒ L1-convergence ⇒

convergence in σ(L1, L∞

)to Kt − Kt0 .

I The result follows by uniqueness of the limit in the weak topologyσ(L1, L∞

).



Let us focus on the first part of the proof:

limh↓0

(K h

t − K ht0)

= limh↓0

t∧τˆ

t0∧τ

1h

( ´ s+hs ϕs (r , βs) f (r) dr´ +∞

s ϕs (v , βs) f (v) dv

)ds

= limh↓0

t∧τˆ

t0∧τ

1h

( ´ s+hs ϕs (r , βs) dr´ +∞


)f (s) ds (7)

+ limh↓0

t∧τˆ

t0∧τ

1h

(´ s+hs ϕs (r , βs) [f (r)− f (s)] dr´ +∞


)ds. (8)

With a procedure analogous to that used in the computation of the limit (7), onecan prove that the limit (8) is equal to 0 P-a.s. by using the uniform continuity off on [t0, t + 1].



It remains to compute:

limh↓0

t∧τˆ

t0∧τ

1h

( ´ s+hs ϕs (r , βs) dr´ +∞


)f (s) ds

= limh↓0

t∧τˆ

t0∧τ

1h

( ´ h0 ϕs (u, βs) du´ +∞


)f (s) ds

= limh↓0

t∧τˆ

t0∧τ

1h

hˆ

0

p(

sus + u , βs , 0

)du g (s, βs)f (s) ds

= limh↓0

t∧τˆ

t0∧τ

1h

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:

limh↓0

t∧τˆ

t0∧τ

1h

hˆ

0

∣∣∣∣p( sus + 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:

limh↓0

t∧τˆ

t0∧τ

1h

hˆ

0

p (u, βs , 0) du g (s, βs) f (s) ds = limh↓0

t∧τˆ

t0∧τ

q (h, βs) g (s, βs) f (s) ds

= limh↓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 probabilitymeasure Qh that converges weakly to the Dirac measure δ0 at 0 as h ↓ 0. Sincethe integrand is continuous and bouned, one can pass to the limit and using thedefinition of g (s, x) we obtain:

limh↓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 apredictable stopping time are considered in another paper, [BH].

Other topics related with Brownian bridges on stochastic intervals areconcerned with:

I the problem of studying the progressive enlargement of a referencefiltration F by the filtration Fβ generated by the information process,

I further applications to Mathematical Finance.


Bibliography

[BBE] M. L. Bedini, R. Buckdahn, H.-J. Engelbert. Brownian Bridges onRandom Intervals. Teor. Veroyatnost. i Primenen., 61:1, 129–157,2016.

[BH] M. L. Bedini, M. Hinz. Credit Defalt Prediction and ParabolicPotential 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 ofdefaultable 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 EconomicDynamics 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 InvestmentManagement, 2004.

[K] O. Kallenberg. Foundation of Modern Probability. Springer- Verlag,New-York, Second edition, 2002.

[KS] I. Karatzas and S. Shreve. Brownian Motion and StochasticCalculus. Springer- Verlag, Berlin, Second edition, 1991.

[M74] R. Merton. On the pricing of Corporate Debt: The Risk Structureof Interest Rates. Journal of Finance, 3:449-470, 1974.

[M66] P.-A. Meyer. Probability and Potentials. Blaisdail PublishingCompany, 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 andMartingales. Vol. 2: Itô Calculus. Cambridge University Press,Second edition, 2000.


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