Brownian bridges on random intervals

Brownian Bridges on Random Intervals
Dr. Matteo L. BEDINI
Intesa Sanpaolo - DRFM, Derivatives
Pisa, 29 January 2016

Summary
The work describes the basic properties of a Brownian bridge starting from
0 at time 0 and conditioned to be equal to 0 at the random time τ. Such
a process is used to model the ﬂow of information about a credit event
occurring at 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, Brownian Bridges on Random
Intervals, Preprint, 2015 (Submitted) [BBE]. Available at
http://arxiv.org/abs/1601.01811.
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 Preliminaries on Brownian Bridges
3 The Information Process
4 Bayes Estimate and Conditional Expectations
5 Semimartingale Decomposition of the Information Process
6 Pricing a Credit Default Swap

Objective and Motivation
Outline
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 4 / 26

Objective and Motivation The ﬂow of information on a default
From the Financial Highlights Archives of the Federal Reserve Bank of Atlanta [FED]
(see also Dwyer, Flavin, 2010 on the impact of news on the Irish spread):
May 12, 2010: 750 billion EU/IMF package
May 26, 2010: Naked short selling banned by German government
Figure: Impact of market information. Source: [FED], Report of June 2, 2010.

Objective and Motivation Previous approaches to credit risk
Let G be a filtration modeling the flow of information on the market and τ a
default time (G-stopping time).
Problem
Which information on the default time τ is available before it occurs?
Structural Approach (Merton, 1974). G is a Brownian filtration.
τ is predictable (Jarrow, Protter, 2004).
Intensity-based Models (Duffie, Schroder, Skiadas, 1996). Assumption:
I{τ≤t} −
´ t
0
λG
s ds, t ≥ 0 is G-martingale.
Difficult pricing formulas (Jeanblanc, Le Cam, 2007).
Hazard-process Approach (Elliot, Jeanblanc, Yor, 2000). G = F ∨ H.
Information on τ may be too poor.
Information-based Approach (Brody, Hughston, Macrina, 2007). G
generated by ξt = αtDT + βT
t .
τ is not modeled explicitly.

Objective and Motivation Blending the Hazard-process with the Information-based
Objective
Our approach aims to give a qualitative description of the information on τ before
the default, thus making τ “a little bit less inaccessible".
Figure: Information on the default is generated by β = (βt, t ≥ 0). The market
ﬁltration G = Fβ
.

Preliminaries on Brownian Bridges
Outline

Preliminaries on Brownian Bridges Brownian bridges and Brownian motion
Let Ω, F, P, F = (Ft)t≥0 be a ﬁltered probability space (usual
condition), N collection of (P, F)-null sets, W = (Wt, t ≥ 0) a Brownian
motion, r ∈ (0, +∞). A Brownian bridge between 0 and 0 is a Brownian
motion conditioned to be equal to 0 at time r (see, e.g., Karatzas, Shreve,
1991). Examples:
Xt := Wt −
t
r
Wr , Yt := (r − t)
t∧rˆ
0
dWs
s − r
ds, t ∈ [0, r] .
Properties:
Markov process, Gaussian process, Semimartingale.
If Γ ∈ B (R) then
P (Xt ∈ Γ) =
ˆ
Γ
ϕt (x, r)dx,
where ϕt (x, r) is the Gaussian density centered in 0 and with
variance t(r−t)
r .
E [Xt] = 0, for all t ∈ [0, r]. E [XtXs] = s ∧ t − st
r , for all s, t ∈ [0, r]...

Preliminaries on Brownian Bridges Extended Brownian bridges
Consider the map
(r, t, ω) → βr
t (ω) := Wt (ω) −
t
r ∨ t
Wr∨t (ω) , t ≥ 0, ω ∈ Ω, r ∈ (0, +∞) .
E [βr
t ] = 0, E [βr
t βr
s ] = s ∧ t ∧ r − (s∧r)(t∧r)
r , s, t ≥ 0, ...
Markov process.
The process Br
t := βr
t +
´ r∧t
0
βr
s
r−s ds, t ≥ 0 is an r-stopped Brownian
motion.
Let τ : Ω → (0, +∞) a random time. Consider the map
(t, ω) → βt (ω) := β
τ(ω)
t (ω) , t ≥ 0, ω ∈ Ω.
If τ is independent of W ⇒ E [G (τ, W ) |τ] = E [G (r, W )] |r=τ .
Corollary: E [G (τ, β) |τ] = E [G (r, βr )] |r=τ .

The Information Process
Agenda - I

The Information Process Definition of the Information process
Let τ : Ω → (0, +∞) be a strictly positive random variable. Notation:
F (t) := P (τ ≤ t) , t ≥ 0.
Assumption
τ is independent of the Brownian motion W .
Definition
The process β = (βt, t ≥ 0) is called information process:
βt := Wt −
t
τ ∨ t
Wτ∨t, t ≥ 0.
F0 = F0
t := σ (βs, 0 ≤ s ≤ t) t≥0 natural filtration of β.
FP = FP
t := F0
t ∨ N
t≥0
natural, completed filtration of β.
Fβ = Fβ
t := F0
t+ ∨ N
t≥0
, smallest complete and right-continuous
filtration containing F0.

The Information Process First key property
Lemma
For all t ≥ 0, {βt = 0} = {τ ≤ t}, P-a.s. In particular, τ is an
FP-stopping time.
Proof.
Easy:{τ ≤ t} ⊆ {βt = 0}.
Also:
P (βt = 0, τ > t) =
ˆ
(t,+∞)
P (βt = 0|τ = r) dF (r)
=
ˆ
(t,+∞)
P (βr
t = 0) dF (r) = 0,
and, hence, {βt = 0} ⊆ {τ ≤ t} , P-a.s.

The Information Process Markov property with respect to FP
Theorem
The information process is a Markov process w.r.t. FP.
Proof.
Let 0 < t0 < t1 < ... < tn = t.
1 Note FP
t generated by βt,
βti
ti
−
βti−1
ti−1
n
i=1
, n ∈ N.
2
βti
ti
−
βti−1
ti−1
=
Wti
ti
−
Wti−1
ti−1
=: ηi .
3 Let h > 0, the random vector η1, .., ηn, βr
t , βr
t+h is Gaussian and
cov(ηi , βr
t ) = cov(ηi , βr
t+h) = 0, i = 1, ..., n,
hence (η1, .., ηn) is independent of βr
t , βr
t+h .
4 Conditioning w.r.t. τ and using step 3 gives the result.

Bayes Estimate and Conditional Expectations
Outline

Bayes Estimate and Conditional Expectations The Bayes formula
By observing the information process β we can update the a-priori
probability of τ using the Bayes theorem (see, e.g., Shiryaev 1991),
obtaining a sharper estimate of the time of bankruptcy, i.e. the
a-posteriori probability of the default time.
Recall that F denotes the (a-priori) distribution function of τ and that
ϕt (r, x) :=
r
2πt (r − t)
exp −
x2r
2t (r − t)
, x ∈ R, 0 < t < r.
Theorem
Let 0 < t ≤ u ≤ T. Then, P-a.s.
P u ≤ τ ≤ T|FP
t = (Markov prop. & stopping time)
= P (u ≤ τ ≤ T|βt)I{t<τ} =
ˆ
[u,T]
ϕt (r, βt)
´
(t,+∞) ϕt (s, βt) dF (s)
dF (r)I{t<τ}.

Bayes Estimate and Conditional Expectations First generalization
The following result will be used to price a Credit Default Swap in a simple
market model.
Theorem
Let t > 0, g s.t. E [|g (τ)|] < +∞. Then, P-a.s.
E g (τ) I{t<τ}|FP
t = (Markov prop. & stopping time)
= E [g (τ) |βt] I{t<τ} =
ˆ
(t,+∞)
g (r)
ϕt (r, βt)
´
(t,+∞) ϕt (s, βt) dF (s)
dF (r)I{t<τ}.

Bayes Estimate and Conditional Expectations Further conditional expectations and Markov property w.r.t. Fβ
Further generalization:
E [g (τ, βt) |βt] = ...
E [g (βu) |βt] = ...
E [g (τ, βu) |βt] = ...
Used together with the Dominated Convergence Theorem (Lebesgue) to
prove
Theorem
The information process β is a Fβ, P -Markov process. Furthermore
Fβ
t = FP
t , t ≥ 0.

Semimartingale Decomposition of the Information Process
Outline

Semimartingale Decomposition of the Information Process Optional projection and the Innovation lemma
Let F be a ﬁltration satisfying the usual condition, T set of F-stopping times.
F-optional projection o
X of a non-negative process X:
E XT I{T<+∞}|FT = o
XT I{T<+∞}, P-a.s., ∀T ∈ T ,
(see, e.g. [RY]). For an arbitrary process X: o
Xt (ω) := o
X+
t (ω) − o
X−
t (ω) if
o
X+
t (ω) ∧ o
X−
t (ω) < +∞ (+∞ otherwise).
Innovation Lemma (see, e.g., [JYC, RW])
Let T ∈ T , B an F-Brownian motion stopped at T and Z an F-optional process
s.t. E
´ t
0
|Zs| ds < +∞. Deﬁne Xt :=
´ t
0
Zsds + Bt, t ≥ 0, and let o
Z be the
FX
-optional projection of Z. Then, the process b given by
bt := Xt −
tˆ
0
o
Zsds, t ≥ 0
is an FX
-Brownian motion stopped at T.

Semimartingale Decomposition of the Information Process Brownian motion in the enlarged filtration G
Recalling that
Br
t := βr
t +
r∧tˆ
0
βr
s
r − s
ds, t ≥ 0
is an r-stopped Brownian motion and that E [G (τ, β) |τ] = E [G (r, βr
)] |r=τ (plus
some technical conditions) we obtain:
Theorem
Let G = (Gt)t≥0 be the filtration defined by
Gt :=
u>t
Fβ
u ∨ σ (τ)
The process B defined by
Bt := βt +
tˆ
0
βt
τ − t
ds, t ≥ 0
is a G-Brownian motion stopped at τ.

Semimartingale Decomposition of the Information Process Semimartingale decomposition of β
We are in the position of applying the Innovation Lemma where F = G,
X = β and Zt = βt
τ−t I{t<τ}, t ≥ 0:
Theorem
The process b = (bt, t ≥ 0) given by
bt := βt +
tˆ
0
E
βs
τ − s
I{s<τ}|Fβ
s ds
= βt +
tˆ
0
βsE
1
τ − s
|βs I{s<τ}ds = ... (m)
is an Fβ-Brownian motion stopped at τ. Thus the information process β is
an Fβ-semimartingale whose decomposition (loc. mart. + BV) is
determined by (m).

Pricing a Credit Default Swap
Outline

Pricing a Credit Default Swap A Credit Default Swap in a toy market model
Assume deterministic default-free spot interest rate r = 0.
A Credit Default Swap (CDS) with maturity T ∈ (0, +∞) is a ﬁnancial
contract between a buyer and a seller.
The buyer wants to insure the risk of default. Protection leg:
δ (τ) I{t≤τ≤T}.
The seller is paid by the buyer to provide such insurance. Fee leg:
I{t<τ}κ [(τ ∧ T) − t] .
The price St (κ, δ, T) of the CDS is given by:
St (κ, δ, T) := E δ (τ) I{t≤τ≤T} − I{t<τ}κ [(τ ∧ T) − t] |Ft
where F = (Ft)t≥0 is the market ﬁltration.

Pricing a Credit Default Swap Pricing a CDS
We compare the pricing formula obtained in the information based approach,
where F = Fβ
, with that obtained in the framework described in [BJR] (see also
[JYC], Section 7.8), where F = H.
A-priori survival probability function: G (v) := P (τ > v) , t ≥ 0.
A-posteriori survival probability function: Ψt (v) := P τ > v|Fβ
t , t, v ≥ 0.
Market ﬁltration F Price St (κ, δ, T)
H I{t<τ}
1
G(t) −
´ T
t
δ (v) dG (v) − κ
´ T
t
G (v)dv (♦)
Fβ
I{t<τ} −
´ T
t
δ (v) dv Ψt (v) − κ
´ T
t
Ψt (v)dv (p)
Table: Comparison of pricing formulas: H is the minimal ﬁltration making τ a
stopping time.
Formal and computational (if you can compute G you can compute Ψ)
analogy between two formulas.
Knowing {τ > t}, formula (♦) provides a deterministic price, while the price
provided by formula (p), through the Bayesian estimate of τ, depends on
the available market information (βt).

Pricing a Credit Default Swap Fair spread of a CDS
The so-called fair spread of a CDS is the value κ∗ such that
St (κ∗
, δ, T) = 0.
The fair spread of a CDS is an observable market quantity describing the
market’s feelings about a default (see Figure 1).
Market filtration F Fair spread κ∗
H −
´ T
t δ(r)dG(r)
´ T
0 G(r)dr
Fβ −
´ T
t δ(r)dr Ψt (r)
´ T
0 Ψt (r)dr
Table: Comparing the fair spread.
Market filtration = H: the fair spread of a CDS is a deterministic
function of time.
Market filtration = Fβ: the fair spread of a CDS depends on the
available market information.

References
[BBE] M. L. Bedini, R. Buckdahn, H.-J. Engelbert. Brownian Bridges on
Random Intervals. Preprint (Submitted), 2015. Available at
http://arxiv.org/abs/1601.01811.
[BJR] T.R. Bielecki, M. Jeanblanc and M. Rutkowski. Hedging of basket
of credit derivatives in a credit default swap market. Journal of Credit
Risk, 3: 91-132, 2007.
[BHM] D. Brody, L. Hughston and A. Macrina. Beyond Hazard Rates: A
New Framework for Credit-Risk Modeling. Advances in Mathematical
Finance: Festschrift Volume in Honour of Dilip Madan (Basel:
Birkhäuser, 2007).
[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.
[DF] G. P. Dwyer, T. Flavin. Credit Default Swaps on Government Debt:
Mindless Speculation? Notes from the Vault, Center for Financial
Innovation and Stability, September 2010, available at
https://www.frbatlanta.org/-/media/Documents/cenfis/
publications/nftv0910.pdf

References
[EJY] R.J. Elliott, M. Jeanblanc and M. Yor. On models of default risk.
Mathematical Finance, 10:179-196, 2000.
[FED] Financial Highlight archives of the Federal Reserve Bank of Atlanta
https://www.frbatlanta.org/economy-matters/
economic-and-financial-highlights/charts/archives/
finhighlights/archives-1.aspx.
Report of May 12, 2010 https://www.frbatlanta.org/~/media/
Documents/research/highlights/finhighlights/fh051210.ashx,
Documents/research/highlights/finhighlights/FH051910.ashx,
Documents/research/highlights/finhighlights/FH052610.ashx
Report of June 2, 2010 https://www.frbatlanta.org/~/media/
Documents/research/highlights/finhighlights/fh060210.ashx.
[JP] R. Jarrow and P. Protter. Structural versus Reduced Form Models: A
New Information Based Perspective. Journal of Investment
Management, 2004.
[JLC] Jeanblanc M., Le Cam Y. Reduced form modeling for credit risk.
Preprint 2007, availabe at: http://ssrn.com/abstract=1021545.

Bibliography
[JYC] M. Jeanblanc, M. Yor and M. Chesney. Mathematical Methods for
Financial Markets. Springer, First edition, 2009.
[KS] I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus.
Springer- Verlag, Berlin, Second edition, 1991.
[M] R. Merton. On the pricing of Corporate Debt: The Risk Structure of
Interest Rates. Journal of Finance, 3:449-470, 1974.
[RY] D. Revuz, M. Yor. Continuous Martingales and Brownian Motion.
Springer-Verlag, Berlin, Third edition, 1999.
[RW] L. C. G. Rogers, D. Williams. Diﬀusions, Markov Processes and
Martingales. Vol. 2: Itô Calculus. Cambridge University Press, Second
edition, 2000.
[S] A. Shiryaev. Probability. Springer-Verlag, Second Edition, 1991.

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