Brownian bridges on random intervals

Brownian Bridges on Random Intervals

Dr. Matteo L. BEDINI

Intesa Sanpaolo - DRFM, Derivatives

Pisa, 29 January 2016

SummaryThe work describes the basic properties of a Brownian bridge starting from0 at time 0 and conditioned to be equal to 0 at the random time τ . Sucha process is used to model the flow of information about a credit eventoccurring at 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, Brownian Bridges on RandomIntervals, Preprint, 2015 (Submitted) [BBE]. Available athttp://arxiv.org/abs/1601.01811.

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".

http://arxiv.org/abs/1601.01811

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 packageMay 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 adefault time (G-stopping time).

ProblemWhich information on the default time τ is available before it occurs?

Structural Approach (Merton, 1974). G is a Brownian filtration.I τ is predictable (Jarrow, Protter, 2004).

Intensity-based Models (Duffie, Schroder, Skiadas, 1996). Assumption:(Iτ≤t −

´ t0 λ

Gs ds, t ≥ 0

)is G-martingale.

I Difficult pricing formulas (Jeanblanc, Le Cam, 2007).Hazard-process Approach (Elliot, Jeanblanc, Yor, 2000). G = F ∨H.

I Information on τ may be too poor.Information-based Approach (Brody, Hughston, Macrina, 2007). Ggenerated by ξt = αtDT + βT

t .I τ is not modeled explicitly.


Objective and Motivation Blending the Hazard-process with the Information-based

ObjectiveOur approach aims to give a qualitative description of the information on τ beforethe default, thus making τ “a little bit less inaccessible".

Figure: Information on the default is generated by β = (βt , t ≥ 0). The marketfiltration 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 Brownianmotion, r ∈ (0,+∞). A Brownian bridge between 0 and 0 is a Brownianmotion conditioned to be equal to 0 at time r (see, e.g., Karatzas, Shreve,1991). Examples:

Xt := Wt −tr Wr , Yt := (r − t)

t∧rˆ

0

dWss − 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 withvariance t(r−t)

r .E [Xt ] = 0, for all t ∈ [0, r ]. E [XtXs ] = s ∧ t − st

r , for all s, t ∈ [0, r ]...M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 9 / 26

Preliminaries on Brownian Bridges Extended Brownian bridges

Consider the map

(r , t, ω) 7→ βrt (ω) := Wt (ω)− t

r ∨ tWr∨t (ω) , t ≥ 0, ω ∈ Ω, r ∈ (0,+∞) .

E [βrt ] = 0, E [βr

tβrs ] = s ∧ t ∧ r − (s∧r)(t∧r)

r , s, t ≥ 0, ...Markov process.The process Br

t := βrt +´ r∧t0

βrs

r−s ds, t ≥ 0 is an r -stopped Brownianmotion.

Let τ : Ω→ (0,+∞) a random time. Consider the map

(t, ω) 7→ βt (ω) := βτ(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.

Assumptionτ is independent of the Brownian motion W .

DefinitionThe process β = (βt , t ≥ 0) is called information process:

βt := Wt −t

τ ∨ tWτ∨t , t ≥ 0.

F0 =(F0

t := σ (βs , 0 ≤ s ≤ t))

t≥0 natural filtration of β.

FP =(FP

t := F0t ∨N

)t≥0

natural, completed filtration of β.

Fβ =(Fβt := F0

t+ ∨N)

t≥0, smallest complete and right-continuous

filtration containing F0.M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 12 / 26

The Information Process First key property

LemmaFor all t ≥ 0, βt = 0 = τ ≤ t, P-a.s. In particular, τ is anFP -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 (βrt = 0) dF (r) = 0,

and, hence, βt = 0 ⊆ τ ≤ t , P-a.s.


The Information Process Markov property with respect to FP

TheoremThe information process is a Markov process w.r.t. FP .

Proof.Let 0 < t0 < t1 < ... < tn = t.

1 Note FPt generated by βt ,

βtiti− 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, β

rt , β

rt+h

)is Gaussian and

cov(ηi , βrt ) = cov(ηi , β

rt+h) = 0, i = 1, ..., n,

hence (η1, .., ηn) is independent of(βr

t , βrt+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-prioriprobability of τ using the Bayes theorem (see, e.g., Shiryaev 1991),obtaining a sharper estimate of the time of bankruptcy, i.e. thea-posteriori probability of the default time.

Recall that F denotes the (a-priori) distribution function of τ and that

ϕt (r , x) :=√

r2πt (r − t) exp

[− x2r2t (r − t)

], x ∈ R, 0 < t < r .

TheoremLet 0 < t ≤ u ≤ T. Then, P-a.s.

P(u ≤ τ ≤ T |FP

t

)= (Markov prop. & stopping time)

= P (u ≤ τ ≤ T |βt)It<τ =ˆ

[u,T ]

ϕt (r , βt)´(t,+∞) ϕt (s, βt) dF (s)dF (r)It<τ.


Bayes Estimate and Conditional Expectations First generalization

The following result will be used to price a Credit Default Swap in a simplemarket model.

TheoremLet t > 0, g s.t. E [|g (τ)|] < +∞. Then, P-a.s.

E[g (τ) It<τ|FP

t

]= (Markov prop. & stopping time)

= E [g (τ) |βt ] It<τ =ˆ

(t,+∞)

g (r) ϕt (r , βt)´(t,+∞) ϕt (s, βt) dF (s)dF (r)It<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) toprove

TheoremThe information process β is a

(Fβ,P

)-Markov process. Furthermore

Fβt = FPt , 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 oX of a non-negative process X :

E[XT IT<+∞|FT

]= oXT IT<+∞,P-a.s., ∀T ∈ T ,

(see, e.g. [RY]). For an arbitrary process X : oXt (ω) := oX+t (ω)− oX−t (ω) if

oX+t (ω) ∧ oX−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 processs.t. E

[´ t0 |Zs | ds

]< +∞. Define Xt :=

´ t0 Zsds + Bt , t ≥ 0, and let oZ be the

FX -optional projection of Z . Then, the process b given by

bt := Xt −tˆ

0

oZsds, 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

Brt := βr

t +r∧tˆ

0

βrs

r − s ds, t ≥ 0

is an r -stopped Brownian motion and that E [G (τ, β) |τ ] = E [G (r , βr )] |r=τ (plussome technical conditions) we obtain:

TheoremLet G = (Gt)t≥0 be the filtration defined by

Gt :=⋂u>tFβu ∨ σ (τ)

The process B defined by

Bt := βt +tˆ

0

βtτ − 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 It<τ, t ≥ 0:

TheoremThe process b = (bt , t ≥ 0) given by

bt := βt +tˆ

0

E[βsτ − s Is<τ|F

βs

]ds

= βt +tˆ

0

βsE[ 1τ − s |βs

]Is<τds = ... («)

is an Fβ-Brownian motion stopped at τ . Thus the information process β isan Fβ-semimartingale whose decomposition (loc. mart. + BV) isdetermined by («).


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 financialcontract between a buyer and a seller.

The buyer wants to insure the risk of default. Protection leg:

δ (τ) It≤τ≤T.

The seller is paid by the buyer to provide such insurance. Fee leg:

It<τκ [(τ ∧ T )− t] .

The price St (κ, δ,T ) of the CDS is given by:

St (κ, δ,T ) := E[δ (τ) It≤τ≤T − It<τκ [(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 |Fβt

), t, v ≥ 0.

Market filtration F Price St (κ, δ,T )

H It<τ 1G(t)

(−´ T

t δ (v) dG (v)− κ´ T

t G (v)dv)(♦)

Fβ It<τ(−´ T

t δ (v) dv Ψt (v)− κ´ T

t Ψt (v)dv)(¨)

Table: Comparison of pricing formulas: H is the minimal filtration making τ astopping 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 priceprovided by formula (¨), through the Bayesian estimate of τ , depends onthe 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 themarket’s feelings about a default (see Figure 1).

Market filtration F Fair spread κ∗

H −´ T

t δ(r)dG(r)´ T0 G(r)dr

Fβ −´ T

t δ(r)dr Ψt (r)´ T0 Ψt (r)dr

Table: Comparing the fair spread.

Market filtration = H: the fair spread of a CDS is a deterministicfunction of time.Market filtration = Fβ: the fair spread of a CDS depends on theavailable market information.


References

[BBE] M. L. Bedini, R. Buckdahn, H.-J. Engelbert. Brownian Bridges onRandom Intervals. Preprint (Submitted), 2015. Available athttp://arxiv.org/abs/1601.01811.

[BJR] T.R. Bielecki, M. Jeanblanc and M. Rutkowski. Hedging of basketof credit derivatives in a credit default swap market. Journal of CreditRisk, 3: 91-132, 2007.

[BHM] D. Brody, L. Hughston and A. Macrina. Beyond Hazard Rates: ANew Framework for Credit-Risk Modeling. Advances in MathematicalFinance: Festschrift Volume in Honour of Dilip Madan (Basel:Birkhäuser, 2007).

[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.

[DF] G. P. Dwyer, T. Flavin. Credit Default Swaps on Government Debt:Mindless Speculation? Notes from the Vault, Center for FinancialInnovation and Stability, September 2010, available athttps://www.frbatlanta.org/-/media/Documents/cenfis/publications/nftv0910.pdf


http://arxiv.org/abs/1601.01811

https://www.frbatlanta.org/-/media/Documents/cenfis/publications/nftv0910.pdf

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 Atlantahttps://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,Report of May 19, 2010 https://www.frbatlanta.org/~/media/Documents/research/highlights/finhighlights/FH051910.ashx,Report of May 26, 2010 https://www.frbatlanta.org/~/media/Documents/research/highlights/finhighlights/FH052610.ashxReport 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: ANew Information Based Perspective. Journal of InvestmentManagement, 2004.

[JLC] Jeanblanc M., Le Cam Y. Reduced form modeling for credit risk.Preprint 2007, availabe at: http://ssrn.com/abstract=1021545.


https://www.frbatlanta.org/economy-matters/economic-and-financial-highlights/charts/archives/finhighlights/archives-1.aspx



https://www.frbatlanta.org/~/media/Documents/research/highlights/finhighlights/fh051210.ashx


https://www.frbatlanta.org/~/media/Documents/research/highlights/finhighlights/FH051910.ashx






http://ssrn.com/abstract=1021545

Bibliography

[JYC] M. Jeanblanc, M. Yor and M. Chesney. Mathematical Methods forFinancial 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 ofInterest 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 andMartingales. Vol. 2: Itô Calculus. Cambridge University Press, Secondedition, 2000.

[S] A. Shiryaev. Probability. Springer-Verlag, Second Edition, 1991.


Science

Brownian bridges on random intervals