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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".
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
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
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.
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 5 / 26
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.
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 6 / 26
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 .
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 7 / 26
Preliminaries on Brownian Bridges
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
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 8 / 26
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 .
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 10 / 26
The Information Process
Agenda - I
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
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 11 / 26
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.
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 13 / 26
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.
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 14 / 26
Bayes Estimate and Conditional Expectations
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
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 15 / 26
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.
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 16 / 26
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.
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 17 / 26
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.
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 18 / 26
Semimartingale Decomposition of the Information Process
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
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 19 / 26
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 .
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 20 / 26
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 .
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 21 / 26
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 («).
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 22 / 26
Pricing a Credit Default Swap
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
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 23 / 26
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.
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 24 / 26
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).
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 25 / 26
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.
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 26 / 26
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
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 26 / 26
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.
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 26 / 26
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.
M. L. Bedini (ISP - DRFM) Brownian Bridges on Random Intervals QF XVII, SNS, 29/01/2016 26 / 26