an information-based approach to credit-risk modelling · 2010-02-26 · an information-based...

AN INFORMATION-BASED

APPROACH TO CREDIT-RISK

MODELLING

by Matteo L. Bedini

Universitè de Bretagne Occidentale

Matteo.Bedini@univ-brest.fr

Agenda

Credit Risk

The Information-based Approach

Defaultable Discount Bond Dynamics

Derivatives and Coupon Bond

Considerations on the Model

Agenda

Credit Risk

The Information-based Approach

Defaultable Discount Bond Dynamics

Derivatives and Coupon Bond

Considerations on the model

Credit Risk

In financial markets credit risk is the risk associated to the

possibility that a counterparty in a financial contract will not

fulfill a contractual commitment to meet her/his obligation

stated in the contract.

EXAMPLES

PARMALAT LEHMAN BROTHERS

Definition

4/31

Credit RiskMathematical Finance and Credit Risk

1. Problem of modelling: How is Credit Risk described?

• Structural Models

• Intensity Models

• Information-based Models

2. Problem of valuating: Given the model, how is a

financial contract valuated?

• Zero-Coupon Bond

• Coupon Bond

• Options

• Credit Default Swap

• …5/31

Credit Risk

1. Default-free interest rate system is deterministic.

Basic Assumptions

2. Financial market is modelled through the specification of a probability space (the probability measure Q is the risk-neutral measure).

3. All processes are adapted to the market filtration.

The existence of a unique risk-neutral measure is ensured,

even if the market may be incomplete.6/31

Credit Risk

Under these hypothesis, if HT represents a cash-flow at time T > 0,

then its value Ht at time t < T is given by:

EXAMPLE: Binary bond.

• Q(HT=h1)=p1 (no default)

• Q(HT=h0)=p0=1-p1 (default)

General settings (1/2)

7/31

Credit RiskGeneral settings (2/2)

• The random variable HT represents the final value of the defaultable bond.

• HT takes value hi with a priori probability pi (i=1,…,n): Q(HT=hi)=pi.

•At time t, the price BtT of a defaultable bond with maturity T>0, is given by:

The purpose is to obtain the bond price process:

8/31

A defaultable bond is a financial contract that, at a pre-specified instant of time

(maturity), delivers to the owner a certain amount of money, if the default

never occurs.

Agenda

Credit Risk

The Information-based Approach

Defaultable Discount Bond Dynamics

Derivatives and Coupon Bond

Considerations on the Model

The Information-based Approach

There exist an Ft-adapted process accessible to market agents,

modelling the flow of information concerning future cash-flow of

the defaultable bond:

The information-process (1/2)

• σ is a constant (information parameter).

• HT is an FT-measurable random variable.

• βtT is a standard Brownian bridge on [0, T] independent from HT (it is FT-

measurable! ).

10/31Theorem: ξt satisfies the Markov property.

The Information-based ApproachThe information-process (2/2)

t=0: all the information is in the a priori

probability distributions

t in (0,T): news, rumors, stories and speculation are mixed together, building the

information about HT arriving on the market.

t=T: the moment of truth.

11/31

The Information-based ApproachBond Price Process

Simplifying assumption: the subalgebra generated by the information

process ξt is the market filtration:

12/31

The Information-based ApproachBayes formula

13/31

The Information-based ApproachBond price process

Next step: obtain the defaultable bond dynamics

dBtT = ?14/31

Agenda

Credit Risk

The Information-based Approach

Defaultable Discount Bond Dynamics

Derivatives and Coupon Bond

Considerations on the Model

Defaultable Discount Bond DynamicsThe Brownian motion

Theorem: Wt is an Ft-Brownian motion.

16/31

The conditional probability:

Defaultable Discount Bond DynamicsDynamics

17/31

Bond price dynamics:

The short rate:

Absolute bond volatility:

Conditional variance:

Defaultable Discount Bond DynamicsSimulations of a digital bond.

18/31

σ=35% σ=55%

σ=75% σ=95%

Agenda

Credit Risk

The Information-based Approach

Defaultable Discount Bond Dynamics

Derivatives and Coupon Bond

Considerations on the Model

Derivatives and Coupon Bond

An European call option is a financial contract that gives the

owner the right to buy a pre-specified asset (the underlying) at a

pre-specified price (the strike price) at a given instant of time.

European call option (1/3)

• T is the maturity of the defaultable bond.

• t is the maturity of the option.

• K is the strike price.

20/31

Derivatives and Coupon BondEuropean call option (2/3)

21/31

Derivatives and Coupon BondEuropean call option (3/3)

22/31

Change of measure by

using factor Φt : from

measure Q to measure B

(the bridge measure).

Binary case (i=1):

Derivatives and Coupon BondNumerical results

23/31

Call option: C0 = f( B0) Put option: P0 = f( B0)

Call option: Δ=∂C0 / ∂ B0 Call option: Vega=∂C0 / ∂σ

Derivatives and Coupon BondThe X-factor Approach

24/31

Modeling more complex situations: how to describe multiple cash-flow?

Idea: if we have n cash-flows, each at time Ti, we can built n information processes

ξ(i) , i=1,…,n, describing the information regarding the corresponding cash-flows.

Derivatives and Coupon BondCredit Default Swap

25/31

A Credit Default Swap (CDS) is a credit derivative between two counterparties,

whereby one makes periodic payments (g) to the other and receives the

promise of a payoff (h) if a third party defaults. The former party receives

credit protection and is said to be the buyer while the other party provides

credit protection and is said to be the seller. The third party is known as the

reference entity. It often happen that the coupon g and the payoff h are chosen

in such way the value Vt of the CDS at time t=0 is V0=0.

(*) In the first formula XtT0 = 1 for convenience

(*)

Derivatives and Coupon BondCoupon Bond

26/30

A Coupon Bond is a contract between a buyer and

a seller in which at time t=0 the buyer gives to the seller p euro (principal). The

seller will pay to the buyer at some pre-specified dates T1 ,…, Tn a pre-specified

amount of money (coupon) ci , i=1,…, n, and at time Tn the seller will pay even

the principal p.

Derivatives and Coupon BondNumerical simulations

27/31

Simulation of the dynamics of a 5-years CDS. Earnings are positive for the seller of protection.

Simulation of the dynamics of a 5-years Coupon Bond.

Agenda

Credit Risk

The Information-based Approach

Defaultable Discount Bond Dynamics

Derivatives and Coupon Bond

Considerations on the Model

Consideration on the ModelFurther development

29/31

Stochastic default-free interest rate system

Final cash-flow (HT) dependent from the “noise”

Generalized noise process

Consideration on the ModelConclusion

30/31

• A new class of models for Credit-risk has been analyzed.

• Central role of the information arriving on the market.

• It is possible to obtain bond price process (relating the a priori

probability with the a posteriori).

• Explicit formula for bond price dynamics.

• Possibility of pricing derivatives (vanilla options, CDS, …).

Bibliography

31/31

• D. C. Brody, L. P. Hughston & A. Macrina. Beyond Hazard rates: a new framework for credit risk

modelling. Advances in Mathematical Finance, Festschrift volume in honour of Dilip Madan.

Birkhauser, Basel, 2007.

• T. R. Bielecki and M. Rutkowski. Credit Risk: Modelling, Valuation and Hedging. Springer, 2002.

• P. J. Schonbucher. Credit Derivatives Pricing Models. John Wiley & Sons, 2003

• T. R. Bielecki, M. Jeanblanc, and M. Rutkowski. Modelling and valuation of credit risk. In Stochastic

Methods in Finance, Bressanone Lectures 2003, eds. M. Frittelli and W. Runggaldier, LNM 1856,

Springer 2004.

• D. Lando. Credit Risk Modelling. Princeton University Press, 2004.

• M. Rutkowski and N. Yu. An extension of the Brody-Hughston-Macrina approach to modelling of

defaultable bonds. Int. J. Theor. Appl. Fin. 10, 557-589, 2007.

• D. C. Brody, M. H. A. Davis, R. L. Friedman, L. P. Hughston, Informed traders. Working paper,

2008..

•D. C. Brody, L. P. Hughston & A. Macrina. Information-based asset pricing. International Journal of

Theoretical and Applied Finance. 2008, vol. II, issue 01, pages 107-142.

THANK YOU VERY MUCH !

Grazie mille !