an information-based approach to credit-risk modelling by matteo l. bedini universitè de bretagne...

AN INFORMATION-BASED AN INFORMATION-BASED APPROACH TO CREDIT-RISK APPROACH TO CREDIT-RISK

MODELLINGMODELLINGby Matteo L. Bedini

Universitè de Bretagne Occidentale

[email protected]

AgendaAgenda

Credit RiskCredit Risk The Information-based Approach The Information-based Approach DefaultableDefaultable Discount Bond Dynamics Discount Bond Dynamics Derivatives and Coupon BondDerivatives and Coupon Bond Considerations on the ModelConsiderations on the Model

AgendaAgenda

Credit RiskCredit Risk The Information-based ApproachThe Information-based Approach Defaultable Discount Bond DynamicsDefaultable Discount Bond Dynamics Derivatives and Coupon BondDerivatives and Coupon Bond Considerations on the modelConsiderations on the model

Credit RiskCredit 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

DefinitionDefinition

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Credit RiskCredit RiskMathematical Finance and Credit RiskMathematical Finance and Credit Risk

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

• Structural ModelsStructural Models• Intensity ModelsIntensity Models• Information-based ModelsInformation-based Models2. Problem of valuating:: Given the model, Given the model,

how is a financial contract valuated?how is a financial contract valuated?• Zero-Coupon BondZero-Coupon Bond• Coupon BondCoupon Bond• OptionsOptions• Credit Default SwapCredit Default Swap• … …

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1. Default-free interest rate system is deterministic.1. Default-free interest rate system is deterministic.

Basic Assumptions

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

3. All processes are adapted to the market filtration.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


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)

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Credit RiskCredit 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:

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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.

AgendaAgenda

Credit RiskCredit Risk The Information-based ApproachThe Information-based Approach Defaultable Discount Bond DynamicsDefaultable Discount Bond Dynamics Derivatives and Coupon BondDerivatives and Coupon Bond Considerations on the ModelConsiderations on the Model

The The Information-based Information-based ApproachApproach

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 The Information-based Information-based ApproachApproachThe 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.

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The The Information-based Information-based ApproachApproachBond Price Process

Simplifying assumption: the subalgebra generated by the information process ξt is the market filtration:

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The The Information-based Information-based ApproachApproachBayes formula

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The The Information-based Information-based ApproachApproachBond price process

Next step: obtain the defaultable bond dynamics

dBtT = ?14/31

AgendaAgenda

Credit RiskCredit Risk The Information-based ApproachThe Information-based Approach Defaultable Discount Bond Defaultable Discount Bond

DynamicsDynamics Derivatives and Coupon BondDerivatives and Coupon Bond Considerations on the ModelConsiderations on the Model

Defaultable Discount Bond Defaultable Discount Bond DynamicsDynamics

The Brownian motion

Theorem: Wt is an Ft-Brownian motion.

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The conditional probability:


Dynamics

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Bond price dynamics:

The short rate:

Absolute bond volatility:

Conditional variance:


Simulations of a digital bond.

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σ=35% σ=55%

σ=75% σ=95%

AgendaAgenda


Derivatives and Coupon Derivatives and Coupon BondBond

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.

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Derivatives and Coupon Derivatives and Coupon BondBondEuropean call option

(2/3)

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Derivatives and Coupon Derivatives and Coupon BondBondEuropean call option

(3/3)

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Change of measure by using factor Φt : from measure Q to

measure B (the bridge measure).

Binary case (i=1):

Derivatives and Coupon Derivatives and Coupon BondBondNumerical results

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Call option: C0 = f( B0)

Put option: P0 = f( B0)

Call option: Δ=∂C0 / ∂ B0

Call option: Vega=∂C0 / ∂σ

Derivatives and Coupon Derivatives and Coupon BondBondThe X-factor Approach

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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 Derivatives and Coupon BondBondCredit Default Swap

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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 Derivatives and Coupon BondBondCoupon Bond

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A Coupon Bond is a contract between a buyer anda 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 Derivatives and Coupon BondBondNumerical simulations

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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.

AgendaAgenda


Consideration on the Consideration on the ModelModelFurther development

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Stochastic default-free interest rate systemStochastic default-free interest rate system

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

Generalized noise processGeneralized noise process

Consideration on the Consideration on the ModelModelConclusion

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• 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, …).

BibliographyBibliography

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• 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 !

an information-based approach to credit-risk modelling by matteo l. bedini universitè de bretagne...

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