an information-based approach to credit-risk modelling · 2010-02-26 · an information-based...
TRANSCRIPT
AN INFORMATION-BASED
APPROACH TO CREDIT-RISK
MODELLING
by Matteo L. Bedini
Universitè de Bretagne Occidentale
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 !