credit derivatives: fundaments and ideasusers.ictp.it/~marsili/fin_math07/credit_ictp.pdf ·...

Trieste, 14-17 Dec 07

Credit Derivatives:fundaments and ideas

Roberto Baviera, Rates & Derivatives Trader & Structurer, Abaxbank

I.C.T.P., 14-17 Dec 2007

Trieste, 14-17 Dec 072R. Baviera

About?

lacked knowledge

Source: WSJ 5 Dec 07

Outline

1. overview

2. bootstrap interbank curve

3. single-name models

4. multi-name models

Outline: overview

1. overview

Outline: overview

Credit event

credit eventcredit risk componentsabsolute priority and an (over)simplified balance sheetbasic definitions (survival probability & hazard rates)building blocks: initial conditions in a simplified model

Credit products

single-name - bonds: floater & fixed coupon- credit derivatives: ASW, CDS

portfolio related - bonds: ABX, CDO- credit derivatives: First-to-default, Loss-layer

Credit markets

market description and structure volumes

credit event

… when a borrower does not pay

i.e. in case of:

bankruptcy

failure to pay (> threshold, after grace period)

obligation default

obligation acceleration

Definition of event now standardized by ISDA (see ISDA Master Agreement).

credit rating

AAAAA+

an (over) simplified balance sheet

assets liabilities

firm value: equity

material & immaterial assets

(loans/bonds/other debt vs suppliers)

V = S + B

default event:

when firm value < total debt

priority rule: in case of default, debt is served in order of seniority

credit risk components

single-name

arrival risk

timing risk

recovery risk

relation with market risks

multi-name

default dependence

timing risk/clustering

probability distribution of time of default

probability distribution of recovery rate

“correlation” with other risk factors

joint default probability/default correlation

clusters of events/ fat tail in loss distribution

Mathematical descriptionRisks

basic notation

: recovery value

: time to default

: survival indicator function

: survival probability between t and T

: stochastic discount

: default free ZC bond (or “initial” discounts)

: defaultable ZC with NO recovery

: hazard rate

: value of 1 in if default in

building blocs in single name credits

: default free ZC bond curve (interbank curve)

: defaultable ZC bond curve with zero recovery

: value of a payoff of 1 in if default occurs in

building blocs in a simple model

a simple model with additional assumptions:

indipendence rates & defaults

constant recovery rate

a set of reset dates: default in (tn, tn+1] … payment in tn+1

no default risk in derivative’s counterpart (ASW & CDS)

In this case:

building blocs:

fixed coupon bond

it 1+itToday 1t

floater

it 1+itToday 1t Nt

1)( 11 −− ii tL )( ii tL)( 00 tL

)( 11 −− NN tLspolsspolsspols

asset swap

pull to par: @ start date

€3m +

A B obligor

)0(1 C−

at start date:

Credit Default Swap (CDS)

it 1+itToday 1t

π−1

fee leg

contingent leg

at start date:

CDS: main relations

(falling angel)

In the simple model:

CDS: mkt quotes

Intesa Sanpaolo CDS (in bps*), 6 Dec 2007

0 2 4 6 8 10

bid ask

%10 2−*1 bp =

maturity (years)

ABS & CDO: waterfall structure

reference

portfolio

senior tranche

mezzanine tranches

equity tranche

Notes issued by SPV

SPV notes have different names depending on the r.p.: ABS (morgage loans), CDO (bonds)...

low risk

high risk

ABS & CDO

basic structuring:

reference portfolio (r.p.) is transferred to a Special Purpose Vehicle (SPV)

SPV issues notes divided in different tranches

r.p. income goes toward paying tranches’ coupons according to their seniority

at maturity r.p. is liquidated and proceeds distributed to the tranches according to their seniority

if a default occurs, recovery payments are reinvested in default-free securities

tranche basic characteristics:

subordination : amount of losses a portfolio can suffer before tranche’s notional is eroded

tranche cumulative loss ( ) ( )[ ] ( )[ ]0,max0,max .... uprdprtranche KtLKtLtL −−−=

ABS & CDO: subordination example

Fig: “Standard” mezzanine ABS CDO subordination

Sr. AAA Jr. AAA AA A BBB BB Equity

Tranche

iTraxx: subordination example

Fig: iTraxx subordination

Sr. AAA Jr. AAA AA A BBB Equity

Tranche

0.001%

0.010%

0.100%

1.000%

10.000%

100.000%0% 5% 10% 15% 20% 25% 30%

loss fraction

10% 1%

24R. Baviera

ABS & CDO: pricing

pricing obtaining portfolio loss distribution at each reset date (“building block”)

Fig: loss distribution in the Vasicek model for two different correlation values (p =5%)

0-5 % 5-20 % 20-100 %

First to Default (FtD)

on a portfolio of reference credits (generally with the same weight)

Loss Layer

on a portfolio of reference credits with notionals

lower and upper notional bounds

r.p. loss cumulative layer loss

25R. Baviera

portfolio credit derivatives: FtD & Loss layer

it 1+itToday 1t

iπ−1

fee leg

contingent leg:

if the i credit defaultsth

( ) ( )[ ] ( )[ ]0,max0,max .... uprdprlayer KtLKtLtL −−−=

ud KK ,

credit market: cash vs derivatives

credit market: derivatives by product

credit market: total CDO issuance

credit market: CDO by reference portfolio

Outline: bootstrap

1. overview

Outline: bootstrap interbank curve

bootstrap interbank market curve

interbank market- deposits- sht futures- swaps

methodology

interest rates dynamics

Credits and bootstrap IR curve

AAAAA+

bootstrap (interbank) IR curve:

how to get from market observables

ingredients

interbank deposits: zero coupon rates up to 6m

Eurodollar “short” futures: first seven contractsForward rate “=“ 100 -

IR swaps:

idea: - choose always the most liquid product- “other” discounts are obtained interpolating

Depos BID ASK0 sn 5-Dec-07 3.980 4.0201 1w 11-Dec-07 4.070 4.1102 1m 4-Jan-08 4.760 4.8203 2m 4-Feb-08 4.770 4.8304 3m 4-Mar-08 4.770 4.8305 6m 4-Jun-08 4.700 4.760

Future BID ASK1 DEC 07 17-Dec-07 95.245 95.2502 MAR 08 17-Mar-08 95.565 95.5703 JUN 08 16-Jun-08 95.765 95.7704 SEP 08 15-Sep-08 95.900 95.9055 DEC 08 15-Dec-08 95.950 95.9556 MAR 09 16-Mar-09 95.975 95.9807 JUN 09 15-Jun-09 95.960 95.9708 SEP 09 14-Sep-09 95.930 95.935

Swap BID ASK1 4-Dec-08 4.622 4.6422 4-Dec-09 4.397 4.4173 6-Dec-10 4.358 4.3774 5-Dec-11 4.356 4.3765 4-Dec-12 4.373 4.3936 4-Dec-13 4.401 4.4217 4-Dec-14 4.432 4.4538 4-Dec-15 4.468 4.4889 5-Dec-16 4.508 4.52810 4-Dec-17 4.548 4.568

Market TARGETToday 30-Nov-07Settlement 4-Dec-07

deposits

zero coupon rates up to 6m

Act/360 Depos MID # days year frac discount0 sn 5-Dec-07 4.000 1 0.002778 0.99989 1 1w 11-Dec-07 4.090 7 0.019444 0.99921 2 1m 4-Jan-08 4.790 31 0.086111 0.99589

futures

Eurodollar “short” futures: forward rates (in %) : 100 – future price

00 =t it 1+it

Today Start End

( ) )( 01 tLtt iii −+

Futures Fixing Start Date End Date1 17-Dec-07 19-Dec-07 19-Mar-082 17-Mar-08 19-Mar-08 19-Jun-083 16-Jun-08 18-Jun-08 18-Sep-084 15-Sep-08 17-Sep-08 17-Dec-085 15-Dec-08 17-Dec-08 17-Mar-096 16-Mar-09 18-Mar-09 18-Jun-097 15-Jun-09 17-Jun-09 17-Sep-09

00 =t it 1+it

Today Start

interpolation

IR swaps:

example (EURO mkt): annual fixed coupon vs 6m Euribor

1st year

next years

interpolation

bootstrap: given we get

... and one should consider the basis adjustment

interpolation rule

The two most common approaches:

5.00in

... and the dynamics a curve dynamics

30 Nov. 07

11:15 C.E.T.

Interest rates dynamics: HJM

the simplest way to model discount factors (HJM)

where and a Brownian motion in

When is a deterministic function of time in , the model is called Gaussian HJM.

In particular, within this frame, the simplest example is Vasicek model:

discounts’ initial condition

(see e.g. Musiela Rutkowsky 1997)

Outline: single-name models

1. overview

2. bootstrap Interbank Curve

Outline: single-name models

firm value (structural) models

motivationsMerton modelBlack Cox modelimplications on credit spreads

intensity based (reduced form) models

deterministic intensity modelsstochastic intensity modelsbuilding blockscalibration & simulation

Firm value models

motivations:

link equity debt instruments

pricing convertible bonds

corporate finance questions e.g. capital structure optimization, strategic defaults…

firm value dynamics:

default trigger:

value @ maturity less than debt : (Merton model)

safety covenants : (Black-Cox model)∃ ( ) ( )tKtV <( ) DTV <D

Merton model

In the model we can intervene only at the maturity T of the debt.

If the value is less than the debt, there is a default. I.e. Condition at maturity is

basic dynamics:

solution:

where is Black-Scholes call solution

Merton model: calibration

Since it almost impossible to infer firm value V from balance sheet....

set of equations for :

where the second equation is obtained observing that

Black-Cox model

In presence of safety covenants, creditors can liquidate the firm if firm value falls below a threshold. Default occurs as soon as

the bond is equal to

Firm value dynamics is Merton’s one. The simplest case is a constant threshold

NO default occurs

default occurs

Black-Cox model: idea

τ Default

NO Default scenario

Black-Cox model: solution

In the case and of constant interest rates r the solution is

... similar solutions can be obtained in the cases:

stochastic IR (Gaussian HJM) impact of correlations rates defaults

0 0.5 1 1.5 2 2.5 3

V=2 V=3 V=4 V=5

48R. Baviera

Black-Cox model: solution

Spread in the zero recovery case

Remark: The bump shape is not in market spreads!

firm value models: summary

Advantages

pricing convertible bonds

link equity debt instrument (e.g. default correlation equity correlation)

analising corporate finance issues (e.g. capital structure optimization in a firm)

Disadvantages

mkt credit spreads have NOT a bump shape

deterministic intensity based models

local probability of default over

inhomogeneous Poisson process, with local probaility to jump =

basic definitions:

counting process with intensity

time of the first jump of

survival probability

deterministic intensity based models: building blocs

main property:

purely discontinuous process zero covariation with continuous martingales(e.g. rates)

building blocks:

deterministic intensity based models: simulation & calibration

simulation

• draw flat

• time of default :

calibration in a simplified example: (flat & , paid continuously)

Jarrow & Turnbull (1995)

... and in general credit bootstrap is straightforward

stochastic intensity based models

Cox processes:

and, conditional on , is a Poisson proces with intensity

basic properties:

stochastic intensity based models: Gaussian HJM example

bond dynamics:

building blocs:

4444 34444 21correlation effect

intensity based models: summary

Advantages

direct calibration to credit mkt spreads

simple simulation

it is possible to see impact of market risks (e.g. effects of IR correlation)

Disadvantages

link equity debt instrument

Outline: multi-name models

1. overview

2. bootstrap Interbank Curve

Outline: multi-name models

Default dependancy and new questions

Single time step model

descriptionfirm’s value model (Vasicek or factor model):

- loss distribution- large homogeneous portfolio (LHP)

generalizations: t-Student, double t-Student, archimedean copula

copula approach

Default time models

calibration & simulation

if this manyborrowers defaulted

on their loan...

and the lenderresold* the

property for*...

the lossseverity

total "pool" losseswould be...

losses on$1.4

trillion

base case30% 60 cents/$ 40% 30% x 40% = 12% = $168 billion

hypothetical stress case

40% 50 cents/$ 50% 40% x 50% = 20% = $280 billion

ABX index (November 20, 2007) 29% = $406 billion

*net of foreclosure and resale expenses

58R. Baviera

Why relevant?

US subprime exposure is just the starting point of a rolling snowball:

- which are the implications on other non-corporate US credits

(Alt-A & prime morgages, credit cards, auto loans, ...)?

- what is the contagion effect on other forms of lending/credit?

losses in the subprime mortgage cash market (simplified example)

Source: FT, 6 Dec 2007

$10121 US$ trillion =

1 US$ billion = $109

outstanding( trio $ )

US home mortgages 10.1 subprime 1.4 Alt-A 1.2 jumbo 1.8 prime (80% agencies) 5.7

commercial Real Estate 3.1

credit cards 0.9

auto loans 1.0

non-corporate US credit 15.1

structured securities 9.0

corporate bonds 5.4

US Treasuries 4.3

source: FT, So le2 4Ore, DB, GS, es t imates

59R. Baviera

orders of magnitude

outstanding( trio $ ) low high low high

US home mortgages 10.1 300 786 subprime 1.4 12.0% 29.0% 168 406 Alt-A 1.2 6.0% 14.0% 72 168 jumbo 1.8 3.0% 7.0% 54 126 prime (80% agencies) 5.7 0.1% 1.5% 6 86

commercial Real Estate 3.1 1.5% 3.5% 47 109

credit cards 0.9 8.0% 11.0% 72 99

auto loans 1.0 3.5% 5.5% 35 55

non-corporate US credit 15.1 total 453 1048

charge-off losses ( bio $ )

60R. Baviera

orders of magnitude: assessing the risk of contagion

objectives

objectives:

reproduce default dependancy of realistic magnitude

reproduce timing of defaults and clustering

... and the model should present:

straight calibrationjoint default information over a fixed time horizon individual bond term structures (for FtD & CDO)

simple implementation

parsimony

e.g. saving & loan US banks 1980-82: 962 bankruptcies

on 4000 existing in 1980

one time step model

Each obligor defaults in the lag T (time window of interest)with a default probability , i.e. a survival probability

Homogeneous Portfolio (HP) assumption:

Large Homogeneous Portfolio (LHP) assumption:

loss fraction

factor models allow simple computation

loss distribution default distribution

factor model: Vasicek

obligor i defaults iff with

where std normal i.i.d.

Remark: given are indipendent Idea: use conditional indipendence

HP default probability given :

HP loss distribution:

multi-name firm value model

factor model: LHP in Vasicek

LHP loss distribution:

0.001%

0.010%

0.100%

1.000%

10.000%

100.000%0% 5% 10% 15% 20% 25% 30%

loss fraction

10% 1%

exponential tail

with p =5%

generalizations: t-Student

where with degrees of freedomstd. Normal i.i.d.

Remark: given are indipendent

LHP loss distribution

(O’Kane & Schloegl 2005)numerically

generalizations: double t-Student

where i.i.d. t-Student

(Hull & White 2004)numerical inversion

generalizations: archimedean copula

survival i defaults iff with

a decreasing function

Remark: given are indipendent

HP survival probability given :

generalizations: examples archimedean copula

Clayton Gumbel

Comparison: implied correlation iTraxx tranches

copula: definition

A function , s.t.

is a distribution function of

Main property: Sklar’s theorem

multi-variate distribution s.t

with the set of univariate marginal distribution functions

Remark. We can separate the two modelling aspects: single obligor default dynamics dependence structure

limit copula

in the 2d case:

example: Gaussian copula

the I-d cumulated normal

example: archimedean copula

with decreasing function with

Main property: Marshall and Olkin theorem

drawing:

have the archimedean copula function with generator

copula: a simple (one-time-step) default model

: individual (marginal) survival probabilities (one time step)

: copula which describes default dependency

Remark:

Probability of no default

Probability of survival of the first k obligors

21DS 21DD

multi-name default time model

: individual (marginal) survival probabilities

: copula which describes default dependency

Remark: for each fixed T the model is a (one-step) copula default model with survival probabilities

simulation:

draw and obtain the default times

given the scenario, evaluate the payoff

calibration:

calibrate the marginal survival probability of each obligor

... the copula ...

conclusions

www.tate.org.uk

77R. Baviera

conclusions

... ...IN CREDITS

which is the financial problem ?

collect the relevant data (for calibration)

select a modeling framework

simulate

critical analysis of the approach: - orders of magnitude- sensitivities vs required precison (mkt bid/asks)- parsimony

…reality is always more complicated

bibliography sketch

P.J. Schonbucher (2003), Credit Derivatives Pricing Models, Wiley

M. Musiela and M. Rutkowsky (1997), Martingale Methods in Financial Modeling, Springer

N. Patel (2002), The vanilla explosion, Risk Magazine 2, 24-31.

D. Li (2000), On default correlation: a copula function approach, J. Fixed Income 9, 43-54

R. Jarrow and S. Turnbull (1995), Pricing derivatives on financial securities subject to credit risk, J. Finance 50, 53-85

L. Schloegl and D. O’Kane (2005), A note on the large homogeneous portfolio approximation with the Student-t copula, Finance and Stochastics 9, 577-584

J. Hull and A. White (2004), Valuation of a CDO and an n-th to default CDS without a Monte carlo simulation, J. Derivatives 2, 8-23

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