credit derivatives: fundaments and ideasusers.ictp.it/~marsili/fin_math07/credit_ictp.pdf ·...
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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
1
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About?
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lacked knowledge
Source: WSJ 5 Dec 07
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Outline
1. overview
2. bootstrap interbank curve
3. single-name models
4. multi-name models
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Outline: overview
1. overview
2. bootstrap interbank curve
3. single-name models
4. multi-name models
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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
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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).
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credit rating
AAAAA+
AA
AA-A+
A
Spre
ad O
ver
Libo
r
BBB
BB
Inte
rban
k m
arke
t
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an (over) simplified balance sheet
assets liabilities
firm value: equity
material & immaterial assets
debt
(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
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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
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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
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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
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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:
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fixed coupon bond
00 =t
it 1+itToday 1t
c
Nt
1c c
c
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floater
00 =t
it 1+itToday 1t Nt
spols
1)( 11 −− ii tL )( ii tL)( 00 tL
)( 11 −− NN tLspolsspolsspols
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asset swap
pull to par: @ start date
€3m +
A B obligor
C
)0(1 C−
with
at start date:
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Credit Default Swap (CDS)
00 =t
it 1+itToday 1t
π−1
fee leg
contingent leg
at start date:
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CDS: main relations
(falling angel)
In the simple model:
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CDS: mkt quotes
Intesa Sanpaolo CDS (in bps*), 6 Dec 2007
30
35
40
45
50
0 2 4 6 8 10
bid ask
%10 2−*1 bp =
maturity (years)
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ABS & CDO: waterfall structure
reference
portfolio
}
senior tranche
mezzanine tranches
equity tranche
risk
Notes issued by SPV
SPV notes have different names depending on the r.p.: ABS (morgage loans), CDO (bonds)...
low risk
high risk
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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 −−−=
( )dK
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ABS & CDO: subordination example
Fig: “Standard” mezzanine ABS CDO subordination
0
5
10
15
20
25
30
35
40
Sr. AAA Jr. AAA AA A BBB BB Equity
Tranche
Sub
ordi
natio
n (%
)
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iTraxx: subordination example
Fig: iTraxx subordination
0
5
10
15
20
25
Sr. AAA Jr. AAA AA A BBB Equity
Tranche
Sub
ordi
natio
n (%
)
not
trad
ed
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0.001%
0.010%
0.100%
1.000%
10.000%
100.000%0% 5% 10% 15% 20% 25% 30%
loss fraction
loss
dis
tribu
tion
10% 1%
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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 %
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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
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portfolio credit derivatives: FtD & Loss layer
00 =t
it 1+itToday 1t
iπ−1
fee leg
contingent leg:
if the i credit defaultsth
( ) ( )[ ] ( )[ ]0,max0,max .... uprdprlayer KtLKtLtL −−−=
ud KK ,
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credit market: cash vs derivatives
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credit market: derivatives by product
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credit market: total CDO issuance
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credit market: CDO by reference portfolio
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Outline: bootstrap
1. overview
2. bootstrap interbank curve
3. single-name models
4. multi-name models
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Outline: bootstrap interbank curve
bootstrap interbank market curve
interbank market- deposits- sht futures- swaps
methodology
interest rates dynamics
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Credits and bootstrap IR curve
AAAAA+
AAAA-
A+
A
Spre
ad O
ver
Libo
r
BBB
BB
Inte
rban
k m
arke
t
bootstrap (interbank) IR curve:
how to get from market observables
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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
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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
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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
End
00 =t it 1+it
Today Start
1
1
interpolation
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swaps
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
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interpolation rule
The two most common approaches:
with
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curve
0.000
0.200
0.400
0.600
0.800
1.000
May
-04
Feb-
07
Nov
-09
Aug
-12
May
-15
Feb-
18
Nov
-20
Jul-2
3
Apr
-26
Jan-
29
Oct
-31
Jul-3
4
Apr
-37
Jan-
40
4.00
4.10
4.20
4.30
4.40
4.50
4.60
4.70
4.80
4.90
5.00in
itia
l dis
coun
ts
Zero
rat
es (
%)
... and the dynamics a curve dynamics
30 Nov. 07
11:15 C.E.T.
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Interest rates dynamics: HJM
the simplest way to model discount factors (HJM)
where and a Brownian motion in
with
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)
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Outline: single-name models
1. overview
2. bootstrap Interbank Curve
3. single-name models
4. multi-name models
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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
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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
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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
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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
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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
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Black-Cox model: idea
t
( )tV
( )tK
T
τ Default
( )0V
NO Default scenario
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Black-Cox model: solution
In the case and of constant interest rates r the solution is
with
where
... similar solutions can be obtained in the cases:
stochastic IR (Gaussian HJM) impact of correlations rates defaults
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0%
5%
10%
15%
20%
0 0.5 1 1.5 2 2.5 3
V=2 V=3 V=4 V=5
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Black-Cox model: solution
Spread in the zero recovery case
Remark: The bump shape is not in market spreads!
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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
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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
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deterministic intensity based models: building blocs
main property:
purely discontinuous process zero covariation with continuous martingales(e.g. rates)
building blocks:
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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
0
1
t
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stochastic intensity based models
Cox processes:
s.t
and, conditional on , is a Poisson proces with intensity
basic properties:
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stochastic intensity based models: Gaussian HJM example
bond dynamics:
with
building blocs:
4444 34444 21correlation effect
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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
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Outline: multi-name models
1. overview
2. bootstrap Interbank Curve
3. single-name models
4. multi-name models
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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
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if this manyborrowers defaulted
on their loan...
and the lenderresold* the
property for*...
the lossseverity
is...
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
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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
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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
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orders of magnitude
$10121 US$ trillion =
1 US$ billion = $109
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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 $ )
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orders of magnitude: assessing the risk of contagion
$10121 US$ trillion =
1 US$ billion = $109
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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
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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
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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:
with
multi-name firm value model
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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
loss
dis
tribu
tion
10% 1%
exponential tail
with p =5%
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generalizations: t-Student
obligor i defaults iff with
where with degrees of freedomstd. Normal i.i.d.
Remark: given are indipendent
HP default probability given :
LHP loss distribution
with
(O’Kane & Schloegl 2005)numerically
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generalizations: double t-Student
obligor i defaults iff with
where i.i.d. t-Student
LHP loss distribution
with
(Hull & White 2004)numerical inversion
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generalizations: archimedean copula
survival i defaults iff with
where
a decreasing function
Remark: given are indipendent
HP survival probability given :
HP default probability given :
LHP loss distribution
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generalizations: examples archimedean copula
Clayton Gumbel
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Comparison: implied correlation iTraxx tranches
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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
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limit copula
in the 2d case:
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example: Gaussian copula
with
the I-d cumulated normal
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example: archimedean copula
with decreasing function with
Main property: Marshall and Olkin theorem
drawing:
have the archimedean copula function with generator
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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
21SD
2q
1q
2u
21SS
1u
0
1
1
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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 ...
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conclusions
www.tate.org.uk
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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
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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