Download A Copula-Based Model of the Term Structure of CDO Tranches

A Copula-Based Model of the
Term Structure of CDO Tranches
U. Cherubini – S. Mulinacci – S. Romagnoli
University of Bologna
International Financial Research Forum
Paris, 27-28 March 2008
Outline
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Motivation
Cross-section and temporal dependence
Copula functions and Markov processes
Models with (in)dependent increments
Application to securitisation structures
Copula applications in finance
• Copula applications to pricing problems in finance are
motivated by the need to price multivariate products
(correlation products) consistently with the prices of
uni-variate products (this is the financial version of
the so called compatibility problem in statistics)
• Pricing applications using copulas have only focussed
on static cross-section applications.
• Econometric applications of copulas have been mainly
on the study of the temporal dynamics of variables
(Ibragimov, 2005, Gagliardini Gourieroux, 2005).
Dependence in finance
• Many correlation products are based on prices of a set
of underlying assets observed at different dates.
• Cross-section compatibility: the price has to be
consistent with those of the univariate assets at any
given time.
• Temporal compatibility: the price has to be consistent
with those of the same underlying asset at different
dates.
• Our research program: using copulas to disentangle
marginal distributions, cross-section dependence and
temporal dependence.
Equity: Barrier Altiplano
• Assume a note paying a set of coupons in a set of
periods, k = 1,2,…P.
• Coupons are digital options indexed to a set of i =
1, 2, …, n assets.
• In each period k the price of assets is monitored at
a set of dates j = 1, 2, …,mk
• Coupons are paid iff all the assets are above a
barrier at all the reset periods.
• The value of each coupons is exposed to n x mk
risk factors and their dependence structure.
Basket credit derivatives and CDOs
• CDO tranches are often quoted (and almost always
involve) premia on a running basis: for this reason they are
intrinsically temporally dependent.
• Denoting EL(ti) the cumulated expected losses on the
tranche as of time EL(ti) and v(t,ti) the risk-free discount
factor we have
n
Premium Leg  Premium  vt , ti 1  ELti 1 
i 1
n
Default Leg   vt , ti ELti   ELti 1 
i 1
Tranches as options on losses
• Pricing equity tranches amounts to price put options
on losses:
max(Ld – L, 0)
where Ld is detachment point.
• Pricing senior tranches amounts to price call options
in losses
max(L – La , 0)
where La is attachment point
• Mezzanine and junior tranches are spread of senior
or equity tranches
Credit: Standard synthetic CDOs
• iTraxx (Europe) and CDX (US) are standardized
CDO deals.
• The underlying portfolio of credit exposures is a
set of 125 CDS deals on primary names, same
nominal exposure, same maturity.
• The tranches of the standard CDO are 5, 7 and 10
year CDS to buy/sell protection on the losses on
the underlying portfolio higher than a given level
(attachment) up to another level (detachment) on a
nominal value equal to the difference between the
two levels.
Cross-section dependence
• The risk involved in the pricing of a CDO are of
course the joint distribution of losses on the
underlying CDS portfolio.
• Again, this could be modelled selecting a specific
distribution, but the distribution should be consistent
with the price of protection of the the uni-variate
CDS contracts, that is the marginal probability of
default of each name.
• For this reason, copula functions have become the
standard pricing tool in the market (the gaussian
copula plays the role of the Black and Scholes
formula in option pricing).
Temporal dependence
• Temporal dependence is an open question in the
pricing of credit correlation products.
• Consider selling protection on a 5 year tranche 0%3% (when attachment is zero this is called equity
tranche). This is like buying a put option on the first
3% of losses.
• Should we charge more or less for selling protection
of the same tranche on a 10 year 0%-3% tranches? Of
course, we will charge more, and how much more
will depend on the losses that will be expected to
occur in the second 5 year period.
Copula applications: literature
• Equity cross-section: Cherubini-Luciano (2002), Rosenberg
(2003), Van der Goorbergh, Genest and Werker (2004)
• Credit cross section: Li (2000), Schonbucher Schubert (2001),
Laurent-Gregory (2003), Andersen-Sidenius (2004).
• Equity temporal and cross-section: Cherubini-Romagnoli (2008)
• In this paper we want to appy copulas to represent the temporal
dynamics of losses.
• Notice: copulas based price dynamics is not exactly the same
concept as dynamic copulas.
Copula product
• The product of a copula has been defined
(Darsow, Nguyen and Olsen, 1992) as
Au, t  Bt , v 
dt
A*B(u,v)  
t
t
0
1
and it may be proved that it is also a copula.
Markov processes and copulas
• Darsow, Nguyen and Olsen, 1992 prove that 1st
order Markov processes (see Ibragimov, 2005 for
extensions to k order processes) can be
represented by the  operator (similar to the
product)
A (u1, u2,…, un) B(un,un+1,…, un+k–1) 
Au1 , u2 ,..., un1 , t  Bt , un1 , um 2 ,..., um k 1 
dt
0
t
t
un
Properties of  products
• Say A, B and C are copulas, for simplicity
bivariate, A survival copula of A, B survival
copula of B, set M = min(u,v) and  = u v
(A  B)  C = A  (B  C) (Darsow et al. 1992)
A M = A, B M = B
(Darsow et al. 1992)
A  = B  = 
(Darsow et al. 1992)
A  B =A  B
(Cherubini Romagnoli, 2008)
Example: Brownian Copula
• Among other examples, Darsow, Nguyen and
Olsen give the brownian copula
 t  1 v   s  1 w 
dw
0  
ts

u
If the marginal distributions are standard normal
this yields a standard browian motion. We can
however use different marginals preserving
brownian dynamics.
Time Changed Brownian
Copulas
• Set h(t,) an increasing function of time t, given state .
The copula
 ht ,   1 v   hs,   1 w 
dw
0  
ht ,    hs,  


u
is called Time Changed Brownian Motion copula
(Schmitz, 2003).
• The function h(t,) is the “stochastic clock”. Cherubini
and Romagnoli (2008) apply this model to barrier multiasset derivatives.
Our approach: dependent increments
• Take three continuous distributions F, G and H. Denote
C(u,v) the copula function linking levels and increments of
the process and D1C(u,v) its partial derivative. Then the
function
u



Cˆ (u, v)   D1C w, F G 1 v   H 1 w dw
0
is a copula iff
 D Cw, F z  H wdw  H * F z   Gz 
1
1
1
0
C
A special class of processes
• F represents the probability distribution of
increments of the process, H represents the
distribution of the level of the process before the
increment and G represents the level of the
process after the increment.
• Distribution G is obtained by an operation that we
denote C-convolution of F and H.
• Lévy processes are obtained as a class in which
– C(u.v) = uv, the operator is the convolution.
– F = G = H: increments are stationary
A temporal aggregation algorithm
•
•
1.
2.
Denote X(ti – 1) level of a variable at time ti – 1 and Hi – 1
the corresponding distribution.
Denote Y(ti ) the increment of the variable in the period
[ti – 1,ti]. The corresponding distribution is Fi.
Start with the probability distribution of increments in
the first period F1 and set F1 = H1.
Numerically compute
1



1
D
C
w
,
F
z

H
1 w dw  H 2  z 
 1 2 2
0
3.
where z is now a grid of values of the variable
Go back to step 2, and using F3 and H2 compute H3…
Time aggregation with
Archimedean copulas: tau = 0.2
20
18
16
14
12
Indep
Clayton
10
Frank
Gumbel
8
Perf dep
6
4
2
0
1
3
5
7
9
11
13
15
17
19
21
Application to credit
• Assume the following data are given
– The cross-section distribution of losses in every time
period [ti – 1,ti] (Y(ti )). The distribution is Fi.
– A sequence of copula functions Ci(x,y) representing
dependence between the cumulated losses at time ti – 1
X(ti – 1), and the losses Y(ti ).
• Then, the dynamics of cumulated losses is recovered by
iteratively computing the convolution-like relationship
1



1
w dw
D
C
w
,
F
z

H
 1
0
Default probability of equity tranches:
LPM, different time horizons
7,00%
6,00%
5,00%
Equity 3%
Equity 7%
Equity 10%
Equity 15%
Equity 30%
4,00%
3,00%
2,00%
1,00%
0,00%
1
2
3
4
5
6
7
8
9
10
“Houston, we have a problem”
• The application of the algorithm to credit leads to a
problem. As the support of the amount of default is
bounded, the algorithm must be modified accordingly,
including constraints.
• Continuous distribution of losses
• D1C (w,FY(K – FX–1(w))) = 1,  w  [0,1]
• Discrete distribution of losses
• C(FX(j),FY(K – j)) – C(FX(j – 1),FY(K – j)) = P(X = j)
j = 0,1,…,K
• These constraints define a recursive system that given the
initial distribution of losses and the temporal dependence
structure yields the distribution of losses in future periods.
Conclusions
• We propose the use of copula functions to represent
the temporal dynamics of losses of CDOs.
• The dynamics is constructed by applying copulas to
model the dependence structure of increments of
losses in a period and cumulated losses at the
beginning of the period.
• When specialized to the multivariate credit problem
this approach induces a recursive algorithm to
compute propagation of the losses in time and a
term structure of tranches premia.