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

• 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

iiii

n

iii

tELtELttv

tELttv

11

11

,LegDefault

1,PremiumLeg Premium

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*B(u,v)

and it may be proved that it is also a copula.

1

0

,,dt

t

vtB

t

tuA

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)

i

nu

kmmnn dtt

uuutB

t

tuuuA

0

121121 ,...,,,,,...,,

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

If the marginal distributions are standard normal this yields a standard browian motion. We can however use different marginals preserving brownian dynamics.

u

dwst

wsvt

0

11

Time Changed Brownian Copulas

• Set h(t,) an increasing function of time t, given state . The copula

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 multi-asset derivatives.

u

dwshth

wshvth

0

11

,,

,,

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

is a copula iff

u

dwwHvGFwCDvuC0

111 ,),(ˆ

zGzFHdwwHzFwCDC

*,1

0

11

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• 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.1. Start with the probability distribution of increments in the first

period F1 and set F1 = H1.2. Numerically compute

where z is now a grid of values of the variable

3. Go back to step 2, and using F3 and H2 compute H3…

zHdwwHzFwCD 2

1

0

11

221 ,

Time aggregation with Archimedean copulas: tau = 0.2

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

0

11 , dwwHzFwCD

Default probability of equity tranches: LPM, different time horizons

0,00%

1,00%

2,00%

3,00%

4,00%

5,00%

6,00%

7,00%

1 2 3 4 5 6 7 8 9 10

Equity 3%

Equity 7%

Equity 10%

Equity 15%

Equity 30%

“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.