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- 1. 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
- 2. Outline <ul><li>Motivation </li></ul><ul><li>Cross-section and temporal dependence </li></ul><ul><li>Copula functions and Markov processes </li></ul><ul><li>Models with (in)dependent increments </li></ul><ul><li>Application to securitisation structures </li></ul>
- 3. Copula applications in finance <ul><li>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) </li></ul><ul><li>Pricing applications using copulas have only focussed on static cross-section applications. </li></ul><ul><li>Econometric applications of copulas have been mainly on the study of the temporal dynamics of variables (Ibragimov, 2005, Gagliardini Gourieroux, 2005). </li></ul>
- 4. Dependence in finance <ul><li>Many correlation products are based on prices of a set of underlying assets observed at different dates. </li></ul><ul><li>Cross-section compatibility: the price has to be consistent with those of the univariate assets at any given time. </li></ul><ul><li>Temporal compatibility: the price has to be consistent with those of the same underlying asset at different dates. </li></ul><ul><li>Our research program: using copulas to disentangle marginal distributions, cross-section dependence and temporal dependence. </li></ul>
- 5. Equity: Barrier Altiplano <ul><li>Assume a note paying a set of coupons in a set of periods, k = 1,2,… P . </li></ul><ul><li>Coupons are digital options indexed to a set of i = 1, 2, …, n assets. </li></ul><ul><li>In each period k the price of assets is monitored at a set of dates j = 1, 2, …, m k </li></ul><ul><li>Coupons are paid iff all the assets are above a barrier at all the reset periods. </li></ul><ul><li>The value of each coupons is exposed to n x m k risk factors and their dependence structure. </li></ul>
- 6. Basket credit derivatives and CDOs <ul><li>CDO tranches are often quoted (and almost always involve) premia on a running basis: for this reason they are intrinsically temporally dependent. </li></ul><ul><li>Denoting EL ( t i ) the cumulated expected losses on the tranche as of time EL ( t i ) and v ( t , t i ) the risk-free discount factor we have </li></ul>
- 7. Tranches as options on losses <ul><li>Pricing equity tranches amounts to price put options on losses: </li></ul><ul><li>max( L d – L , 0) </li></ul><ul><li>where L d is detachment point. </li></ul><ul><li>Pricing senior tranches amounts to price call options in losses </li></ul><ul><li>max( L – L a , 0) </li></ul><ul><li>where L a is attachment point </li></ul><ul><li>Mezzanine and junior tranches are spread of senior or equity tranches </li></ul>
- 8. Credit: Standard synthetic CDOs <ul><li>iTraxx (Europe) and CDX (US) are standardized CDO deals. </li></ul><ul><li>The underlying portfolio of credit exposures is a set of 125 CDS deals on primary names, same nominal exposure, same maturity. </li></ul><ul><li>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. </li></ul>
- 9. Cross-section dependence <ul><li>The risk involved in the pricing of a CDO are of course the joint distribution of losses on the underlying CDS portfolio. </li></ul><ul><li>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. </li></ul><ul><li>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). </li></ul>
- 10. Temporal dependence <ul><li>Temporal dependence is an open question in the pricing of credit correlation products. </li></ul><ul><li>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. </li></ul><ul><li>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. </li></ul>
- 11. Copula applications: literature <ul><li>Equity cross-section: Cherubini-Luciano (2002), Rosenberg (2003), Van der Goorbergh, Genest and Werker (2004) </li></ul><ul><li>Credit cross section: Li (2000), Schonbucher Schubert (2001), Laurent-Gregory (2003), Andersen-Sidenius (2004). </li></ul><ul><li>Equity temporal and cross-section: Cherubini-Romagnoli (2008) </li></ul><ul><li>In this paper we want to appy copulas to represent the temporal dynamics of losses. </li></ul><ul><li>Notice: copulas based price dynamics is not exactly the same concept as dynamic copulas. </li></ul>
- 12. Copula product <ul><li>The product of a copula has been defined (Darsow, Nguyen and Olsen, 1992) as </li></ul><ul><li>A * B ( u , v ) </li></ul><ul><li>and it may be proved that it is also a copula. </li></ul>
- 13. Markov processes and copulas <ul><li>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) </li></ul><ul><li>A ( u 1 , u 2 ,…, u n ) B ( u n , u n+ 1 ,…, u n+k –1 ) </li></ul><ul><li>i </li></ul>
- 14. Properties of products <ul><li>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 </li></ul><ul><li>(A B) C = A ( B C) (Darsow et al. 1992) </li></ul><ul><li>A M = A, B M = B (Darsow et al. 1992) </li></ul><ul><li>A = B = (Darsow et al. 1992) </li></ul><ul><li>A B = A B (Cherubini Romagnoli, 2008) </li></ul>
- 15. Example: Brownian Copula <ul><li>Among other examples, Darsow, Nguyen and Olsen give the brownian copula </li></ul><ul><li>If the marginal distributions are standard normal this yields a standard browian motion. We can however use different marginals preserving brownian dynamics. </li></ul>
- 16. Time Changed Brownian Copulas <ul><li>Set h ( t , ) an increasing function of time t, given state . The copula </li></ul><ul><li>is called Time Changed Brownian Motion copula (Schmitz, 2003). </li></ul><ul><li>The function h(t, ) is the “stochastic clock”. Cherubini and Romagnoli (2008) apply this model to barrier multi-asset derivatives. </li></ul>
- 17. Our approach: dependent increments <ul><li>Take three continuous distributions F , G and H . Denote C ( u , v ) the copula function linking levels and increments of the process and D 1 C ( u , v ) its partial derivative. Then the function </li></ul><ul><li>is a copula iff </li></ul><ul><li> </li></ul>
- 18. A special class of processes <ul><li>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. </li></ul><ul><li>Distribution G is obtained by an operation that we denote C-convolution of F and H . </li></ul><ul><li>Lévy processes are obtained as a class in which </li></ul><ul><ul><li>C(u.v) = uv, the operator is the convolution. </li></ul></ul><ul><ul><li>F = G = H : increments are stationary </li></ul></ul>
- 19. A temporal aggregation algorithm <ul><li>Denote X ( t i – 1 ) level of a variable at time t i – 1 and H i – 1 the corresponding distribution. </li></ul><ul><li>Denote Y ( t i ) the increment of the variable in the period [ t i – 1 , t i ]. The corresponding distribution is F i . </li></ul><ul><li>Start with the probability distribution of increments in the first period F 1 and set F 1 = H 1 . </li></ul><ul><li>Numerically compute </li></ul><ul><li>where z is now a grid of values of the variable </li></ul><ul><li>3. Go back to step 2, and using F 3 and H 2 compute H 3 … </li></ul>
- 20. Time aggregation with Archimedean copulas: tau = 0.2
- 21. Application to credit <ul><li>Assume the following data are given </li></ul><ul><ul><li>The cross-section distribution of losses in every time period [ t i – 1 , t i ] ( Y ( t i )). The distribution is F i . </li></ul></ul><ul><ul><li>A sequence of copula functions C i ( x , y ) representing dependence between the cumulated losses at time t i – 1 X ( t i – 1 ), and the losses Y ( t i ). </li></ul></ul><ul><li>Then, the dynamics of cumulated losses is recovered by iteratively computing the convolution-like relationship </li></ul>
- 22. Default probability of equity tranches: LPM, different time horizons
- 23. “ Houston, we have a problem” <ul><li>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. </li></ul><ul><li>Continuous distribution of losses </li></ul><ul><li>D 1 C ( w , F Y ( K – F X –1 ( w ))) = 1, w [0,1] </li></ul><ul><li>Discrete distribution of losses </li></ul><ul><li>C ( F X ( j ), F Y ( K – j )) – C ( F X ( j – 1 ), F Y ( K – j )) = P( X = j ) </li></ul><ul><li>j = 0,1,…, K </li></ul><ul><li>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. </li></ul>
- 24. Conclusions <ul><li>We propose the use of copula functions to represent the temporal dynamics of losses of CDOs. </li></ul><ul><li>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. </li></ul><ul><li>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. </li></ul>

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