Barbara Choroś Wolfgang Härdle Institut für Statistik and Ökonometrie CASE - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin
Motivation - Dependence Matters! The normal world is not enough..6. itraxx S, Jun 6 A Senior Correlation.... Equity Senior Mezzanine Senior Mezz Mezzanine Junior 6 Tranches Figure : Gaussian one factor model with constant correlation.
Motivation - Multidimensional Bermudan Option Copulae are flexible tools!
Motivation - Log Returns of DC and VW VW - - VW - - - - DC - - DC Figure : Standardized log returns, DaimlerChrysler (DC) and Volkswagen (VW), -8 (left) and -7 (right).
Motivation - VaR Estimation Figure : Estimated copula parameter ˆθ t with Local Change Point method, Clayton copula.
Motivation - Grid-Type Copulae
Motivation -6 Outline. Motivation. Collateralized Debt Obligation. Multidimensional Bermudan Option. Grid-Type Copulae
CDO - Definitions Collateralized debt obligation (CDO): credit risk on a pool of assets is tranched and sold to investors. Originator buys protection and pays a premium. Tranche holder sells credit risk protection and receives a premium. Tranche is defined by a lower and an upper attachment points.
CDO - Attachment Points Attachment points (%) Tranche number Tranche name Lower K L Upper K U Equity Mezzanine Junior 6 Mezzanine 6 9 Super Senior Junior 9 Super Senior 6 Senior Table : Example of a CDO tranche structure.
CDO - Homogeneous Gaussian Copula Model Assumptions: (,t,..., d,t ) are random variables determining whether firm i, i =,..., d, defaults at maturity. The individual default probabilities of each firm defaulting in the collateral are equal p. Default occurs when i < D, where D is a threshold value. Then p = Φ(D). Loss given default is equal for all credits in the portfolio.
CDO - Assumptions i = ρ + ρz i, where (systematic risk factor), Z,..., Z d (idiosyncratic risk factors) are i.i.d. N(, ). The correlation coefficient ρ between each pair of random variables i and l is the same for any two firms. where (,t,..., d,t ) N(, Σ), ρ ρ ρ ρ Σ =...... ρ ρ
CDO - Loss Distribution Loss variable of i-th firm L i = I { ρ + ρzi } Portfolio loss L = d i= L i
CDO -6. Implied Correlation Smile Portfolio Loss fixed. This Distribution demonstrated the Figure. (Please, note that the scale for y axis was chosen different in every case for convenience of the reader, the x axis represents loss as a percentage of exposure). Fig..: Portfolio Loss Distribution for different correlation parameters and fixed probability of default %. Figure : Portfolio Loss Distribution for different correlation parameters and fixed probability of default %. Further we provide intuition on how default correlation affects tranche spreads through its influence on collateral loss distribution. First, let us look at the equity tranche, that suffers the first losses, absorbing any losses below its upper attachment point of %. Because of this upper limit on losses, the equity tranche is not much influenced by occurence of many defaults while equity investors are better off with few defaults occured as payments to them are not redirected and their investment is not totally lost. This reduces the
CDO -7 CDO Spreads CDO spread depends on loss distribution. Hence CDO spreads in Homogeneous Gaussian Copula Model depend only on one parameter ρ.
CDO -8 Bloomberg Figure : ITRA Europe, series 6EU with maturity years.
CDO -9 itraxx Europe A static portfolio of equally weighted CDS on European entities New series of itraxx Europe issued every 6 months (March and September) and the underlying reference entities are reconstituted Sub-portfolio: HiVol (), Non-Financials () and Financials () Maturities:,, 7,
CDO - Implied Correlation Smile! The market prices correlations..6. itraxx S, Jun 6 A Senior Correlation... Equity Mezzanine Senior. Senior Mezz Mezzanine Junior 6 Tranches Figure 6: Implied Correlation Smile.
CDO - Implied Correlation Smile? Demand and supply conditions Segmentation among investors Model weakness Therefore copulae! Research Project: Copulae in tempore varientes.
Multidimensional Bermudan Option - Bermudan Option A Bermudan option allows the buyer to exercise the security on one of several specific dates before expiration. Figure 7: Whither the fates lead us.
Multidimensional Bermudan Option - Model Assumptions Bermudan option on d underlying assets: ds l,t S l,t = (r q l )dt + σ l dw l,t, l =,..., d. t = (,t,..., d,t ) are the standardized log price per strike price l,t = log(s l,t /K) N{ l, + (r q l σ l )t, σ l t}. Copula connects the asset marginals to their joint distribution.
Multidimensional Bermudan Option - Notation Consider d-dimensional Bermudan option T expiration date = t < t <... < t n = T dates when option can be exercised = t i t i V i value of the Bermudan option at time t i S i = (S,i,..., S d,i ) underlying asset values g(s i ) option payoff function
Multidimensional Bermudan Option - Dynamic Semiparametric Method The no arbitrage option values on possible early exercise dates are { Vn (S n ) = g(s n ) V i (S i ) = max{g(s i ), e r E Q (V i+ S i )} if i < n. The option value on each possible early exercise date t i is set to be the maximum of g(s i ) the payoff associated with immediate exercise, called the intrinsic value, e r E(V i+ S i ) the discounted conditional expectation of the future option value, called the continuation value.
Multidimensional Bermudan Option - Dynamic Semiparametric Approch nonparametric step function to approximate the option value at time t n, V n (S n ) the conditional joint distribution of t given t is modeled by the copula function C{F (,t,t ),..., F d ( d,t d,t )}
Multidimensional Bermudan Option -6 Grid-Type Copulae
Grid-Type Copulae - Grid-Type Copula Grid-type copula with 9 subsquares: dim = and n =. a b / a b c d / c d / a c / b d / + a + b + c with suitable real numbers a, b, c [, /] and For this choice corr(, ) =! d = a b c.
Grid-Type Copulae - Dependent Risks Fig. 7: empirical copula for windstorm vs. flooding Figure 8: Empirical copula for windstorm vs. flooding. Performing a χ -test here with three cells (one marked in green and two in red), and hence degrees of freedom, gives a test statistic of T =,776 corresponding to a p-value of,7. So it is reasonable to assume that there is some dependence between these risks. The data can be well fitted to a grid-type copula represented by the following weight matrix (see section ):
Grid-Type Copulae - DSFM for Time Varying Copulae The model has the form: t,j = L Z t,l m l ( t,j ) + ε t,j = Zt m( t,j ) + ε t,j l= j =,..., J t, t,j R d, m( ) is a tuple of functions (m, m,..., m L ) m j : R d R Z t = (, Z t,,..., Z t,l ) is a multivariate time series
Grid-Type Copulae - Series Estimator Z t m( ) = L K Z t,l l= k= a l,k ψ k ( ) = Z t Aψ( ) where ψ( ) = (ψ,..., ψ K ) ( ) is a vector of known basis functions, A R (L+) K is a coefficient matrix. K plays the role of the bandwidth h. Define the least squares estimators Ẑ t = (Ẑ,..., ẐL) and  = (â l,k) l=,...,l;k=,...,k (Ẑt, Â) = arg min T J Z t,a t= j= { } t,j Zt Aψ( t,j )
References C. Bluhm, L. Overbeck and C. Wagner An Introduction to Credit Risk Modeling CRC Press, A. Elizalde Credit Risk Models IV: Understanding and pricing CDOs M. Feld Implied Correlation Smile MSc Thesis, http://edoc.hu-berlin.de/ S. Huang and M. Guo Valuation of Multidimensional Bermudan Options R. Nelsen An intoduction to copulas Springer, 999 D. Straßburger and D. Pfeifer Dependence Matters!