Sklar theorem Sklar theorem each joint distribution HX
Sklar theorem • Sklar theorem: each joint distribution H(X, Y) can be written as a copula function C(FX, FY) taking the marginal distributions as arguments, and vice versa, every copula function taking univariate distributions as arguments yields a joint distribution.
Copula functions and dependence • Copula functions allow to separate specification of marginal distributions and the dependence structure. • Say two risks A and B have joint probability H(X, Y) and marginal probabilities FX and FY. We have that H(X, Y) = C(FX , FY), and C is a copula function. • Examples C(u, v) = uv, independence C(u, v) = min(u, v), perfect positive dependence C(u, v) = max (u + v - 1, 0) perfect negative dependence • Perfect dependence cases are Fréchet bounds.
Example: guarantee on credit • Assume a credit exposure with probability of default of Pa = 20% in a year. • Say the credit exposure is guaranteed by another party with default probability equal to Pb = 1%. • The probability of default on the exposure is now the joint probability DP = C(Pa , Pb) • The worst case is perfect dependence between default of the two counterparties leading to DP = min(Pa , Pb )
Dependence measures • Copula functions are linked to non-parametric dependence statistics, as in example Kendall’s or Spearman’s S • Notice that differently from non-parametric estimators, the linear correlation depends on the marginal distributions and may not cover the whole range from – 1 to + 1, making the assessment of the relative degree of dependence involved.
Flipped copulas • Consider a copula corresponding to the probability of the event A and B, Pr(A, B) = C(Pa, Pb). Define the marginal probability of the complements Ac, Bc as Fa=1 – Pa and Pb=1 – P b. • The following duality relationships hold among copulas Pr(A, B) = C(Pa, Pb) Pr(Ac, B) = Pb – C(Ha, Hb) = Ca(Fa, Pb) Pr(A, Bc) = Pa – C(Pa, Pb) = Cb(Pa, Fb) Pr(Ac, Bc) =Fa + Fb – 1 + C(1 – Fa, 1 – Fb) = C(Fa, Fb) = Survival copula • Notice. This property of copulas is paramount to ensure put-call parity relationships in option pricing applications.
Radial symmetry • Take a copula function C(u, v) and its survival version C(u, v) = v + u – 1 + C( 1 – u, 1 – v) • A copula is said to be endowed with the radial symmetry (reflection symmetry) property if C(u, v) = C(u, v)
Radial simmetry example • Take u = v = 20%. Take the gaussian copula and compute N(u, v; 0, 3) = 0, 06614 • Verify that: N(1 – u, 1 – v; 0, 3) = 0, 66614 = = 1 – u – v + N(u, v; 0, 3) • Try now the Clayton copula and compute Clayton(u, v; 0, 2792) = 0, 06614 and verify that Clayton(1 – u, 1 – v; 0, 2792) = 0, 6484 0, 66614
Exchangeability • Another concept of symmetry is «exchageability» H(X, Y) = H(Y, X) • Notice that this implies: • Equal distribution: FX(t) = FY(t) • Exchangeable copula: C(u, v) = C(v, u) • Almost all the copula functions that are used are exchangeable. • Exceptions: • Marshall Olkin copula • Hierarchical copulas
Economic interpretation • Consider two obligors A and B with DPA and DPB. Assume A is much riskier than B, that is DPA > DPB. • Exchangeability means C(DPA, DPB) = C(DPB, DPA) • In other terms, the effect of the higher riskiness of A is the same that we would have is B were riskier. • If the copula is non-exchageable, that is for example C(DPA, DPB) > C(DPB, DPA) than this would mean that a financial crisis of A has a higher impact on the joint probability than it would be if B were riskier.
Conditional probability I • The dualities above may be used to recover the conditional probability of the events.
Tail dependence • Copula functions may be used to compute an index of tail dependence assessing the evidence of simultaneous booms and crashes on different markets • In the case of left tail events:
Fréchet copula • C(x, y) =b. Cmin +(1 – a – b)Cind + a. Cmax , a, b [0, 1] Cmin= max (x + y – 1, 0), Cind= xy, Cmax= min(x, y) • The parameters a, b are linked to non-parametric dependence measures by particularly simple analytical formulas. For example S = a - b • Mixture copulas (Li, 2000) are a particular case in which copula is a linear combination of Cmax and Cind for positive dependent risks (a>0, b =0), Cmin and Cind for the negative dependent (b>0, a =0).
Elliptical copulas • Elliptical multivariate distributions, such as multivariate normal or Student t, can be used as copula functions. • Normal copulas are obtained C(u 1, … ud ) = = Nd (N – 1 (u 1 ), N – 1 (u 2 ), …, N – 1 (ud ); ) and extreme events are indipendent. • For Student t copula functions with v degrees of freedom C (u 1, … ud) = = Td (T – 1 (u 1 ), T – 1 (u 2 ), …, T – 1 (ud ); , v) extreme events are dependent, and the tail dependence index is a function of v.
Monte Carlo simulation Gaussian Copula 1. Cholesky decomposition A of the correlation matrix R 2. Simulate a set of n independent random variables z = (z 1, . . . , zn)’ from N(0, 1), with N standard normal 3. Set x = Az 4. Determine ui = N(xi) with i = 1, 2, . . . , n 5. (y 1, . . . , yn)’ =[F 1 -1(u 1), . . . , Fn-1(un)] where Fi denotes the i-th marginal distribution.
Monte Carlo simulation Student t Copula 1. Cholesky decomposition A of the correlation matrix R 2. Simulate a set of n independent random variables z = (z 1, . . . , zn)’ from N(0, 1), with N standard normal 3. Simulate a random variable s from 2 indipendent from z 4. Set x = Az 5. Set x = ( /s)1/2 y 6. Determine ui = Tv(xi) with Tv the Student t distribution 7. (y 1, . . . , yn)’ =[F 1 -1(u 1), . . . , Fn-1(un)] where Fi denotes the i -th marginal distribution.
Archimedean copulas • Archimedean copulas are build from a suitable generating function from which we compute C(u 1, …, un) = – 1 [ (u 1)+…+ (un)] • The function (x) must have precise properties. Obviously, it must be (1) = 0. Furthermore, it must be decreasing and convex. As for (0), if it is infinite the generator is said strict. • In n dimension a simple rule is to select the inverse of the generator as a completely monotone function (infinitely differentiable and with derivatives alternate in sign). This identifies the class of Laplace transform.
Example: Clayton copula • Take (t) = [t – – 1]/ such that the inverse is – 1(s) =(1 – s) – 1/ the Laplace transform of the gamma distribution. Then, the copula function C(u 1, …, un) = – 1 [ (u 1)+…+ (un)] is called Clayton copula. It is not symmetric and has lower tail dependence (no upper tail dependence).
Example: Gumbel copula • Take (t) = (–log t) such that the inverse is – 1(s) =exp(– s – 1/ ) the Laplace transform of the positive stable distribution. Then, the copula function C(u 1, …, un) = – 1 [ (u 1)+…+ (un)] is called Gumbel copula. It is not symmetric and has upper tail dependence (no lower tail dependence).
Absolutely continuous copulas • Most of the copulas that are used in applied work are absolutely continuous, meaning that the probability mass on every curve in the unit square is zero. • In an application of default times, this means that the probability of two names defaulting at the same time is zero. • Copula functions for which the probability mass is concentrated on some curve are called «singular copulas» . Examples are the perfect dependence copulas (min(u, v)) or so called «shuffles of min» . • Some copulas have both a singular and absolutely continuous part.
Marshall-Olkin copula • The Marshall-Olkin copula has both an absolutely continuous and a singular part u v min(u–α 1, v–α 2) • Kendall’s tau and the value of the singular part have the same value = 0 /( 1 + 2 + 0)
Extended Marshall-Olkin • We may extend Marshall-Olkin assuming Archimedean dependence of the hidden factors (Cherubini and Mulinacci, 2015)
Securitization deals Senior Tranche Originator Sale of Assets Special Purpose Vehicle SPV Junior 1 Tranche Junior 2 Tranche … Tranche Equity Tranche
The economic rationale • Arbitrage (no more available): by partitioning the basket of exposures in a set of tranches the originator used to increase the overall value. • Regulatory Arbitrage: free capital from low-risk/low-return to high return/high risk investments. • Funding: diversification with respect to deposits • Balance sheet cleaning: writing down non performing loans and other assets from the balance sheet. • Providing diversification: allowing mutual funds to diversify investment
Structuring securitization deals • Securitization deal structures are based on three decisions • Choice of assets (well diversified) • Choice of number and structure of tranches (tranching) • Definition of the rules by which losses on assets are translated into losses for each tranches (waterfall scheme)
Choice of assets • The choice of the pool of assets to be securitized determines the overall scenarios of losses. • Actually, a CDO tranche is a set of derivatives written on an underlying asset which is the overall loss on a portfolio L = L 1 + L 2 +…Ln • Obviously the choice of the kinds of assets, and their dependence structure, would have a deep impact on the probability distribution of losses.
Tranche • A tranche is a bond issued by a SPV, absorbing losses higher than a level La (attachment) and exausting principal when losses reach level Lb (detachment). • The nominal value of a tranche (size) is the difference between Lb and La. Size = Lb – La
Kinds of tranches • Equity tranche is defined as La = 0. Its value is a put option on tranches. v(t, T)EQ[max(Lb – L, 0)] • A senior tranche with attachment La absorbs losses beyond La up to the value of the entire pool, 100. Its value is then v(t, T)(100 – La) – v(t, T)EQ[max(L – La, 0)]
Arbitrage relationships • If tranches are traded and quoted in a liquid market, the following no-arbitrage relationships must hold. • Every intermediate tranche must be worth as the difference of two equity tranches EL(La, Lb) = EL(0, Lb) – EL(0, La) • Buyng an equity tranche with detachment La and buyng the corresponding senior tranche (attachment La) amounts to buy exposure to the overall pool of losses. v(t, T)EQ[max(La – L, 0)] + v(t, T)(100 – La) – v(t, T)EQ[max(L – La, 0)] = v(t, T)[100 – EQ (L)]
Risk of different “tranches” • Different “tranches” have different risk features. “Equity” tranches are more sensitive to idiosincratic risk, while “senior” tranches are more sensitive to systematic risk factors. • “Equity” tranches used to be held by the “originator” both because it was difficult to place it in the market and to signal a good credit standing of the pool. In the recent past, this job has been done by private equity and hedge funds.
Securitization zoology • Cash CDO vs Synthetic CDO: pools of CDS on the asset side, issuance of bonds on the liability side • Funded CDO vs unfunded CDO: CDS both on the asset and the liability side of the SPV • Bespoke CDO vs standard CDO: CDO on a customized pool of assets or exchange traded CDO on standardized terms • CDO 2: securitization of pools of assets including tranches • Large CDO (ABS): very large pools of exposures, arising from leasing or mortgage deals (CMO) • Managed vs unmanaged CDO: the asset of the SPV is held with an asset manager who can substitute some of the assets in the pool.
Synthetic CDOs Senior Tranche Originator Protection Sale CDS Premia Interest Payments Collateral AAA Special Purpose Vehicle SPV Investment Junior 1 Tranche Junior 2 Tranche … Tranche Equity Tranche
CDO 2 Originator Senior Tranche 1, j Tranche 2, j Tranche i, j Tranche … Special Purpose Vehicle SPV Junior 1 Tranche Junior 2 Tranche … Tranche Equity Tranche
Standardized CDOs • Since June 2003 standardized securitization deals were introduced in the market. They are unfunded CDOs referred to standard set of “names”, considered representative of particular markets. • The terms of thess contracts are also standardized, which makes them particularly liquid. They are used both to hedged bespoke contracts and to acquire exposure to credit. • 125 American names (CDX) o European, Asian or Australian (i. Traxx), pool changed every 6 months • Standardized maturities (5, 7 e 10 anni) • Standardized detachment • Standardized notional (250 millions)
i-Traxx and CDX quotes, 5 year, September 27 th 2005 i-Traxx CDX Tranche Bid Ask 0 -3% 23. 5* 24. 5* 0 -3% 44. 5* 45* 3 -6% 71 73 3 -7% 113 117 6 -9% 19 22 7 -10% 25 30 9 -12% 8. 5 10 -15% 13 16 12 -22% 4. 5 5. 5 15 -30% 4. 5 5. 5 (*) Amount to be paid “up-front” plus 500 bp on a running basis Source: Lehman Brothers, Correlation Monitor, September 28 th 2005.
Gaussian copula and implied correlation • The standard technique used in the market is based on Gaussian copula C(u 1, u 2, …, u. N) = N(N – 1 (u 1 ), N – 1 (u 2 ), …, N – 1 (u. N ); ) where ui is the probability of event i T and i is the default time of the i-th name. • The correlation used is the same across all the correlation matrix. The value of a tranche can either be quoted in terms of credit spread or in term of the correlation figure corresponding to such spread. This concept is known as implied correlation. • Notice that the Gaussian copula plays the same role as the Black and Scholes formula in option prices. Since equity tranches are options, the concept of implied correlation is only well defined for them. In this case, it is called base correlation. The market also use the term compound correlation for intermediate tranches, even though it does not have mathematical meaning (the function linking the price of the intermediate tranche to correlation is NOT invertible!!!)
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