Equity and Credit Correlation Products and Copula Functions

1. Equity and Credit Correlation Products and Copula Functions Umberto Cherubini Matemates – University of Bologna Birbeck College, London 04/02/2009

2. Outline Motivation: structured finance products and risk management issues Copula functions: main concepts Radial symmetry and non-exchangeability Estimation and calibration Copula pricing: cross-section dependence Copula pricing: temporal dependence A general dynamic model for equity markets A general dynamic model for credit products Risk management applications

3. Motivation Structured Finance Products and Risk Management Issues

4. Barrier Altiplano Assume a set of coupon periods, k = 1,2,…P. In each period k a set of dates j = 1, 2, …,mk A basket of i = 1, 2, …, n assets Coupon paid: ck if Si(tj)> Ki for all i and all j 0 otherwise

5. European Altiplano Investment period: March 2000 - March 2005 Principal repaid at maturity Coupon paid on march 15th every year. Coupon determination Coupon = 10% if (i = 1,2,3,4,5) Nikkei (15/3/200i) > Nikkei (15/3/2000) e Nasdaq 100 (15/3/200i) > Nasdaq 100 (15/3/2000) Coupon = 0% otherwise Digital note = ZCB + bivariate digital call options

6. Barrier Products Consider a product that at time tm pays 1 unit of cash if the value of the underlying asset X(ti) remains above, a given barrier B level on a set of dates {t1, t2,…, tm}. This is a no-touch option, which is also called digital barrier option, or uni-variate Altiplano. Like the European digital option represents the pricing kernel of European options, the barrier digital product is the pricing kernel of barrier options.

7. “First-to-default” derivatives Consider a credit derivative product that pays protection, to keep things simple in a fixed sum L, the first time that a company in a reference basket of credit risks gets into default. The reference credit risks included in the basket are called “names” in the structured finance jargon. Again to keep things simple, let us assume that payment occurs at expiration date T of the derivative contract If Q(?1 > T, ?2 > T…) denotes the joint survival probability of all the names in the basket, it is straightforward to check that the value of the derivative contract, named “first-to-default” turns out to be “First to default” = LP(t,T)(1 – Q(?1 > T, ?2 > T…))

8. Synthetic CDO

9. 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.

10. Term structure of CDX

11. Risks Products like these are made to provide exposure to specific equity or credit risk sources, with particular reference to correlation among them (correlation products). We will refer to this correlation as dependence or association (more general terms) and we will denote it cross-section dependence (that is dependence among a set of assets or risk factors evaluated at the same time).

12. Cross-section dependence Any pricing strategy for these products requires to select specific joint distributions for the risk-factors or assets. Notice that a natural requirement one would like to impose on the multivariate distributions would be consistency with the price of the uni-variate products observed in the market (digital options for multivariate equity and CDS for multivariate credit) In order to calibrate the joint distribution to the marginal ones one will be naturally led to use of copula functions.

13. Temporal dependence Barrier Altiplanos: the value of a barrier Altiplano depends on the dependence structure between the value of underlying assets at different times. Should this dependence increase, the price of the product will be affected. CDX: consider selling protection on a 5 or on a 10 year tranche 0%-3%. 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.

14. Hybrids Correlation among different risk factors is the main risk in “hybrids”, or “risky swaps” as are called in the market. These are contracts between two parties, A and B, indexed to interest rates, currencies, or credit and contingent on survival of a third counterparty C.

15. Risk management applications Cross-section aggregation of risk measures (risk measure integration). The problem is to compute the joint distribution of losses due to different risk factors (typically, market, credit, operational) Temporal aggregation of risk measures. The problem is to compute the distribution of losses over different time horizons. Temporal aggregation is often a prerequisite to cross-section aggregation since risk measures for market risk are often computed at different horizons with respect to those for credit and operational risk.

16. Risk management of funds Temporal aggregation is a key feature of risk measurement of returns on managed funds. You are not interest in a 10 day risk measure if you are evaluating investment in a mutual fund. You are even less interested if you are planning to invest in a closed-end fund or a hedge fund. The issue is to compute VaR or other risk measures over different time horizons, an issue called “long term VaR”. The state of the art is: i) to include a drift in the specification of the dynamics and ii) to apply the “square-root” rule, that is to multiply the risk measure by T^1/2. The “square root” law is obviously subject to very strict assumptions. The process must have i.i.d. gaussian innovations.

17. Copula functions Main Concepts

18. Compatibility problems In statistics compatibility refers to the relationship between joint distributions (say of dimension n) and marginal distributions (for all dimensions k < n). In finance compatibility means that the price of multi-variate derivatives, namely the value of contingent claims written on a set of events have to be consistent with the values of derivatives written on subsets of the events.

19. Single bets Assume you get 1000 $ if 3 months from now the US stock market is at least 2% lower than today and zero otherwise. How much are you willing to pay for this bet? Say 200 $. This price is linked to a 20% probability of success. Assume you get 1000 $ if 3 months from now the Canadian stock market is at least 3% lower than today and zero otherwise. How much are you willing to pay for this bet? Say 200 $. This price is linked to a 20% probability of success.

20. Multiple bets (Altiplano) Assume you get 1000 $ iff 3 months from now the US stock market is at least 2% lower than today AND the Canadian market is at least 3% lower than today. How much are you willing to pay for this bet? Say 66.14 $. This price is linked to a 6.614% probability of success. Of course 6.614% =v(t,T)C(20%,20%)

21. Copula functions Copula functions are based on the principle of integral probability transformation. Given a random variable X with probability distribution FX(X). Then u = FX(X) is uniformly distributed in [0,1]. Likewise, we have v = FY(Y) uniformly distributed. The joint distribution of X and Y can be written H(X,Y) = H(FX –1(u), FY –1(v)) = C(u,v) Which properties must the function C(u,v) have in order to represent the joint function H(X,Y) .

22. Copula function Mathematics A copula function z = C(u,v) is defined as 1. z, u and v in the unit interval 2. C(0,v) = C(u,0) = 0, C(1,v) = v and C(u,1) = u 3. For every u1 > u2 and v1 > v2 we have VC(u,v) ? C(u1,v1) – C (u1,v2) – C (u2,v1) + C(u2,v2) ? 0 VC(u,v) is called the volume of copula C

23. Copula functions: Statistics 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.

24. Copula functions and dependence structure in risks Copula functions represent a tool to separate the 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 The perfect dependence cases are called Fréchet bounds.

25. Positive orthant dependence Copula functions are clearly linked to dependence. The first measure of dependence we could think of refers to the sign. Positive (negative) orthant dependency determines whether variables co-move in the same direction or in opposite directions. In the previous example C(20%,20%) = 6,614% > 0.2*0.2 = 4%

26. Copula function and dependence structure 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.

27. Dualities among copulas Consider a copula corresponding to the probability of the event A and B, Pr(A,B) = C(Ha,Hb). Define the marginal probability of the complements Ac, Bc as Ha=1 – Ha and Hb=1 – Hb. The following duality relationships hold among copulas Pr(A,B) = C(Ha,Hb) Pr(Ac,B) = Hb – C(Ha,Hb) = Ca(Ha, Hb) Pr(A,Bc) = Ha – C(Ha,Hb) = Cb(Ha,Hb) Pr(Ac,Bc) =1 – Ha – Hb + C(Ha,Hb) = C(Ha, Hb) = Survival copula Notice. This property of copulas is paramount to ensure put-call parity relationships in option pricing applications.

28. Conditional probability I The dualities above may be used to recover the conditional probability of the events.

29. Conditional probability II The conditional probability of X given Y = y can be expressed using the partial derivative of a copula function.

30. Tail dependence in crashes… 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 crashes…

31. …and in booms In the case of booms, we have instead It is easy to check that C(u,v) = uv leads to lower and upper tail dependence equal to zero. C(u,v) = min(u,v) yields instead tail indexes equal to 1.

32. Examples of copula functions The Fréchet family C(x,y) =bCmin +(1 – a – b)Cind + aCmax , 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).

33. Fréchet Family: C(0.3,0,3)

34. Fréchet Family: C(0.3,0.7)

35. Examples of copula functionsEllictical copulas Ellictal multivariate distributions, such as multivariate normal or Student t, can be used as copula functions. Normal copulas are obtained C(u1,… un ) = = N(N – 1 (u1 ), N – 1 (u2 ), …, N – 1 (uN ); ?) and extreme events are indipendent. For Student t copula functions with v degrees of freedom C (u1,… un ) = = T(T – 1 (u1 ), T – 1 (u2 ), …, T – 1 (uN ); ?, v) extreme events are dependent, and the tail dependence index is a function of v.

36. Gaussian copula vs mixture

37. Examples of copula functionsArchimedean copulas Archimedean copulas are build from a suitable generating function ? from which we compute C(u,v) = ? – 1 [?(u)+?(v)] 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.

38. Examples: Frank copula Take ?(t)= – log[(1 – exp(– ?t)/(1 – exp(– ?) ] such that the inverse is ? – 1(s) = – ? – 1 log[(1 –(1 –exp(– ?)exp(– s)] the Laplace transform of the logarithmic series. Then, the copula function C(u1,…, un) = ? – 1 [?(u1)+…+?(un)] is called Frank copula. It is symmetric and does not have tail dependence (either lower or upper).

39. Frank copula vs gaussian

40. Examples: 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(u1,…, un) = ? – 1 [?(u1)+…+?(un)] is called Clayton copula. It is not symmetric and has lower tail dependence (no upper tail dependence).

41. Clayton copula vs Frank copula

42. Examples: 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(u1,…, un) = ? – 1 [?(u1)+…+?(un)] is called Gumbel copula. It is not symmetric and has upper tail dependence (no lower tail dependence).

43. Gumbel copula vs Frank copula

44. Kendall function For the class of Archimedean copulas, there is a multivariate version of the probability integral transfomation theorem. The probability t = C(u,v) is distributed according to the distribution KC (t) = t – ?(t)/ ?’(t) where ?’(t) is the derivative of the generating function. There exist extensions of the Kendall function to n dimensions. Constructing the empirical version of the Kendall function enables to test the goodness of fit of a copula function (Genest and McKay, 1986).

45. Kendall function: Clayton copula

46. Hedge Funds: Market Neutral

47. Hierarchical copulas Consider a set of copula functions C(u1,…, un+m) = = ?21– 1 [?21 ??11– 1(?11(u1)+…+?11(un)) + + ?21 ??12– 1(?12(u1)+…+?12(um)] where ?21 , ?11 and ?12 are generators. This is a copula iff the dependence of the ?21 generator is lower or equal to the dependence of the ?11 and ?12 generators This is called Archimedean hierarchical copula and allows to extend Archimedean copula to arbitrary dimensions with different bivariate dependence

48. Radial symmetry and exchangeability

49. Radial symmetry Take a copula function C(u,v) and its survival version C(1 – u, 1 – v) = 1 – v – u + C( u, v) A copula is said to be endowed with the radial symmetry (reflection symmetry) property if C(u,v) = C(u, v)

50. Radial symmetry: 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

51. Radial symmetry: economics In economics and econometrics, radial symmetry has led to discover phenomena of correlation asymmetry. Empirical evidence have been found that correlation is higher for downward moves of the stock market than for upward moves (Longin and Solnik, Ang and Chen among others).

52. Exceedance correlation Longin and Solnik have first proposed the concept of exceedance correlation: correlation measured on data sampled in the tails. Step 1. Standardize data si = (Si – ?) /? Step 2. Select sub-samples: si > ?, si < – ? Step 3. Corr (si > ?, sj > ?),Corr (si < – ?, sj < – ?) For (radial) symmetric distributions Corr (si > ?, sj > ?) = Corr (si < – ?, sj < – ?)

53. Exceedance rank-correlation Schmid and Schmidt propose a similar concept of conditional rank-correlation.

54. Exchangeable copulas Most of the copula functions used in finance are symmetric or “exchangeable”, meaning C(u,v) = C(v,u) In a recent paper, Nelsen proposes a measure of non-exchangeability 0 ? 3 sup |C(u,v) – C(v,u)| ? 1 Nelsen also identifies a class of maximum non-exchangeable copulas.

55. Non exchangeable copulas A way to extend copula functions to account for non-exchangeability was suggested by Khoudraji (Phd dissertation, 1995). Take copula functions C*(.,.) and C(.,.), and 0 < ?, ? < 1 and define C?, ?(u,v) = C*(u1 – ?, v1 – ?) C(u ?, v ?) The copula function obtained is in general non-exchangeable. In particular, this was used by Genest, Ghoudi and Rivest (1998) taking C*(.,.) the product copula and C(.,.) the Gumbel copula C?, ?(u,v) = u1 – ?v1 – ?C(u ?, v ?)

56. Non-exchangeable copulas: example Take u = 0,2 and v = 0,7 and the Gaussian copula. Verify that N(u, v; 30%) = N (v, u; 30%) = 16,726% Compute now C(u,v) = u0,5 N(u0,5, v; 30%) = 15,511% and C(v,u) = v0,5 N(v0,5, u; 30%) = 15,844%

57. Non-Exchangeability: economics To have an idea of the ecnomic meaning of exchangeability, consider the following case: Lehmann Bros provides capital insurance to life insurance policies to Mediolanum, an Italian bank-insurance company. Assume that in case of default of Lehmann, Mediolanum takes over the cost of insurance (as it actually happened after September 15 2008) Consider now the joint probability of default of Lehmann and Mediolanum. If the marginal probability of default of Lehmann is high, the joint distribution will also be high. But if the marginal probability of Mediolanum is high, the joint distribution might not be as high as in the previous case.

58. C(u,v) – C(v,u)

59. Partial exchangeability Notice that the Archimedean hierarchical copula allows for some non-exchangeability among variables in different groups. For example, assume a hierarchical representation in which assets are grouped by sector at the lower level and between sector at the higher level. Of course, the dependence structure within the group is exchangeable, but between the groups it is not. This case is called partial exchangeability. Econometricians would recognize similarities with the within and between estimators.

60. Estimation and simulation

61. Copula function calibration A first straightforward way of determining the copula function representing the dependence structure between two variables was proposed by Genest and McKay, 1986. The algorithm is particularly simple Change the set of variables in ranks Measure the association between the ranks Determine the parameter of the copula (a family of copula has to be chosen) in order to obtain the same association measure. Plot the Kendall function of each copula and select the copula which is closest to the empirical Kendall function.

62. Nasdaq 100 vs Nikkey 225

63. Gumbel fit

64. Clayton fit

65. Copula density The cross derivative of a copula function is its density. The copula density times the marginal density yields the joint density The density is also called the canonical representation of a copula.

66. Copula function likelihood Using the canonical representation of copulas one can write the log-likelihood of a set of data. Notice that the likelihood may be partitioned in two parts: one only depends on the copula density and the other only on the marginal densities.

67. Maximum Likelihood Estimation Maximum Likelihood Estimation (MLE). The Likelihood is written and maximised with respect to both the parameters of the marginal distributions and those of the copula function simultaneously Inference from the margin (IFM). The likelihood is maximised in two stages by first estimating the parameters of the marginal distributions and then maximizing it with respect to the copula function parameter Canonical Maximum Likelihood (CML). The marginal distribution is not estimated but the data are transformed in uniform variates.

68. Conditional copula functions A problem with specification of the copula function is that both dependence parameters and the marginal distributions can change as time elapses The conditional copula proposal (Patton) is a solution to this problem The key feature is that Sklar’s theorem can be extended to conditional distribution if both the margins and the copula function are function of the same information set.

69. Conditional copula estimation. The conditional copula is C(H(S1t|It), H(S2t|It); ? | It) Step 1. Estimate Garch models for variables S1 and S2 Step 2. Apply the probability integral transform to both S1 and S2 and test the specification Step 3. Estimate the dynamics of the dependence parameter in a ARMA model ?t = ?(f(?t - 1,u t - 1,…u t – p , v t - 1,…,v t – p )) with ?: ?? (0, 1)

70. Dynamic copula functions An alternative, proposed by Van der Goorberg, Genest Verker is based on the estimation, for Archimedean copulas, of the dependence non-parametric statistic as a function of marginal conditional volatilities In particular, they specify Kendall’s ? as ?t = ?0 + ?1 log (max(h1t,h2t))

71. Monte Carlo simulationGaussian Copula Cholesky decomposition A of the correlation matrix R Simulate a set of n independent random variables z = (z1,..., zn)’ from N(0,1), with N standard normal Set x = Az Determine ui = N(xi) with i = 1,2,...,n (y1,...,yn)’ =[F1-1(u1),...,Fn-1(un)] where Fi denotes the i-th marginal distribution.

72. Monte Carlo simulationStudent t Copula Cholesky decomposition A of the correlation matrix R Simulate a set of n independent random variables z = (z1,..., zn)’ from N(0,1), with N standard normal Simulate a random variable s from ?2? indipendent from z Set x = Az Set x = (?/s)1/2y Determine ui = Tv(xi) with Tv the Student t distribution (y1,...,yn)’ =[F1-1(u1),...,Fn-1(un)] where Fi denotes the i-th marginal distribution.

73. Other simulation techniques Conditional sampling: the first marginal is obtained by generating a random variable from the uniform distribution. The others are obtained by generating other uniform random variables and using the inverse of the conditional distribution. Marshall-Olkin: Laplace transforms and their inverse are used to generate the joint variables.

74. Copula pricing: cross-section dependence

75. Digital Binary Note Example Investment period: March 2000 - March 2005 Principal repaid at maturity Coupon paid on march 15th every year. Coupon determination Coupon = 10% if (i = 1,2,3,4) Nikkei (15/3/200i) > Nikkei (15/3/2000) e Nasdaq 100 (15/3/200i) > Nasdaq 100 (15/3/2000) Coupon = 0% otherwise Digital note = ZCB + bivariate digital call options

76. Coupon determination

77. Super-replication It is immediate to check that Max[DCNky + DCNsd – v(t,T),0] = Coupon and Coupon = Min[DCNky,DCNsd ] otherwise it will be possible to exploit arbitrage profits. Fréchet bounds provide super-replication prices and hedges, corresponding to perfect dependence scenarios.

78. Copula pricing It may be easily proved that in order to rule out arbitrage opportunities the price of the coupon must be Coupon = v(t,T)C(DCNky/v(t,T),DCNsd /v(t,T)) where C(u,v) is a survival copula representing dependence between the Nikkei and the Nasdaq markets. Intuition.Under the risk neutral probability framework, the risk neutral probability of the joint event is written in terms of copula, thanks to Sklar theorem,the arguments of the copula being marginal risk neutral probabilities, corresponding to the forward value of univariate digital options. Notice however that the result can be prooved directly by ruling out arbitrage opportunities on the market. The bivariate price has to be consistent with the specification of the univariate prices and the dependence structure. Again by arbitrage we can easily price…

79. …a “bearish” coupon

80. Bivariate digital put options No-arbitrage requires that the bivariate digital put option, DP with the same strikes as the digital call DC be priced as DP = v(t,T) – DCNky – DCNsd + DC = = v(t,T)[1 – DCNky /v(t,T)– DCNsd /v(t,T) + C(DCNky /v(t,T),DCNsd /v(t,T)) ] =v(t,T)C(1 – DCNky /v(t,T),1 – DCNsd /v(t,T)) = v(t,T)C(DPNky /v(t,T),DPNsd /v(t,T)) where C is the copula function corresponding to the survival copula C, DPNky and DPNsd are the univariate put digital options. Notice that the no-arbitrage relationship is enforced by the duality relationship among copulas described above.

81. Radial symmetry: finance Now consider the following pricing problem. Bivariate digital call on Nikkei and Nasdaq with marginal probability of exercise equal to u and v respectively. Bivariate digital put on Nikkei and Nasdaq with marginal probability of exercise equal to u and v respectively. Radial symmetry means that C(u,v) = C(u,v) so that DP(u,v) = DC(u,v) Imagine to recover implied correlation from call and put prices: you would recover a symmetric correlation smile

82. Pricing strategies The pricing of call and put options whose pay-off is dependent on more than one event may be obtained by Integrating the the value of the pay-off with respect to the copula density times the marginal density Integrating the conditional probability distribution times the marginal distribution of a risk factor Integrating the joint probability distribution

83. From the pricing kernel to options The idea relies on Breeden and Litzenberger (1978) By integrating the pricing kernel (i.e. the cumulative or decumulative risk neutral distribution) we may recover put and call prices From simple digital call and put options we can recover call and put prices: simply set Pr(S(T) = u) = Q(u)

84. Joint probability distribution approach Assume a product with pay-off Max[f(S1(T), S2(T)) – K, 0 ] The price can be computed as

85. Conditional probability distribution approach Assume a product with pay-off Max[f(S1(T), S2(T)) – K, 0 ] The price can be computed as

86. AND/OR operators Copula theory also features more tools, which are seldom mentioned in financial applications. Example: Co-copula = 1 – C(u,v) Dual of a Copula = u + v – C(u,v) Meaning: while copula functions represent the AND operator, the functions above correspond to the OR operator.

87. Equity-linked bonds Assume a coupon which is defined and paid at time T. Assume a basket of n = 1,2,…N assets, whose prices are Sn(T). Denote Sn(t0) the reference prices, typically registered at the origin of the contract, and used as strike prices. The coupon of a basket option is max[Average(Sn(T)/Sn(0),1+k] = = (1 + k) + max[Average(Sn(T)/Sn(0) – (1+k),0] with n = 1,2,…,N and a minimum guaranteed return equal to k.

88. Everest Assume a basket of n = 1,2,…N assets, whose prices are Sn(T). Denote Sn(t0) the reference prices, typically registered at the origin of the contract, and used as strike prices. The coupon of an Everest note is max[min(Sn(T)/Sn(0),1+k] = = (1 + k) + max[min(Sn(T)/Sn(0) – (1+k),0] with n = 1,2,…,N and a minimum guaranteed return equal to k. The replicating portfolio is Everest note = ZCB + 2-colour rainbow (call on minimum)

89. Exercises Verify that a product giving a call on the maximum of a basket is short correlation Hint 1: ask whether the pay-off includes a AND or OR operator Hint 2: verify the result writing the replicating portfolio in a bivariate setting Verify that a long position in a “first to default swap” is short correlation

90. Copula pricing:temporal dependence

91. 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.

92. 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

93. 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)

94. Symmetric Markov processes Definition. A Markov process is symmetric if Marginal distributions are symmetric The ? product T1,2(u1, u2) ? T2,3(u2,u3)… ? Tj – 1,j(uj –1 , uj) is radially symmetric Theorem. A ? B is radially simmetric if either i) A and B are radially symmetric, or ii) A ? B = A ? A with A exchangeable and A survival copula of A.

95. 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.

96. Time Changed Brownian Copulas Set h(t,?) an increasing function of time t, given state ?. The copula is called Time Changed Brownian Motion copula (Schmidz, 2003). The function h(t,?) is the “stochastic clock”. If h(t,?)= h(t) the clock is deterministic (notice, h(t,?) = t gives standard Brownian motion). Furthermore, as h(t,?) tends to infinity the copula tends to uv, while as h(s,?) tends to h(t,?) the copula tends to min(u,v)

97. Copula martingale processes A problem for pricing applications is to impose the martingale restriction in the Markov process representation. Cherubini, Mulinacci and Romagnoli, 2008 propose a particular strategy to use copula functions to build martingale process. The novelty of the idea is to use copula functions to model the dependence structure between the increment of a stochastic process and its level before the increment.

98. CheMuRo Model 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

99. 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

100. Dependent increment with gaussian marginals As an example, consider the increments to be standard normal F = ?. We have Notice that marginals are gaussian, since they refer to the sum of gaussian variables The dependence structure is instead given by the copula function C(u,v).

101. Temporal dependence and scaling law Notice that given a marginal distribution and a copula describing the link between increment and the level before it, we simultaneously derive the marginal distribution of the level following the increment and the dependence structure with the level before. Notice that once distribution of the increments and dependence with the levels have been selected, the dynamics of the process is completely specified. So, there is a relationship between dependence of the increments and scaling law of the process.

102. Clayton dependence and gaussian marginals: 5% perc

103. Dependent increments? S&P 500

104. Dependent increments? USD/Euro

105. HFIndex: convertible arbitrage

106. HFIndex: Dedicated short bias

107. HFIndex: Emerging Markets

108. Copula based dynamics Given the evidence above the class of model that seems more appropriate to describe the dynamics of assets seems to be with

109. Martingale restrictions In the model with independent increments the martingale requirement is very easy to implement. In fact, it suffices to choose zero mean distributions for increments. Notice that independent increments are not a necessary requirement. In fact, the martingale requirement may be ensured for symmetric distributions of increments given specific dependence structures.

110. A general dynamic model for equity markets

111. The model of the market Our task is to model jointly cross-section and time series dependence. Setting of the model: A set of ?S1, S2, …,Sm? assets conditional distribution A set of ?t0, t1, t2, …,tn? dates. We want to model the joint dynamics for any time tj, j = 1,2,…,n. We assume to sit at time t0, all analysis is made conditional on information available at that time. We face a calibration problem, meaning we would like to make the model as close as possible to prices in the market.

112. Assumptions Assumption 1. Risk Neutral Marginal Distributions The marginal distributions of prices Si(tj) conditional on the set of information available at time t0 are Qi j Assumption 2. Markov Property. Each asset is generated by a first order Markov process. Dependence of the price Si(tj -1) and asset Si(tj) from time tj-1 to time tj is represented by a copula function Tij – 1,j(u,v) Assumption 3. No Granger Causality. The future price of every asset only depends on his current value, and not on the current value of other assets. This implies that the m x n copula function admits the hierarchical decomposition C(G1 (Q11, Q12,… Q1n)…, Gm(Qm1, Qm2,… Qmn))

113. No-Granger Causality The no-Granger causality assumption, namely P(Si(tj)? S1(tj –1),…, Sm(tj –1)) = P(Si(tj)? Si(tj –1)) enables the extension of the martingale restriction to the multivariate setting. In fact, we assuming Si(t) are martingales with respect to the filtration generated by their natural filtrations, we have that E(Si(tj)?S1(tj –1),…, Sm(tj –1)) = = E(Si(tj)?Si(tj –1)) = S(t0) Notice that under Granger causality it is not correct to calibrate every marginal distribution separately.

114. Running maxima and minima Due to the first order Markov assumption the dynamics of each asset can be represented as a function of the bivariate copulas Tij – 1,j(u,v) Running Maximum: Gij (u1, u2,… uj) = = Ti1,2(u1, u2) ? Ti2,3(u2,u3)… ? Ti j – 1,j(uj –1 , uj) Running Minimum: Gij (u1, u2,… uj) = = Ti1,2(u1, u2) ? Ti2,3(u2,u3)… ? Ti j – 1,j(uj –1 , uj) Notice that the operator implies recursions like Gij (u1, u2,… uj) = Gij – 1(u1, u2,… uj –1) ? Ti j – 1,j(uj –1 , uj)

115. Univariate barrier Altiplanos Risk neutral valuation implies compatibility restrictions between barrier and European options. In fact, set DC(K, tk) : forward price of the European digital option paying one unit of cash iff S(tk) > K. NT(K, tk) : forward price of the barrier digital option paying one unit of cash iff S(tp) > K, for all p =1,2,…,k. Price compatibility requires then NT(K, tk) = NT(K,tk–1) ? Ti k – 1,k(DC(K, tk–1) , DC(K, tk)) Notice that the Markov property assumption implies this recursive structure (bootstrapping)

116. Barrier BootstrappingBrownian motion and O-U clock

117. Cross-section compatibility Assume: Q(Si(tm) > H, Sj(tm) > K) = ?(Q(Si(tm) > H),Q(Sj(tm) > K)), then Q(mink?m Si(tk) > H, mink?m Sj(tk) > K) = = ?(Q(mink?m Si(tk) > H),Q(mink?mSj(tk) > K))) Sketch of proof for the symmetric case. Use the reflection principle Pr(mink?m Si(tk) > K) = 2 Pr(Si(tm) > K) – 1 to prove Pr(mink?m Si(tk) > H, mink?m Si(tk) > K) = C(Pr(mink?m Si(tk) > H), Pr(mink?m Sj(tk) > K) = C(Pr(Ui > ui), Pr(Uj > uj)) = C(Pr[(Ui +1)/2> (ui +1)/2], Pr[(Uj +1)/2 > (uj +1)/2] = Pr(Si(tm) > H, Sj(tm) > K) =?((ui +1)/2, (uj +1)/2)

118. European and Barrier Altiplanos DCi(K, tm) : forward price of the European digital option paying one unit of cash iff Si(tm) > K. NTi(K, tm) : forward price of the barrier digital option paying one unit of cash iff Si(tp) > K, for all p =1,2,…,m. The price of a European Altiplano is EA = ?(DC1(K, tm), DC2(K, tm),…, DCn(K, tm)) The price of a barrier Altiplano is BA = ?(NT1(K, tm), NT2(K, tm),…, NTn(K, tm))

119. European and barrier AltiplanosTemporal Dependence Shocks

120. European and barrier AltiplanosCross-Section Dependence Shocks

121. Multivariate credit products

122. Application to credit market 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

123. 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. Start with the probability distribution of increments in the first period F1 and set F1 = H1. Numerically compute where z is now a grid of values of the variable 3. Go back to step 2, using F3 and H2 compute H3…

124. Distribution of losses: 10 y

125. Temporal dependence

126. Equity tranche: term structure

127. Senior tranche: term structure

128. “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.

129. Risk management applications

130. Value-at-Risk Aggregation Assume you want to compute the Value at Risk of an investment on a hedge funds over different time horizons. Applying the “square root” law would imply independent increments, which is inconsistent with findings for hedge funds. One could estimate the dependence structure between increments and levels and apply the aggregation algorithm described above.

131. Value-at-Risk for Hedge Funds

132. Counterpart risk in derivatives Most of the derivative contracts, particularly options, forward and swaps, are traded on the OTC market, and so they are affected by credit risk Credit risk may have a relevant impact on the evaluation of these contracts, namely, The price and hedge policy may change Linear contracts can become non linear Dependence between the price of the underlying and counterparty default should be accounted for

133. The replicating portfolio approach The idea is to design a replicating portfolio to hedge and price counterparty derivatives Goes back to Sorensen and Bollier (1994) approach: counterparty risk in swaps represented as a sequence of swaptions Copula functions may be used to extend the idea to dependence between counterparty risk and the underlying

134. Dependence structure A more general approach is to account for dependence between the two main events under consideration Exercise of the option Default of the counterparty Copula functions can be used to describe the dependence structure between the two events above.

135. Vulnerable digital call option Consider a vulnerable digital call (VDC) option paying 1 euro if S(T) > K (event A). In this case, if the counterparty defaults (event B), the option pays the recovery rate RR. The payoff of this option is VDC = v(t,T)[H(A,Bc)+RR H(A,B)] = v(t,T) [Ha – H(A,B)+RR H(A,B)] = v(t,T)Ha – (1 – RR)H(A,B) = DC – v(t,T) Lgd C(Ha, Ha)

136. Vulnerable digital put option Consider a vulnerable digital put (VDP) option paying 1 euro if S(T) = K (event Ac). In this case, if the counterparty defaults (event B), the option pays the recovery rate RR. The payoff of this option is VDP = DP – v(t,T)(1 – RR)H(Ac,B) = P(t,T)Ha – v(t,T)(1 – RR)H(Ac,B) = P(t,T)Ha – v(t,T)(1 – RR)[Hb – C(Ha, Hb)] = v(t,T)(1 – Ha) – v(t,T) Lgd [Hb – C(Ha, Hb)] = v(t,T) – VDC – v(t,T) Lgd Hb

137. Vulnerable digital put call parity Define the expected loss EL = Lgd Hb. If D(t,T) is a defaultable ZCB issued by the counterparty we have D(t,T) = v(t,T)(1 – EL) Notice that copula duality implies a clear no-arbitrage relationship VDC + VDP = v(t,T) – v(t,T) EL = D(t,T) Buying a vulnerable digital call and put option from the same counterparty is the same as buying a defaultable zero-coupon bond

138. Vulnerable call and put options

139. Vulnerable put-call parity

140. Example: swap credit risk Counterparty BBB

141. Reference Bibliography I Nelsen R. (2006): Introduction to copulas, 2nd Edition, Springer Verlag Joe H. (1997): Multivariate Models and Dependence Concepts, Chapman & Hall Cherubini U. – E. Luciano – W. Vecchiato (2004): Copula Methods in Finance, John Wiley Finance Series. Cherubini, U. (2004): “Pricing Swap Credit Risk with Copulas”, working paper Cherubini U. – E. Luciano (2003) “Pricing and Hedging Credit Derivatives with Copulas”, Economic Notes, 32, 219-242. Cherubini U. – E. Luciano (2002) “Bivariate Option Pricing with Copulas”, Applied Mathematical Finance, 9, 69-85 Cherubini U. – E. Luciano (2002) “Copula Vulnerability”, RISK, October, 83-86 Cherubini U. – E. Luciano (2001) “Value-at-Risk Trade-Off and Capital Allocation with Copulas”, Economic Notes, 30, 2, 235-256

142. Reference bibliography II Cherubini U. – S. Mulinacci – S. Romagnoli (2008): “Copula Based Martingale Processes and Financial Prices Dynamics”, working paper. Cherubini U. – Mulinacci S. – S. Romagnoli (2008): “A Copula-Based Model of the Term Structure of CDO Tranches”, in Hardle W.K., N. Hautsch and L. Overbeck (a cura di) Applied Quantitative Finance,,Springer Verlag, 69-81 Cherubini U. – S. Romagnoli (2008): “The Dependence Structure of Running Maxima and Minima: Results and Option Pricing Applications”, Mathematical Finance, forthcoming Cherubini U. – S. Romagnoli (2008): “Computing Copula Volume in n Dimensions”, Applied Mathematical Finance, forthcoming