Copula Models and Speculative Price Dynamics

1. Copula Models and Speculative Price Dynamics Umberto Cherubini University of Bologna RMI Workshop National University of Singapore, 5/2/2010

2. Outline Copula functions: main concepts Copula functions and Markov processes Application to credit (CDX) Application to equity Application to managed funds

3. Copula functions and Markov processes

4. 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) .

5. 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

6. 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.

7. 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.

8. 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.

9. The Fr�chet 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).

10. Ellictical 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.

11. Archimedean 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.

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

13. 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.

14. 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

15. 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, 2010)

16. 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.

17. 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.

18. 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)

19. 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

20. 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.

21. 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.

22. Credit market applications

23. 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

24. 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�

25. Distribution of losses: 10 y

26. Temporal dependence

27. Equity tranche: term structure

28. Senior tranche: term structure

29. A general dynamic model for equity markets

30. 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.

31. 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))

32. 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.

33. Multivariate equity derivatives Pricing algorithm: Estimate the dependence structure of log-increments from time series Simulate the copula function linking levels at different maturities. Draw the pricing surface of strikes and maturities Examples: Multivariate digital notes (Altiplanos), with European or barrier features Rainbow options, paying call on min (Everest Spread options

37. Performance measurement of managed funds

38. Performance measurement Denote X the return on the market, Y the return due to active fund management and Z = X + Y the return on the managed fund In performance measurement we may be asked to determine The distribution of Z given the distribution od X and that of Y The distribution of Y given the distribution of Z and that of X measures from historical data.

39. Asset management style The asset management style is entirely determined by the distribution Y and its dependence with X. Stock picking: the distribution of Y (alpha) Market timing: the dependence of X and Y The analysis of the return Z can be performed as a basket option on X and Y. Passive management: X and Y are independent and Y has zero mean Pure stock picking: X and Y are independent

40. Henriksson Merton copula In the Heniksson Merton approach, it is Y = ? + ? max(0, � X) + ? and the market timing activity results in a �protective put strategy� Notice that market timing does not imply positive dependence between the return on the strategy Y and the benchmark X HM copula is particularly cumbersome to write down (see paper), but it is only a special case of market timing. In general market timing means association (positive or negative) of X and Y

41. Hedge funds Market neutral investment is part of the picture, considering that market neutral investment means H(Z, X) = FZ FX For this reason the distribution of the investment return FY is computed by

42. Multicurrency equity fund

43. Corporate bond fund

44. Reference Bibliography Cherubini U. � E. Luciano � W. Vecchiato (2004): Copula Methods in Finance, John Wiley Finance Series. 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 (2010): �The Dependence Structure of Running Maxima and Minima: Results and Option Pricing Applications�, Mathematical Finance, Cherubini U. � F. Gobbi � S. Mulinacci � S. Romagnoli (2010): �A Copula-Based Model For Spatial and Temporal Dependence of Equity Makets�, in Durante et. Al. Cherubini U. � F. Gobbi � S. Mulinacci � S. Romagnoli (2010): �A Copula-Based Model For Spatial and Temporal Dependence of Equity Makets�, in Durante et. Workshop on Copula Theory and Its Applications, Proceedings, Springer, forthcoming Cherubini U. � F. Gobbi � S. Mulinacci � S. Romagnoli (2010): �On the Term Structure of Multivariate Equity Derivatives� working paper Cherubini U. � F. Gobbi � S. Mulinacci (2010): �Semi-parametric Estimation and Simulation of Actively Managed Portfolios� working paper