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Copula Models and Speculative Price Dynamics

Copula Models and Speculative Price Dynamics. Umberto Cherubini University of Bologna RMI Workshop National University of Singapore, 5/2/2010. Outline. Copula functions: main concepts Copula functions and Markov processes Application to credit (CDX) Application to equity

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Copula Models and Speculative Price Dynamics

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  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 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 parametersa,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 – 1X(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

  34. Performance measurement of managed funds

  35. 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.

  36. 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

  37. 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

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

  39. Multicurrency equity fund

  40. Corporate bond fund

  41. 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

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