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Copula Functions and Markov Processes for Equity and Credit Derivatives

Copula Functions and Markov Processes for Equity and Credit Derivatives . Umberto Cherubini Matemates – University of Bologna Birbeck College, London 24/02/2010. Outline. Copula functions: main concepts Copula functions and Markov processes Application to credit (CDX) Application to equity

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Copula Functions and Markov Processes for Equity and Credit Derivatives

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  1. Copula Functions and Markov Processes for Equity and Credit Derivatives Umberto Cherubini Matemates – University of Bologna Birbeck College, London 24/02/2010

  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. Coupon determination

  10. 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.

  11. 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…

  12. …a “bearish” coupon

  13. 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.

  14. 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.

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

  16. 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…

  17. …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.

  18. 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).

  19. 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.

  20. 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.

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

  22. 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.

  23. 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

  24. 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)

  25. 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.

  26. 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.

  27. 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)

  28. 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

  29. 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.

  30. 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.

  31. Credit market applications

  32. Top down vs bottom up • In credit risk applications, top down approaches denote models that specify the joint distribution of the default events and the marginal probability of default of each “name” as an outcome. The main shortcoming of the approach is to calibrate the model to single name derivatives products. • Bottom up approaches model the term structure of single name default in the first place and joint default probability after that. That can be done either using copula functions or multivariate intensity models (Marshal Olkin, for example). The main flaw of this approach is to ensure temporal consistency (particularly if one uses copula functions).

  33. 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

  34. 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…

  35. Distribution of losses: 10 y

  36. Temporal dependence

  37. Equity tranche: term structure

  38. Senior tranche: term structure

  39. A general dynamic model for equity markets

  40. Top-down vs bottom up • When pricing multivariate equity derivatives one is required to satisfy two conditions: • Multivariate prices must be consistent with univariate prices • Prices must be temporally consistent and must be martingale • One approach, that we call top down, consists in the specification of the multivariate distribution and the determination of univariate distributions • On another approach, that we call bottom up, one first specifies the univariate distributions and then the joint distribution in the second stage.

  41. Top down vs bottom up • Top down approaches include: Wishart processes, Jacobi processes for average correlation, Radon transform to recover the multivariate density form option prices (multivariate Breeden and Litzenberger). In this approach it may be difficult to calibrate all univariate prices simultaneously. • Bottom up approaches include copula functions. For copula functions it may be very difficult to ensure the martingale requirement.

  42. 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.

  43. SCOMDY dynamics • The analysis is based on a very flexible multivariate asset dynamics called SCOMDY (Semi-Parametric Copula-based Multivariate Dynamics) due to Chen and Fan (2006). • The idea is a multivariate setting in which the price increments are linked by copula functions. • We build a model with this structure and we build into it the features that enable to ensure the martingale requirement. In a single world, we design a market wiht SCOMDY dynamics and independent increments.

  44. 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. • Notice: the independent increment property guarantees both the Markov property and no-Granger-causality

  45. 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, 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.

  46. H-condition • H-condition denotes the case in which a process which is a martingale with respect to a filtration remains a martingale with respect to an enlarged filtration • H-condition and no-Granger-causality are very close concepts. No Granger causality enables to say that if a process is Markov with respect to an enlarged filtration it remains Markov with respect to rhe natural filtration. Based on this, a result due to Bremaud and Yor states that the H-condition holds. • Notice that the H-condition allows to obtain martingales by linking martingale processes with copulas. It justifies mixing cross-section analysis (to calibrate martingale prices) and time series analysis (to estimate dependence).

  47. 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

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