Copulas from Fokker-Planck equation

1. Copulas from Fokker-Planck equation Hi Jun Choe Dept of Math Yonsei University Seoul, Korea

2. Financial Crisis in 2007 Wired Magazine 02.23.09 Recipe for Disaster:The Formula That Killed Wall Street by S. Salmon Gaussian Copula by Davis X. Li “On Default Correlation:A Copula Function Approach”, The Journal of Fixed Income, 2000.

3. Bond Market Investors needed clear probability concept to manage risks. Quants at Wall Street were excited by the convenience, elegance and tractability of Gaussian copula and adopted universally in risk management. The amount of CDS(credit default swap) increased from 920 billion dollar in 2001 to 62 trillion dollar by 2007. The amount of CDO(collateral debt obligation) increased from 275 billion dollar in 2000 to 4.7 trillion by 2006.

4. Portfolio selection Efficient portfolios are given by the mean variance optimization; Sup a x with a ∑ a < c and a 1=1, t t t where x is expected return vector and ∑ is covariance matrix . The variance corresponds to the risk measure, but it implies the world is Gaussian. There arise two problems: Gaussian assumption and joint distribution modeling.

5. Danger of Uncertainty Structure of Decision Makers: Quant-Trader-Sales The correlation of financial quantities are notoriously unstable and highly volatile. The market is stable with 99% probability although the 1% failure produces huge impact. Thus everybody ignored the warning signal.

6. Introduction P6 Black-Scholes formula is challenged in two aspects Non-normality of asset return that appears as volatility smile and structure form of Implied volatility(When there is smile effect, the return shows non-normality and the linear correlation shows bias). 2. Market incompleteness. Decides the asset value by a general stochastic differential equation(SDE). The complexity of financial market causes a significant difficulty in hedging a large variety of different risks for a financial institute. The derivative products are mutually connected and often exotic. Copulas from Fokker-Planck equation

7. Introduction(cont’d) P7 Chapman-Kolmogrov equation One focuses on the marginal distributions of each product and considers the correlation of them. The Copulas are of great help to evaluation and hedging of Derivative products. A filtered probability space generated by the stochastic process is Markov and the transition probability density Function satisfies Chapman-Kolmogorov equation Copulas from Fokker-Planck equation

8. Introduction(cont’d) P8 Sklar’s Theorem Let H be a two-dimensional distribution function with marginal distribution functions F and G. Then there exists a copula C such that Conversely, for any univariate distribution functions F and G and any copula C, the function H is a two-dimensional distribution function with marginals F and G. Furthermore, if F and G are continuous, then C is unique. Copulas from Fokker-Planck equation

9. Fokker-Planck equation for copula P9 Copula function is Copula if is continuous function satisfying for all and . From condition (1), (2) and (3) we could prove that and for all . Copulas from Fokker-Planck equation

10. Concordance Definition: D is a measure of concordance for two random variables X and Y whose copula is C if -1 =K(X,-X)=< K(C) =< K(X,X)=1 K(X,Y)=K(Y,X) If X and Y are independent, K(X,Y)=0 K(-X,Y)=K(X,-Y)=-K(X,Y) If C1 < C2, then K(C1) < K(C2) Example: Kendall’s tau, Spearman’s rho and Gini indices

11. index Τ= 4∫C(u,v) dC(u,v) -1 ρ= 12 ∫uvdC(u,v) -3 Г= 2 ∫|u+v-1| -|u-v| dC(u,v)

12. Dependence Definition: D is a measure of dependence for two randon variables X and Y whose copula is C if 0=D(uv) =< D(C)=<D(Min(u,v)) =1 D(X,Y)=D(Y,X) D(uv)=D(X,Y)=0 if and only if X and Y are independent D(X,Y)=D(Min(u,v))=1 if and onlly if each of X and Y Is almost surely monotone increasing function of the other 6. D(h1(X),h2(X))=h(X,Y) for increasing functions h1 and h2 Example: Schweitzer and Wolff’s sigma and Hoeffding’s phi

13. index Σ= 12 ∫ |C(u,v)-uv| dudv Φ = 90 ∫ |C(u,v) – uv|^2 dudv

14. Introduction(cont’d) P14 Example of copula(Gaussian copula) Gaussian copulafunction : : the standard bivariate normal cumulative distribution function with correlationρ : the standard normal cumulative distribution function Differentiating C yields the copula density function: is the density function for the standard bivariate Gaussian. is the standard normal density. Copulas from Fokker-Planck equation

15. Introduction(cont’d) P15 Example of copula(Archimedian copula) Unlike elliptical copulas (e.g. Gaussian), most of the Archimedean copulas have closed-form solutions and are not derived from the multivariate distribution functions using Sklar’s theorem. One particularly simple form of a n-dimensional copula is where is known as a generator function. Any generator function which satisfies the properties below is the basis for a valid copula: Copulas from Fokker-Planck equation

16. Introduction(cont’d) P16 Example of copula(Archimedian copula) Gumbel copula : Frank copula : Periodic copula : In 2005 AurélienAlfonsi and DamianoBrigo introduced new families of copulas based on periodic functions. They noticed that if ƒ is a 1-periodic non-negative function that integrates to 1 over [0, 1] and F is a double primitive of ƒ, then both are copula functions, the second one not necessarily exchangeable. This may be a tool to introduce asymmetric dependence, which is absent in most known copula functions. Copulas from Fokker-Planck equation

17. Introduction(cont’d) P17 Example of copula(Empirical copulas) Empirical copulas : When analysing data with an unknown underlying distribution, one can transform the empirical data distribution into an "empirical copula" by warping such that the marginal distributions become uniform. Mathematically the empirical copula frequency function is calculated by where x(i) represents the ith order statistic of x. Less formally, simply replace the data along each dimension with the data ranks divided by n. Copulas from Fokker-Planck equation

18. Introduction(cont’d) P18 Example of copula(Bernstein copula) Let 2-dimension case : Copulas from Fokker-Planck equation

19. Introduction(cont’d) P19 Example of copula(Student-t copula) Student-t copula : Copulas from Fokker-Planck equation

20. Introduction(cont’d) P20 Example of copula(Marshall-Olkin copula) Marshall-Olkin copula: The Marshall-Olkin copula is a function With an appropriate extension of its domain to , the copula is a joint distribution function with marginals uniform on [0,1]. This copula depends on a parameter θ∈[0,1](we consider the case in which the variables are exchangeable) that reflexes the dependent structure existing between the marginals, from the stochastic independent situation (θ=0) to the situation of co-monotonicity (θ=1). Copulas from Fokker-Planck equation

21. Introduction(cont’d) P21 Maximum and Minimum copulas Maximum copula: M(u,v) = Min (u,v) Minimum copula: W(u,v) = Max (u+v-1,0) W (u,v) =< C(u,v) =< M(u,v) Copulas from Fokker-Planck equation

22. Fokker-Planck equation for copula(cont’d) P22 Fokker-Planck equation and joint pdf : joint pdf of at time t. where By integrating Copulas from Fokker-Planck equation

23. Fokker-Planck equation for copula(cont’d) P23 Fokker-Planck equation and joint pdf . Copulas from Fokker-Planck equation

24. Fokker-Planck equation for copula(cont’d) P24 Fokker-Planck equation and joint pdf . , where the distribution function is . . Copulas from Fokker-Planck equation

25. Fokker-Planck equation for copula(cont’d) P25 Fokker-Planck equation and joint pdf Hence . Copulas from Fokker-Planck equation

26. Inference Function of Margin In market, we have to deal with hundreds or thounds financial data which are correlated. Finding the joint probability density function is very difficult. Further, if time is a main parameter, it is almost impossible to find their joint pdf. Therefore, we only Consider each data separately, namely, find the marginal distribution of each data. The correlation is obtained using the marginal distributions.

27. Fokker-Planck equation for copula(cont’d) P27 Fokker-Planck equation and joint pdf Relation between copula and marginal distribution function satisfies the Fokker-Planck equation. . From inference function of margin, we consider separable structure SDE Under Markov property, the joint pdf satisfies Fokker-Planck equation Copulas from Fokker-Planck equation

28. Fokker-Planck equation for copula(cont’d) P28 Fokker-Planck equation and joint pdf We find that the marginal distribution functions satisfy . where and from the separable structure of SDE, the marginal distribution functions and can be solved independently. Copulas from Fokker-Planck equation

29. Fokker-Planck equation for copula(cont’d) P29 Fokker-Planck equation and joint pdf , If we define then . Distribution function is and satisfies with the boundary condition and the initial condition Copulas from Fokker-Planck equation

30. Fokker-Planck equation for copula(cont’d) P30 Fokker-Planck equation and Copula Change variable to new variables and thus The Copula satisfies the Fokker-Planck equation : . Copulas from Fokker-Planck equation

31. Fokker-Planck equation for copula(cont’d) P31 Fokker-Planck equation and Copula Considering the equation for marginal distributions . In with the boundary condition For all and and the initial condition Copulas from Fokker-Planck equation

32. Fokker-Planck equation for copula(cont’d) P32 Fokker-Planck equation and copula Conversely, if C is a solution to with the copula boundary condition, then C is copula. The maximum principle for the derivatives of C is key ingredient for proof. Copulas from Fokker-Planck equation

33. Fokker-Planck equation for copula(cont’d) P33 Fokker-Planck equation and theorem Theorem. We consider the solution to for a large k. Then we find that are independent standard Brownian processes. Copulas from Fokker-Planck equation

34. Numerical Study P34 Marginal distribution function Stochastic differential equations : . Copulas from Fokker-Planck equation

35. Numerical Study (cont’d) P35 Quantile-Quantile . Copulas from Fokker-Planck equation

36. Numerical Study (cont’d) P36 Copular(independent SDE) . Copulas from Fokker-Planck equation

37. Numerical Study (cont’d) P37 Copular(dependent SDE) . Copulas from Fokker-Planck equation

38. Thank you!! P38 Thank you !! Copulas from Fokker-Planck equation