Copula methods in Finance

Transcription

1 Wolfgang K. Härdle Ostap Okhrin Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Center for Applied Statistics and Economics Humboldt-Universität zu Berlin

2 Motivation 1-1 In the mid-'80s, Wall Street turned to the quants brainy nancial engineers to invent new ways to boost prots. Their methods for minting money worked brilliantly... until one of the them devastated the global economy.

3 Motivation 1-2 Here's what killed your 401(k). David X. Li's Gaussian copula function as rst published in Investors exploited it as a quick and fatally awed way to assess risk. Probability - Specically, this is a joint default probabilitythe likelihood that any two members of the pool (A and B) will both default. It's what investors are looking for, and the rest of the formula provides the answer.

4 Motivation 1-3 Here's what killed your 401(k). David X. Li's Gaussian copula function as rst published in Investors exploited it as a quick and fatally awed way to assess risk. Survival times - The amount of time between now and when A and B can be expected to default. Li took the idea from a concept in actuarial science that charts what happens to someone's life expectancy when their spouse dies.

5 Motivation 1-4 Here's what killed your 401(k). David X. Li's Gaussian copula function as rst published in Investors exploited it as a quick and fatally awed way to assess risk. Distribution functions - The probabilities of how long A and B are likely to survive. Since these are not certainties, they can be dangerous: Small miscalculations may leave you facing much more risk than the formula indicates.

6 Motivation 1-5 Here's what killed your 401(k). David X. Li's Gaussian copula function as rst published in Investors exploited it as a quick and fatally awed way to assess risk. Copula - This couples (hence the Latinate term copula) the individual probabilities associated with A and B to come up with a single number. Errors here massively increase the risk of the whole equation blowing up.

7 Motivation 1-6 Here's what killed your 401(k). David X. Li's Gaussian copula function as rst published in Investors exploited it as a quick and fatally awed way to assess risk. Gamma - The all-powerful correlation parameter, which reduces correlation to a single constant-something that should be highly improbable, if not impossible. This is the magic number that made Li's copula function irresistible.

8 Motivation 1-7 Example we pay 200 EUR for the chance to win 1000 EUR, if DAX returns decrease by 2% P DAX (r DAX 0.02) = F DAX ( 0.02) = 0.2

9 Motivation 1-8 Example we pay 200 EUR for the chance to win 1000 EUR, if DAX returns decrease by 2% P DAX (r DAX 0.02) = F DAX ( 0.02) = 0.2 we pay 200 EUR for the chance to win 1000 EUR, if DJ returns decrease by 1% P DJ (r DJ 0.01) = F DJ ( 0.01) = 0.2

10 Motivation 1-9 Example we get 1000 EUR if DAX and DJ indices decrease simultaneously by 2% and 1% respectively. how much are we ready to pay in this case? P{(r DAX 0.02) (r DJ 0.01)} = F DAX,DJ ( 0.02, 0.01) = C{F DAX ( 0.02), F DJ ( 0.01)} = C(0.2, 0.2).

11 Motivation 1-10 Outline 1. Motivation 2. Univariate Distributions and their Estimation 3. Multivariate Distributions and their Estimation 4. Copulae 5. Classical Approach 6. Our Findings

12 Univariate Distributions 2-1 Univariate Case Let x 1,..., x n be realizations of the random variable X X F, where F is unknown Example 1 x i are returns of the asset for one rm at the day t i x i are numbers of sold albums The Man Who Sold the World by David Bowie at day t i What is a good approximation of F? traditional or modern approach

13 Univariate Distributions 2-2 DAX (P t ) P t date

14 Univariate Distributions 2-3 Histogram of DAX Density 0e+00 1e 04 2e 04 3e 04 4e 04 5e 04 6e P

15 Univariate Distributions 2-4 DAX returns (r t = log P t P t 1 ) r t date

16 Univariate Distributions 2-5 Histogram of DAX returns Density r

17 Univariate Distributions 2-6 Traditional approach: F 0 known distribution parameters of F 0 are estimated from the sample x 1,..., x n F 0 = N(μ, σ 2 ) (μ, σ), here ˆμ = x, ˆσ 2 = ŝ 2 F 0 = St(α, β, μ, σ 2 ) (α, β, μ, σ) are estimated by Hull Estimator, Tail Exponent Estimation, etc. check the appropriateness of F 0 by a test (KS type) if test conrm F 0, use ˆF 0 H 0 : F = F 0 vs H 1 : F = F 0

18 Univariate Distributions 2-7 Fit of the Normal distribution to DAX returns (ˆμ = , ˆσ 2 = ) Density R

19 Univariate Distributions 2-8 Modern approach: calculate the edf ˆF (x) = 1 n n I{X i x}, i=1 or the nonparametric kernel smoother ˆfh (x) = 1 n ( ) x Xi K nh h i=1 name Uniform Epanechnikov Gaussian K(u) 1 I{ u 1} (1 u2 )I{ u 1} 1 exp { 2π 1 u2} 2

20 Univariate Distributions 2-9 Kernel smoothing with UNI kernel x = ( 0.475, 1.553, 0.434, 1.019, 0.395)

21 Univariate Distributions 2-10 Kernel smoothing with EPA kernel x = ( 0.475, 1.553, 0.434, 1.019, 0.395)

22 Univariate Distributions 2-11 Kernel smoothing with GAU kernel x = ( 0.475, 1.553, 0.434, 1.019, 0.395)

23 Univariate Distributions 2-12 The estimated density of DAX returns Density Histogram Epanechnikov Normal t Vert., 5 df R

24 Univariate Distributions 2-13 Multivariate Case {x 1i,..., x di } i=1,...,n is the realization of the vector (X 1,..., X d ) F, where F is unknown. Example 2 {x 1i,..., x di } i=1,...,n are returns of the d assets in the portfolio at day t i (x 1i, x 2i ) are numbers of sold albums The Man Who Sold The World by David Bowie and singles I Saved The World Today by Eurythmics at day t i

25 Univariate Distributions 2-14 Multivariate Case What is a good approximation of F? traditional or modern approach Very exible approximation to F is challenging in high dimension due to curse of dimensionality.

26 Univariate Distributions 2-15 Traditional approach: Mainly restricted to the class of elliptical distributions: Normal or t distributions 1 f N (x 1,..., x d ) = { exp 1 } Σ (2π) d 2 (x μ) Σ 1 (x μ) Drawbacks of the elliptical distributions: 1. does not often describe nancial data properly 2. huge number of parameters to be estimated d(d 1) f.e. for Normal distribution: + }{{ 2 }{{} 2d } in margins in dependency 3. ellipticity

27 Univariate Distributions 2-16 Simulate X N(μ, Σ) with the sample size n = 1000 and estimate the parameters (ˆμ, ˆΣ) Σ = ˆΣ = μ = (0, 0, 0) ˆμ = (0.0175, , ) ˆΣ and Σ are not close to each other for only 3 dimensions and quiet big sample

28 Univariate Distributions 2-17 Correlation Gaussian Gumbel Figure 1: Scatterplots for two distribution with ρ = 0.4 same marginal distributions same linear correlation coecient

29 Univariate Distributions 2-18 Extreme, synchronized rises and falls in nancial markets occur infrequently but they do occur. The problem with the models is that they did not assign a high enough chance of occurrence to the scenario in which many things go wrong at the same time - the perfect storm scenario (Business Week, September 1998)

30 Univariate Distributions 2-19 Correlation bmw siemens Black Monday Berlin Wall Kremlin , , , , , Krise

31 Copulae 3-1 Copula For a distribution function F with marginals F X1,..., F Xd, there exists a copula C : [0, 1] d [0, 1], such that F (x 1,..., x d ) = C{F X1 (x 1 ),..., F Xd (x d )}.

32 Copulae 3-2 A little bit of history 1940s: Wassilij Hoeding studies properties of multivariate distributions , b. Mustamäki, Finland; d. Chapel Hill, NC gained his PhD from U Berlin in work in U Berlin Wassilij Hoeding on BBI

33 Copulae 3-3 A little bit of history 1940s: Wassilij Hoeding studies properties of multivariate distributions 1959: The word copula appears for the rst time (Abe Sklar) 1999: Introduced to nancial applications (Paul Embrechts, Alexander McNeil, Daniel Straumann in RISK Magazine) 2000: Paper by David Li in Journal of Derivatives on application of copulae to CDO 2006: Several insurance companies, banks and other nancial institutions apply copulae as a risk management tool

34 Copulae 3-4 Applications Practical Use: 1. medicine (Vandenhende (2003)) 2. hydrology (Genest and Favre (2006)) 3. biometrics (Wang and Wells (2000, JASA), Chen and Fan (2006, CanJoS)) 4. economics portfolio selection (Patton (2004, JoFE), Xu (2004, PhD thesis), Hennessy and Lapan (2002, MathFin)) time series (Chen and Fan (2006a, 2006b, JoE), Fermanian and Scaillet (2003, JoR), Lee and Long (2005, JoE)) risk management (Junker and May (2002, EJ), Breyman et. al. (2003, QF))

35 Copulae 3-5 Applications Bourdeau-Brien (2007) covers 871 publications

36 Copulae 3-6 Special Copulas Theorem Let C be a copula. Then for every (u 1, u 2 ) [0, 1] 2 max (u 1 + u 2 1, 0) C(u 1, u 2 ) min(u 1, u 2 ), where bounds are called lower and upper Fréchet-Hödings bounds. When they are copulas they represent perfect negative and positive dependence respectively. The simplest copula is product copula Π(u 1, u 2 ) = u 1 u 2 characterize the case of independence.

37 Copulae 3-7 Copula Classes 1. elliptical implied by well-known multivariate df's (Normal, t), derived through Sklar's theorem do not have closed form expressions and are restricted to have radial symmetry 2. Archimedean C(u 1, u 2 ) = φ 1 {φ(u 1 ) + φ(u 2 )} allow for a great variety of dependence structures closed form expressions several useful methods for multivariate extension not derived from mv df's using Sklar's theorem

38 Copulae 3-8 Copula Examples 1 Gaussian copula C G δ (u 1, u 2 ) = Φ δ {Φ 1 (u 1 ), Φ 1 (u 2 )} = Φ 1 (u 1) Φ 1 (u 2) 1 2π 1 δ 2 exp Gaussian copula contains the dependence structure { } (s 2 2δst + t 2 ) ds dt, 2(1 δ 2 ) normal marginal distribution + Gaussian copula = multivariate normal distributions non-normal marginal distribution + Gaussian copula = meta-gaussian distributions allows to generate joint symmetric dependence, but no tail dependence

39 Copulae 3-9 Copula Examples 2 Gumbel copula { [ Cθ Gu (u 1, u 2 ) = exp ( log u 1 ) 1/θ + ( log u 2 ) 1/θ] } θ. for θ > 1 allows to generate dependence in the upper tail for θ = 1 reduces to the product copula for θ obtain Frèchet-Hoeding upper bound C θ (u 1, u 2 ) θ min(u 1, u 2 )

40 Copulae 3-10 Copula Examples 3 Clayton copula C Cl θ (u 1, u 2 ) = [ max ( u θ 1 + u θ 2 1, 0 )] 1 θ dependence becomes maximal when θ independence is achieved when θ = 0 the distribution tends to the lower Frèchet-Hoeding bound when θ 1 allows to generate asymmetric dependence and lower tail dependence, but no upper tail dependence

41 Copulae 3-11 x y z Normal Copula x y z x y z Gumbel Copula x y z x y z Clayton Copula x y z

42 Copulae 3-12 Dependencies, Linear Correlation δ(x 1, X 2 ) = Cov(X 1, X 2 ) Var(X1 )Var(X 2 ). Sensitive to outliers Measures the 'average dependence' between X 1 and X 2 Invariant under strictly increasing linear transformations May be misleading in situations where multivariate df is not elliptical

43 Copulae 3-13 Dependencies, Kendall's tau Denition If F is continuous bivariate cdf and let (X 1, X 2 ), (X 1, X 2 ) be independent random pairs with distribution F. Then Kendall's tau is τ = P{(X 1 X 1)(X 2 X 2) > 0} P{(X 1 X 1)(X 2 X 2) < 0} Less sensitive to outliers Measures the 'average dependence' between X and Y Invariant under strictly increasing transformations Depends only on the copula of (X 1, X 2 ) For elliptical copulae: δ(x 1, X 2 ) = sin ( π 2 τ)

44 Copulae 3-14 Dependencies, Spearmans's rho Denition If F is a continuous bivariate cumulative distribution function with marginal F 1 and F 2 and let (X 1, X 2 ) F. Then Spearmans's rho is a correlation between F 1 (X 1 ) and F 2 (X 2 ) ρ = Cov{F 1 (X 1 ), F 2 (X 2 )} Var{F1 (X 1 )}Var{F 2 (X 2 )}. Less sensitive to outliers Measures the 'average dependence' between X 1 and X 2 Invariant under strictly increasing transformations Depends only on the copula of (X 1, X 2 ) For elliptical copulae: δ(x 1, X 2 ) = 2 sin ( π 6 ρ)

45 Copulae 3-15 δ = 0.892, τ = 0.956, ρ = δ = 0.659, τ = 0.888, ρ = x y x y

46 Copulae 3-16 Dependencies, Examples Gaussian copula ρ = 6 π arcsin δ 2, τ = 2 arcsin δ, π where δ is a linear correlation coecient. Gumbel copula ρ no closed form, τ = 1 1 θ.

47 Copulae 3-17 Multivariate Copula Denition Denition The copula is a multivariate distribution with all univariate margins being U(0, 1). Theorem (Sklar, 1959) Let X 1,..., X k be random variables with marginal distribution functions F 1,..., F k and joint distribution function F. Then there exists a k-dimensional copula C : [0, 1] k [0, 1] such that x 1,..., x k R = [, ] F (x 1,..., x k ) = C{F 1 (x 1 ),..., F k (x k )} (1) If the margins F 1,..., F k are continuous, then C is unique. Otherwise C is uniquely determined on F 1 (R) F k (R) Conversely, if C is a copula and F 1,..., F k are distribution functions, then the function F dened in (1) is a joint distribution function with margins F 1,..., F k.

48 Copulae 3-18 Copula Density Several theorems provides existence of derivatives of copulas, having them copula density is dened as c(u 1,..., u k ) = n C(u 1,..., u k ) u 1... u k. Joint density function based on copula cf (x 1,..., x k ) = c{f 1 (x 1 ),..., F k (x k )} f 1 (x 1 )... f k (x k ), where f 1 ( ),..., f k ( ) are marginal density functions.

49 Copulae 3-19 Special Copulas Theorem Let C be a copula. Then for every (u 1,..., u k ) [0, 1] k ( k ) max u i + 1 k, 0 C(u 1,..., u k ) min(u 1,..., u k ), i=1 where bounds are called lower and upper Fréchet-Hödings bounds. When they are copulas they represent perfect negative and positive dependence respectively. The simplest copula is product copula k Π(u 1,..., u k ) = characterize the case of independence. i=1 u i

50 Simulation 4-1 Simulation Frees and Valdez, (1998, NAAJ), Whelan, (2004, QF), Marshal and Olkin, (1988, JASA) Conditional inversion method: Let C = C(u 1,..., u k ), C i = C(u 1,..., u i, 1,..., 1) and C k = C(u 1,..., u k ). Conditional distribution of U i is given by C i (u i u 1,..., u i 1 ) = P{U i u i U 1 = u 1... U i 1 = u i 1 } Generate i.r.v. v 1,..., v k U(0, 1) Set u 1 = v 1 u i = C 1 k (v i u 1,..., u i 1 ) i = 2, k = i 1 C i (u 1,..., u i ) u 1... u i 1 / i 1 C i 1 (u 1,..., u i 1 ) u 1... u i 1

51 Simulation 4-2 Estimations: Empirical Copula Let (x(1) i,..., x i (T )) be the order statistics if i-th stock and (r i 1,..., r T i ) corresponding rank statistics such that x (r = x i t i for t )i all i = 1,..., d. Any function Ĉ is an empirical copula ( t1 T,..., t ) d = 1 T T T t=1 i=1 d I{rt i t i }

52 Simulation 4-3 Estimation: bivariate case based on Kendall's τ estimator τ n = 4 n(n 1) P n 1, where P n is the number of concordant pairs. For Gumbel copula ˆθ n = 1 1 τ n based on Spearman's ρ estimator ρ n = n n i=1 (R i R) 2 (S i S) 2 i=1 (R i R) 2 n i=1 (S i S) 2, where (R i, S i ) i = 1, n are pairs of ranks. For Gaussian Copula δ n = 2 sin πρ n 6

53 Simulation 4-4 Copula Estimation The distribution of X = (X 1,..., X d ) with marginals F Xj (x j, δ j ) j = 1,..., d is given by F X (x 1,..., x d ) = C{F X1 (x 1, δ 1 ),..., F Xd (x d, δ d ); θ} and its density is given by f (x 1,..., x d, δ 1,..., δ d, θ) = c{f X1 (x 1, δ 1 ),..., F Xd (x d, δ d ); θ} d f j (x j, δ j ) j=1

54 Simulation 4-5 Copula Estimation For a sample of observations {x t } t=1 and θ = (δ 1,..., δ d ; θ) R d+1 the likelihood function is L(θ; x 1,..., x T ) = T f (x 1,t,..., x d,t ; δ 1,..., δ d ; θ) t=1 and the corresponding log-likelihood function l(θ; x 1,..., x T ) = + T log c{f X1 (x 1,t, δ 1 ),..., F Xd (x d,t, δ d ); θ} t=1 T t=1 j=1 d log f j (x j,t, δ j )

55 Simulation 4-6 Full Maximum Likelihood (FML) FML estimates vector of parameters θ in one step through θ FML = arg maxl(θ) θ the estimates θ FML = ( δ 1,..., δ d, θ) solve ( l/ δ 1,..., l/ δ d, l/ θ) = 0 Drawback:with an increasing dimension the algorithm becomes too burdensome computationally

56 Simulation 4-7 Inference for Margins (IFM) 1. estimate parameters δ j from the marginal distributions: { T } ˆδ j = arg max log f j (x j,t ; δ j ) δ 2. estimate the dependence parameter θ by minimizing the pseudo log-likelihood function l(θ; ˆδ 1,..., ˆδ T d ) = log c{f X1 (x 1,t ; ˆδ 1 ),..., F Xd (x d,t ; ˆδ d ); θ} t=1 t=1 3. the estimates ˆθ IFM = (ˆδ 1,..., ˆδ d, ˆθ) solve 4. Advantage: numerically stable ( l/ δ 1,..., l/ δ d, l/ θ) = 0

57 Simulation 4-8 Canonical Maximum Likelihood (CML) CML maximizes the pseuso log-likelihood function with empirical marginal distributions T l(θ) = log c{ˆf X1 (x 1,t ),..., ˆF Xd (x d,t ); θ} where t=1 ˆθ CML = arg maxl(θ) θ ˆF Xj (x) = 1 T + 1 T I{X j, t x} Advantage: no assumptions about the parametric form of the marginal distributions t=1

58 Simulation 4-9 (X 1, X 2 ) Cθ Gu, with θ = 1.5 and F 1 = F 2 = N (μ 1, σ1 2) = N (μ 2, σ2 2 ) = N (0, 1) estimate std. error μ σ μ σ θ

59 Simulation 4-10 Attractive Features A copula describes how the marginals are tied together in the joint distribution The joint df is decomposed into the marginal dfs and a copula The marginal dfs and the copula can be modelled and estimated separately, independent of each other Given a copula, we can obtain many multivariate distributions by selecting dierent marginal dfs The copula is invariant under increasing and continuous transformations