Lecture Quantitative Finance Spring Term 2015
|
|
- Vincent Leonard
- 5 years ago
- Views:
Transcription
1 on bivariate Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas Lecture 07: April 2, / 54
2 Outline on bivariate 1 2 bivariate 3 Distribution Comments and conclusions 2 / 54
3 on bivariate key words: modeling dependencies and the study of and their applications (in risk management, in option pricing) is a rather modern phenomenon! several international conferences in the last 15 years! why are of interest to students? Fisher, Encyclopedia of Statistical Sciences (1997): firstly: as a way of studying scale-free measures of dependence secondly: as a starting point for constructing families of bivariate s, sometimes with a view to simulation 3 / 54
4 on bivariate the word copula is a Latin noun that means a link, tie, bond (Casell s Latin Dictionary) is used in grammar and logic to describe that part of a proposition which connects the subject and predicate (Oxford English Dictionary) aim of Quantitative Risk Management: find good joint models F (x 1,..., x n ), e.g. N n (µ, Σ); tk n (µ, Σ) the whole idea of is to go from individual models to the joint model if we don t have a basic joint model then we can try to use! 4 / 54
5 bivariate : Outline on bivariate Grounded Margins 2-increasing Definition of Frechet bounds for 5 / 54
6 on bivariate Grounded let S 1 and S 2 be nonempty subsets of [, ] Definition: suppose S 1 has a least element a 1 and S 2 has a least element a 2 a function H : S 1 S 2 R is grounded if Example: H(x, a 2 ) = 0 = H(a 1, y) for all (x, y) S 1 S 2. H : [ 1, 1] [0, ] R H(x, y) = (x + 1)(ey 1) x + 2e y 1 is grounded! 6 / 54
7 Margins on bivariate let S 1 and S 2 be nonempty subsets of [, ] Definition: suppose S 1 has a greatest element b 1 and b 2 has a greatest element b 2 a function H : S 1 S 2 R has margins and the margins of H are given by: F (x) = H(x, b 2 ) for all x S 1 G(y) = H(b 1, y) for all y S 2. 7 / 54
8 Margins: Example on bivariate Example: the function H : [ 1, 1] [0, ] R has margins: H(x, y) = (x + 1)(ey 1) x + 2e y 1 F (x) = H(x, ) = x G(y) = H(1, y) = 1 e y 8 / 54
9 on bivariate 2-increasing let S 1 and S 2 be nonempty subsets of [, ] Definition: let B = [x 1, x 2 ] [y 1, y 2 ] S 1 S 2 the H-volume of the rectangle B is or V H (B) = x2 x 1 y2 y 1 H(x, y) V H (B) = H(x 2, y 2 ) H(x 2, y 1 ) H(x 1, y 2 ) + H(x 1, y 1 ) a function H : S 1 S 2 R is 2-increasing if V H (B) 0 for all rectangles B S 1 S 2 9 / 54
10 2-increasing : Examples on bivariate Example 1: H : [0, 1] [0, 1] R H(x, y) = (2x 1)(2y 1) is 2-increasing! however it is decreasing in x for any y (0, 1/2) and decreasing in y for any x (0, 1/2)! 10 / 54
11 on bivariate Example 2: H : [0, 1] [0, 1] R Grounded and 2-increasing H(x, y) = max (x, y) is a nondecreasing function of x and a nondecreasing function of y however V H ([0, 1] [0, 1]) = 1 thus H is NOT 2-increasing grounded and 2-increasing implies non-decreasing in each argument! 11 / 54
12 (Bivariate) Copulas: Definition on bivariate a function C : [0, 1] [0, 1] [0, 1] is a copula if 1 C is grounded, 2 for every u, v [0, 1] 3 C is 2-increasing. C(u, 1) = u and C(1, v) = v 12 / 54
13 Bivariate : Examples on bivariate fundamental examples Π(x, y) = xy W (x, y) = max (x + y 1, 0) M(x, y) = min (x, y) let α, β [0, 1] with α + β 1; then C α,β (x, y) = αm(x, y) + (1 α β)π(x, y) + βw (x, y) is a copula (Frechet-Mardia)! 13 / 54
14 Bivariate : Frechet bounds on bivariate lower Frechet-Hoeffding bound (copula only for n = 2): W (u, v) = max (u + v 1, 0) upper Frechet-Hoeffding bound (a copula also for n 2): M(u, v) = min (u, v) note that for any copula C W (u, v) C(u, v) M(u, v) 14 / 54
15 Distribution on bivariate Definition: a function is a function F : [, ] [0, 1] such that F is nondecreasing F ( ) = 0 and F (+ ) = 1 Example: (unit step at a) 0, x [, a) ε a (x) = 1, x [a, ] 15 / 54
16 Distribution : Example on bivariate the uniform on [a, b]: 0, x [, a) x a U ab (x) = b a, x [a, b] 1, x (b, ] 16 / 54
17 Joint on bivariate a function H : [, ] [, ] [0, 1] is a joint function if H is 2-increasing H(x, ) = 0 = H(, y) and H(, ) = 1 Remark: note that H is grounded and has margins F (x) = H(x, ) and G(y) = H(, y) which are 17 / 54
18 on bivariate Joint : Examples (x+1)(e y 1) x+2e y 1, (x, y) [ 1, 1] [0, ) H(x, y) = 1 e y, (x, y) (1, ) [0, ] 0,, elsewhere. margins: 0, x [, 1) x+1 F (x) = U 1,1 (x) = 2, x [ 1, 1] 1, x (1, ] 0, y [, 0) G(y) = 1 e y, y [0, ] 18 / 54
19 on bivariate 1 let H be a joint function with margins F and G then there exists a copula C such that for all x, y [, ] H(x, y) = C(F (x), G(y)) if F and G are continuous, then C is unique; otherwise, C is uniquely determined on RanF RanG 2 conversely, if C is a copula and F and G are, then the function H defined as a above is a joint function with margins F and G 19 / 54
20 : Example on bivariate consider (x+1)(e y 1) x+2e y 1, (x, y) [ 1, 1] [0, ) H(x, y) = 1 e y, (x, y) (1, ) [0, ] 0,, elsewhere. then the associated copula is C(u, v) = uv u + v uv check it in the class! 20 / 54
21 on bivariate let F be a function Quasi-inverse of a function a quasi-inverse of F is any function F ( 1) : [0, 1] R such that 1 if t Ran F then F ( 1) (t) is any number in [, ] such that F (x) = t, i.e. for all t Ran F 2 if t / Ran F, then F (F ( 1) (t)) = t; F ( 1) (t) = inf {x : F (x) t} = sup {x : F (x) t}. if F is strictly increasing, then it has a single quasi-inverse, which is of course the ordinary inverse, for which we use the customary notation F 1 21 / 54
22 Quasi-inverse of a function: Example on bivariate the quasi-inverses of ε a, the unit step at a are the given by: a 0, t = 0 ε ( 1) a (t) = a, t (0, 1) a 1,, t = 1 where a 0 and a 1 are any numbers in [, ] such that a 0 < a a 1 22 / 54
23 using quasi-inverse of on bivariate let H be a joint function with margins F and G let the copula C given by, i.e. such that for all x, y [, ] H(x, y) = C(F (x), G(y)) let F ( 1) and G ( 1) be quasi-inverses of F and G respectively then for any u, v [0, 1] C(u, v) = H(F ( 1) (u), G ( 1) (v)) 23 / 54
24 on bivariate Gumbel s bivariate exponential let θ [0, 1] and let H θ be the joint function given by 1 e x e y + e (x+y+θxy), x 0, y 0 H θ (x, y) = 0, otherwise the marginal are exponentials, with quasi-inverses given for u, v [0, 1] by F ( 1) (u) = log(1 u) and G ( 1) (v) = log(1 v) hence the corresponding copula is given by θ log(1 u) log(1 v) C θ (u, v) = u + v 1 + (1 u)(1 v) e 24 / 54
25 on bivariate Copulas as joint with uniform margins let C be a copula and define H C : [, ] [, ] [0, 1] 0, x < 0 and y < 0, C(x, y), x, y [0, 1] H C (x, y) = x, y > 1, x [0, 1] y, x > 1, y [0, 1] 1, x > 1 and y > 1 then H C is a function both of whose margins are readily seen to be U 01 Interpretation: are restrictions to [0, 1] [0, 1] of joint s whose margins are U / 54
26 on bivariate consider X and Y two with F and G respectively, i.e. F (x) = P[X x] function H, i.e. G(y) = P[Y y], H(x, y) = P[X x, Y y] then the copula C given by Sklar s Theorem is called the copula of X and Y and is denoted C XY Theorem: let X and Y two with continuous then X and Y are independent if, and only if, C XY = Π 26 / 54
27 dependence structure on bivariate Sklars theorem shows how a unique copula C describes in a sense the dependence structure of the multivariate function of a random vector X = (X 1, X 2 ) This motivates a further Definition: The copula of (X 1, X 2 ) is the function C of (F 1 (X 1 ), F 2 (X 2 )) 27 / 54
28 dependence structure on bivariate Theorem: let X and Y two with continuous then C XY is invariant under strictly increasing transformations of X and Y, i.e. if α and β are strictly increasing on RanF and RanG then C α(x )β(y ) = C XY. 28 / 54
29 Construction of bivariate s on bivariate if we have a collection of, then, as a consequence of we automatically have a bivariate or multivariate s with whatever marginal s we desire by the invariance of the copula under strictly increasing transformations of the if follows that the nonparametric nature of the dependence between two random variables is expressed by a copula we need to have a variety of at our disposal! 29 / 54
30 on bivariate Definition: : pseudo-inverses let ϕ : [0, 1] [0, ] be a continuous strictly decreasing function such that ϕ(1) = 0 the pseudo-inverse of ϕ is the function ϕ [ 1] : [0, ] [0, 1] ϕ 1 (t), 0 t ϕ(0), given by: ϕ [ 1] (t) = 0, ϕ(0) t. note that ϕ [ 1] is continuous and non-increasing on [0, ] and strictly decreasing on [0, ϕ(0)]. note that if ϕ(0) = then ϕ [ 1] = ϕ 1 30 / 54
31 on bivariate Theorem: : definition let ϕ : [0, 1] [0, ] be a continuous strictly decreasing function such that ϕ(1) = 0 ϕ [ 1] be the pseudo-inverse of ϕ let C : [0, 1] [0, 1] [0, 1] given by: C(u, v) = ϕ [ 1] (ϕ(u) + ϕ(v)) then C is a copula if, and only if, ϕ is convex those are called and the function ϕ is called the generator of the copula if ϕ(0) = we say ϕ is a strict generator and C is a strict copula 31 / 54
32 : Example 1 on bivariate let ϕ : [0, 1] [0, ], ϕ(t) = log t then ϕ(0) = and ϕ is strict thus ϕ [ 1] (t) = ϕ 1 (t) = exp( t) and the generated copula is C(u, v) = exp(log u + log v) = uv = Π(u, v) consequently Π is a strict copula 32 / 54
33 : Example 2 on bivariate let ϕ : [0, 1] [0, ], ϕ(t) = 1 t then ϕ [ 1] (t) = 1 t and 0 for t > 1; i.e. ϕ [ 1] (t) = max(1 t, 0) consequently the generated copula is C(u, v) = max(u + v 1, 0) = W (u, v). this means W is also an copula note that the copula M(u, v) = min(u, v) is not 33 / 54
34 : Example 3 on bivariate let θ (0, 1] and ϕ θ : [0, 1] [0, ], ϕ θ (t) = log(1 θ log t) then ϕ θ (0) =, ϕ θ is strict, and ϕ [ 1] θ (t) = ϕ 1 θ (t) = exp[(1 et )/θ] consequently the generated copula is C θ (u, v) = uv exp( θ log u log v) 34 / 54
35 : further examples on bivariate two-parameter family of α > 0, β 1 C α,β (x, y) = { } 1/α [(x α 1) β + (y α 1) β ] 1/β / 54
36 Looking forward on bivariate The next slides are optional material!!! 36 / 54
37 multivariate : Outline on bivariate Definition in n-dimensions, Gumbel copula, Clayton copula Implicit 37 / 54
38 on bivariate : definition A function C : [0, 1] [0, 1] [0, 1] is a copula if 1 C is grounded, 2 for every i = 1,..., n and any u i [0, 1] C(1,..., 1, u i, 1,..., 1) = u i 3 C is n-increasing (i.e. for all (x 1,...x n ), (y 1,..., y n ) [0, 1] n with x j y i we have 2 i 1=1 2 ( 1) i1+...+in C(u 1i1,..., u nin ) 0 i d =1 where u j1 = x j and u j2 = y j for all j = 1,..., n.) 38 / 54
39 on bivariate n-dimensional are Lipschitz : further properties C(v 1,..., v n ) C(u 1,..., u n ) n v k u k. k=1 Definition: n-dimensional are H : [, ] [, ] R such that H is n-increasing H(x 1,..., x n ) = 0 for all such that x k = for at least one k and H(,..., ) = 1 thus H is grounded and the one-dimensional margins are : F 1,..., F n 39 / 54
40 on bivariate in n-dimensions let H be an n-dimensional function with margins F 1,..., F n then there exists an n-copula C such that for all x 1,..., x n [, ] H(x 1,..., x n ) = C(F 1 (x 1 ),..., F n (x n )). if F 1,..., F n are continuous, then C is unique; otherwise, C is uniquely determined on RanF 1... RanF n conversely, if C is an n-copula and F 1,..., F n are then the function H defined as a above is an n-dimensional function with margins F 1,..., F n 40 / 54
41 dependence structures on bivariate Sklars theorem shows how a unique copula C describes in a sense the dependence structure of the multivariate function of a random vector X = (X 1,..., X n ) this motivates the further Definition: the copula of (X 1,..., X n ) is the function C of (F 1 (X 1 ),..., F n (X n ))! 41 / 54
42 on bivariate Theorem: let ϕ : [0, 1] [0, ] be a continuous strictly decreasing function such that ϕ(1) = 0 and ϕ(0) = let ϕ 1 be the inverse of ϕ let C : [0, 1]... [0, 1] [0, 1] given by: C(u 1,..., u n ) = ϕ 1 (ϕ(u 1 ) ϕ(u n )). then C is a copula if, and only if, ϕ 1 is completely monotone on [0, ), i.e. has derivatives of all orders that alternate in sign. those are called and the function ϕ is called the generator of the copula. 42 / 54
43 : Examples on bivariate Gumbel copula: θ 1, ϕ θ (t) = ( log t) θ Cθ Gu (u 1,..., u n ) = exp ( [ ( log u 1 ) θ +...( log u n ) θ] ) 1/θ θ = 1 gives independence, θ gives comonotonicity Clayton copula: θ > 0, ϕ θ (t) = t θ 1. Cθ Cl (u 1,..., u n ) = ( u θ un θ d + 1 ) 1/θ θ 0 gives independence, θ gives comonotonicity 43 / 54
44 : some remarks on bivariate Pro: multivariate can be generated fairly simple Con: all the k-margins of an n- copula are identical Con: there are only one or two parameters and this limits the nature of the dependence structure in these families 44 / 54
45 on bivariate there are essentially two possibilities: Parametric 1 implicit in well-known parametric s states that we can always find a copula in a parametric function let H be the function and let F 1,..., F n its continuous margins then the implied copula is C(u 1,..., u n ) = H(F ( 1) 1 (u 1 ),..., F ( 1) n (u n )) such a copula may not have a simple closed form 45 / 54
46 Parametric on bivariate closed form parametric copula families generated by some explicit construction that is known to yield the best example is the copula family these generally have limited numbers of parameters 46 / 54
47 Implicit : Example on bivariate Gaussian Copula: C Ga P (u 1,..., u n) = N P ( N 1 (u 1 ),..., N 1 (u n) ) where N denotes the standard univariate function x 1 N(x) = e t2 2 dt, 2π N P denotes the joint function of X N n(0, P) and P is a correlation matrix 47 / 54
48 Key facts for simulation: Probability and Quantile Transform on bivariate Proposition 1: (Probability transform) let X be a random variable with continuous function F then F (X ) U 01 (standard uniform) u (0, 1) P(F (X ) u) = P(X F 1 (u)) = F (F 1 (u)) = u, Proposition 2: (Quantile transform) let U be uniform and F the function of any rv X then F 1 (U) has the same with X so that P(F 1 (U) x) = F (x) 48 / 54
49 Key facts for simulation: Probability and Quantile Transform on bivariate These facts are the key to all statistical simulation and essential in dealing with Simulating Gaussian copula Simulate X N n (0, P) Set U = (Φ(X 1 ),..., Φ(X n )) (probability transformation) 49 / 54
50 on bivariate Meta-s and their simulation by the converse of Sklars Theorem we know that if C is a copula and F 1,..., F d are univariate, then F (x) = C(F 1 (x 1 ),..., F n (x n )) is a multivariate with margins F 1,..., F n we refer to F as a meta- with the dependence structure represented by C for example, if C is a Gaussian copula we get a meta-gaussian and if C is a t copula we get a meta-t if we can sample from the copula C, then it is easy to sample from F : we generate a vector (U 1,..., U n ) with function C and then return (F ( 1) 1 (U 1 ),..., F n ( 1) (U n )) 50 / 54
51 Some additional comments on bivariate correlation is defined only when the variances of the two random variables are finite not ideal when we work with heavy tailed s example: actuaries who model losses in different business lines with infinite variance s may not describe the dependence of their risk using correlation! correlation of two risks does not depend only on their copula correlation is linked to the marginal s of the risks! 51 / 54
52 Some additional comments on bivariate often very difficult (in particular in higher dimensions and in situations where we are dealing with heterogeneous risk factors) to find a good multivariate model that describes both marginal behavior and dependence structure effectively the copula approach to multivariate models allows us to consider marginal modeling and dependence modeling issues 52 / 54
53 on bivariate Some conclusions help in the understanding of dependence at a deeper level they show us potential pitfalls of approaches to dependence that focus only on correlation they allow us to define alternative dependence structures they express dependence on a quantile scale ( QRM!) they facilitate a bottom-up approach to multivariate model building they are easily simulated and thus lend themselves to Monte Carlo risk studies useful in risk management where we often have a much better idea about the marginal behavior of individual risk factors than we do about their dependence structure 53 / 54
54 References on bivariate Books: 1 An to Copulas Author: Roger Nelsen Springer Quantitative Risk Management: Concepts, Techniques and Tools Authors: Paul Embrechts, Rüdiger Frey, Alexander McNeil 54 / 54
Explicit Bounds for the Distribution Function of the Sum of Dependent Normally Distributed Random Variables
Explicit Bounds for the Distribution Function of the Sum of Dependent Normally Distributed Random Variables Walter Schneider July 26, 20 Abstract In this paper an analytic expression is given for the bounds
More informationA Brief Introduction to Copulas
A Brief Introduction to Copulas Speaker: Hua, Lei February 24, 2009 Department of Statistics University of British Columbia Outline Introduction Definition Properties Archimedean Copulas Constructing Copulas
More informationOn the Conditional Value at Risk (CoVaR) from the copula perspective
On the Conditional Value at Risk (CoVaR) from the copula perspective Piotr Jaworski Institute of Mathematics, Warsaw University, Poland email: P.Jaworski@mimuw.edu.pl 1 Overview 1. Basics about VaR, CoVaR
More informationModelling Dependence with Copulas and Applications to Risk Management. Filip Lindskog, RiskLab, ETH Zürich
Modelling Dependence with Copulas and Applications to Risk Management Filip Lindskog, RiskLab, ETH Zürich 02-07-2000 Home page: http://www.math.ethz.ch/ lindskog E-mail: lindskog@math.ethz.ch RiskLab:
More informationCopulas and dependence measurement
Copulas and dependence measurement Thorsten Schmidt. Chemnitz University of Technology, Mathematical Institute, Reichenhainer Str. 41, Chemnitz. thorsten.schmidt@mathematik.tu-chemnitz.de Keywords: copulas,
More informationIntroduction to Dependence Modelling
Introduction to Dependence Modelling Carole Bernard Berlin, May 2015. 1 Outline Modeling Dependence Part 1: Introduction 1 General concepts on dependence. 2 in 2 or N 3 dimensions. 3 Minimizing the expectation
More informationCopulas and Measures of Dependence
1 Copulas and Measures of Dependence Uttara Naik-Nimbalkar December 28, 2014 Measures for determining the relationship between two variables: the Pearson s correlation coefficient, Kendalls tau and Spearmans
More informationLecture 2 One too many inequalities
University of Illinois Department of Economics Spring 2017 Econ 574 Roger Koenker Lecture 2 One too many inequalities In lecture 1 we introduced some of the basic conceptual building materials of the course.
More informationContents 1. Coping with Copulas. Thorsten Schmidt 1. Department of Mathematics, University of Leipzig Dec 2006
Contents 1 Coping with Copulas Thorsten Schmidt 1 Department of Mathematics, University of Leipzig Dec 2006 Forthcoming in Risk Books Copulas - From Theory to Applications in Finance Contents 1 Introdcution
More informationA measure of radial asymmetry for bivariate copulas based on Sobolev norm
A measure of radial asymmetry for bivariate copulas based on Sobolev norm Ahmad Alikhani-Vafa Ali Dolati Abstract The modified Sobolev norm is used to construct an index for measuring the degree of radial
More informationCONTAGION VERSUS FLIGHT TO QUALITY IN FINANCIAL MARKETS
EVA IV, CONTAGION VERSUS FLIGHT TO QUALITY IN FINANCIAL MARKETS Jose Olmo Department of Economics City University, London (joint work with Jesús Gonzalo, Universidad Carlos III de Madrid) 4th Conference
More informationCopulas. Mathematisches Seminar (Prof. Dr. D. Filipovic) Di Uhr in E
Copulas Mathematisches Seminar (Prof. Dr. D. Filipovic) Di. 14-16 Uhr in E41 A Short Introduction 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 The above picture shows a scatterplot (500 points) from a pair
More informationModelling and Estimation of Stochastic Dependence
Modelling and Estimation of Stochastic Dependence Uwe Schmock Based on joint work with Dr. Barbara Dengler Financial and Actuarial Mathematics and Christian Doppler Laboratory for Portfolio Risk Management
More informationExtreme Value Analysis and Spatial Extremes
Extreme Value Analysis and Department of Statistics Purdue University 11/07/2013 Outline Motivation 1 Motivation 2 Extreme Value Theorem and 3 Bayesian Hierarchical Models Copula Models Max-stable Models
More informationON UNIFORM TAIL EXPANSIONS OF BIVARIATE COPULAS
APPLICATIONES MATHEMATICAE 31,4 2004), pp. 397 415 Piotr Jaworski Warszawa) ON UNIFORM TAIL EXPANSIONS OF BIVARIATE COPULAS Abstract. The theory of copulas provides a useful tool for modelling dependence
More informationFinancial Econometrics and Volatility Models Copulas
Financial Econometrics and Volatility Models Copulas Eric Zivot Updated: May 10, 2010 Reading MFTS, chapter 19 FMUND, chapters 6 and 7 Introduction Capturing co-movement between financial asset returns
More informationVaR vs. Expected Shortfall
VaR vs. Expected Shortfall Risk Measures under Solvency II Dietmar Pfeifer (2004) Risk measures and premium principles a comparison VaR vs. Expected Shortfall Dependence and its implications for risk measures
More informationModelling Dependent Credit Risks
Modelling Dependent Credit Risks Filip Lindskog, RiskLab, ETH Zürich 30 November 2000 Home page:http://www.math.ethz.ch/ lindskog E-mail:lindskog@math.ethz.ch RiskLab:http://www.risklab.ch Modelling Dependent
More informationOn Conditional Value at Risk (CoVaR) for tail-dependent copulas
Depend Model 2017; 5:1 19 Research Article Special Issue: Recent Developments in Quantitative Risk Management Open Access Piotr Jaworski* On Conditional Value at Risk CoVaR for tail-dependent copulas DOI
More informationLifetime Dependence Modelling using a Generalized Multivariate Pareto Distribution
Lifetime Dependence Modelling using a Generalized Multivariate Pareto Distribution Daniel Alai Zinoviy Landsman Centre of Excellence in Population Ageing Research (CEPAR) School of Mathematics, Statistics
More informationMultivariate Measures of Positive Dependence
Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 4, 191-200 Multivariate Measures of Positive Dependence Marta Cardin Department of Applied Mathematics University of Venice, Italy mcardin@unive.it Abstract
More informationSimulating Exchangeable Multivariate Archimedean Copulas and its Applications. Authors: Florence Wu Emiliano A. Valdez Michael Sherris
Simulating Exchangeable Multivariate Archimedean Copulas and its Applications Authors: Florence Wu Emiliano A. Valdez Michael Sherris Literatures Frees and Valdez (1999) Understanding Relationships Using
More informationOverview of Extreme Value Theory. Dr. Sawsan Hilal space
Overview of Extreme Value Theory Dr. Sawsan Hilal space Maths Department - University of Bahrain space November 2010 Outline Part-1: Univariate Extremes Motivation Threshold Exceedances Part-2: Bivariate
More informationFitting Archimedean copulas to bivariate geodetic data
Fitting Archimedean copulas to bivariate geodetic data Tomáš Bacigál 1 and Magda Komorníková 2 1 Faculty of Civil Engineering, STU Bratislava bacigal@math.sk 2 Faculty of Civil Engineering, STU Bratislava
More information, respectively, and a joint distribution function H(x,y) = PX [ xy, y]
2 Definitions and Basic Properties In the Introduction, we referred to copulas as functions that join or couple multivariate distribution functions to their one-dimensional marginal distribution functions
More informationThe Instability of Correlations: Measurement and the Implications for Market Risk
The Instability of Correlations: Measurement and the Implications for Market Risk Prof. Massimo Guidolin 20254 Advanced Quantitative Methods for Asset Pricing and Structuring Winter/Spring 2018 Threshold
More informationGoodness-of-Fit Testing with Empirical Copulas
with Empirical Copulas Sami Umut Can John Einmahl Roger Laeven Department of Econometrics and Operations Research Tilburg University January 19, 2011 with Empirical Copulas Overview of Copulas with Empirical
More informationNon parametric estimation of Archimedean copulas and tail dependence. Paris, february 19, 2015.
Non parametric estimation of Archimedean copulas and tail dependence Elena Di Bernardino a and Didier Rullière b Paris, february 19, 2015. a CNAM, Paris, Département IMATH, b ISFA, Université Lyon 1, Laboratoire
More informationBIVARIATE P-BOXES AND MAXITIVE FUNCTIONS. Keywords: Uni- and bivariate p-boxes, maxitive functions, focal sets, comonotonicity,
BIVARIATE P-BOXES AND MAXITIVE FUNCTIONS IGNACIO MONTES AND ENRIQUE MIRANDA Abstract. We give necessary and sufficient conditions for a maxitive function to be the upper probability of a bivariate p-box,
More informationMAXIMUM ENTROPIES COPULAS
MAXIMUM ENTROPIES COPULAS Doriano-Boris Pougaza & Ali Mohammad-Djafari Groupe Problèmes Inverses Laboratoire des Signaux et Systèmes (UMR 8506 CNRS - SUPELEC - UNIV PARIS SUD) Supélec, Plateau de Moulon,
More informationRisk Aggregation with Dependence Uncertainty
Introduction Extreme Scenarios Asymptotic Behavior Challenges Risk Aggregation with Dependence Uncertainty Department of Statistics and Actuarial Science University of Waterloo, Canada Seminar at ETH Zurich
More informationMultivariate Distribution Models
Multivariate Distribution Models Model Description While the probability distribution for an individual random variable is called marginal, the probability distribution for multiple random variables is
More informationA simple graphical method to explore tail-dependence in stock-return pairs
A simple graphical method to explore tail-dependence in stock-return pairs Klaus Abberger, University of Konstanz, Germany Abstract: For a bivariate data set the dependence structure can not only be measured
More informationRobustness of a semiparametric estimator of a copula
Robustness of a semiparametric estimator of a copula Gunky Kim a, Mervyn J. Silvapulle b and Paramsothy Silvapulle c a Department of Econometrics and Business Statistics, Monash University, c Caulfield
More informationSimulation of Tail Dependence in Cot-copula
Int Statistical Inst: Proc 58th World Statistical Congress, 0, Dublin (Session CPS08) p477 Simulation of Tail Dependence in Cot-copula Pirmoradian, Azam Institute of Mathematical Sciences, Faculty of Science,
More informationGENERAL MULTIVARIATE DEPENDENCE USING ASSOCIATED COPULAS
REVSTAT Statistical Journal Volume 14, Number 1, February 2016, 1 28 GENERAL MULTIVARIATE DEPENDENCE USING ASSOCIATED COPULAS Author: Yuri Salazar Flores Centre for Financial Risk, Macquarie University,
More informationNonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution
Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. /2 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution
More informationMultivariate Non-Normally Distributed Random Variables
Multivariate Non-Normally Distributed Random Variables An Introduction to the Copula Approach Workgroup seminar on climate dynamics Meteorological Institute at the University of Bonn 18 January 2008, Bonn
More informationX
Correlation: Pitfalls and Alternatives Paul Embrechts, Alexander McNeil & Daniel Straumann Departement Mathematik, ETH Zentrum, CH-8092 Zürich Tel: +41 1 632 61 62, Fax: +41 1 632 15 23 embrechts/mcneil/strauman@math.ethz.ch
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections Chapter 3 - continued 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions 3.6 Conditional
More informationEXTREMAL DEPENDENCE OF MULTIVARIATE DISTRIBUTIONS AND ITS APPLICATIONS YANNAN SUN
EXTREMAL DEPENDENCE OF MULTIVARIATE DISTRIBUTIONS AND ITS APPLICATIONS By YANNAN SUN A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON
More informationESTIMATING BIVARIATE TAIL
Elena DI BERNARDINO b joint work with Clémentine PRIEUR a and Véronique MAUME-DESCHAMPS b a LJK, Université Joseph Fourier, Grenoble 1 b Laboratoire SAF, ISFA, Université Lyon 1 Framework Goal: estimating
More informationTail Approximation of Value-at-Risk under Multivariate Regular Variation
Tail Approximation of Value-at-Risk under Multivariate Regular Variation Yannan Sun Haijun Li July 00 Abstract This paper presents a general tail approximation method for evaluating the Valueat-Risk of
More informationModelling Dropouts by Conditional Distribution, a Copula-Based Approach
The 8th Tartu Conference on MULTIVARIATE STATISTICS, The 6th Conference on MULTIVARIATE DISTRIBUTIONS with Fixed Marginals Modelling Dropouts by Conditional Distribution, a Copula-Based Approach Ene Käärik
More informationRisk Aggregation. Paul Embrechts. Department of Mathematics, ETH Zurich Senior SFI Professor.
Risk Aggregation Paul Embrechts Department of Mathematics, ETH Zurich Senior SFI Professor www.math.ethz.ch/~embrechts/ Joint work with P. Arbenz and G. Puccetti 1 / 33 The background Query by practitioner
More informationTail Dependence of Multivariate Pareto Distributions
!#"%$ & ' ") * +!-,#. /10 243537698:6 ;=@?A BCDBFEHGIBJEHKLB MONQP RS?UTV=XW>YZ=eda gihjlknmcoqprj stmfovuxw yy z {} ~ ƒ }ˆŠ ~Œ~Ž f ˆ ` š œžÿ~ ~Ÿ œ } ƒ œ ˆŠ~ œ
More informationVaR bounds in models with partial dependence information on subgroups
VaR bounds in models with partial dependence information on subgroups L. Rüschendorf J. Witting February 23, 2017 Abstract We derive improved estimates for the model risk of risk portfolios when additional
More informationOptimization of Spearman s Rho
Revista Colombiana de Estadística January 215, Volume 38, Issue 1, pp. 29 a 218 DOI: http://dx.doi.org/1.156/rce.v38n1.8811 Optimization of Spearman s Rho Optimización de Rho de Spearman Saikat Mukherjee
More informationFRÉCHET HOEFFDING LOWER LIMIT COPULAS IN HIGHER DIMENSIONS
FRÉCHET HOEFFDING LOWER LIMIT COPULAS IN HIGHER DIMENSIONS PAUL C. KETTLER ABSTRACT. Investigators have incorporated copula theories into their studies of multivariate dependency phenomena for many years.
More informationSklar s theorem in an imprecise setting
Sklar s theorem in an imprecise setting Ignacio Montes a,, Enrique Miranda a, Renato Pelessoni b, Paolo Vicig b a University of Oviedo (Spain), Dept. of Statistics and O.R. b University of Trieste (Italy),
More informationProbability Distribution And Density For Functional Random Variables
Probability Distribution And Density For Functional Random Variables E. Cuvelier 1 M. Noirhomme-Fraiture 1 1 Institut d Informatique Facultés Universitaires Notre-Dame de la paix Namur CIL Research Contact
More informationarxiv: v1 [math.pr] 9 Jan 2016
SKLAR S THEOREM IN AN IMPRECISE SETTING IGNACIO MONTES, ENRIQUE MIRANDA, RENATO PELESSONI, AND PAOLO VICIG arxiv:1601.02121v1 [math.pr] 9 Jan 2016 Abstract. Sklar s theorem is an important tool that connects
More informationDEPARTMENT OF ECONOMICS DISCUSSION PAPER SERIES
ISSN 1471-0498 DEPARTMENT OF ECONOMICS DISCUSSION PAPER SERIES DEPENDENCE AND UNIQUENESS IN BAYESIAN GAMES A.W. Beggs Number 603 April 2012 Manor Road Building, Oxford OX1 3UQ Dependence and Uniqueness
More informationOn the Estimation and Application of Max-Stable Processes
On the Estimation and Application of Max-Stable Processes Zhengjun Zhang Department of Statistics University of Wisconsin Madison, WI 53706, USA Co-author: Richard Smith EVA 2009, Fort Collins, CO Z. Zhang
More informationBounds on the value-at-risk for the sum of possibly dependent risks
Insurance: Mathematics and Economics 37 (2005) 135 151 Bounds on the value-at-risk for the sum of possibly dependent risks Mhamed Mesfioui, Jean-François Quessy Département de Mathématiques et d informatique,
More informationStochastic orders: a brief introduction and Bruno s contributions. Franco Pellerey
Stochastic orders: a brief introduction and Bruno s contributions. Franco Pellerey Stochastic orders (comparisons) Among his main interests in research activity A field where his contributions are still
More informationFRÉCHET HOEFFDING LOWER LIMIT COPULAS IN HIGHER DIMENSIONS
DEPT. OF MATH./CMA UNIV. OF OSLO PURE MATHEMATICS NO. 16 ISSN 0806 2439 JUNE 2008 FRÉCHET HOEFFDING LOWER LIMIT COPULAS IN HIGHER DIMENSIONS PAUL C. KETTLER ABSTRACT. Investigators have incorporated copula
More informationApproximation of multivariate distribution functions MARGUS PIHLAK. June Tartu University. Institute of Mathematical Statistics
Approximation of multivariate distribution functions MARGUS PIHLAK June 29. 2007 Tartu University Institute of Mathematical Statistics Formulation of the problem Let Y be a random variable with unknown
More informationProbability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institute of Technology, Kharagpur
Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institute of Technology, Kharagpur Lecture No. # 29 Introduction to Copulas Hello and welcome to this
More informationA Goodness-of-fit Test for Copulas
A Goodness-of-fit Test for Copulas Artem Prokhorov August 2008 Abstract A new goodness-of-fit test for copulas is proposed. It is based on restrictions on certain elements of the information matrix and
More informationCorrelation: Copulas and Conditioning
Correlation: Copulas and Conditioning This note reviews two methods of simulating correlated variates: copula methods and conditional distributions, and the relationships between them. Particular emphasis
More informationOn the Choice of Parametric Families of Copulas
On the Choice of Parametric Families of Copulas Radu Craiu Department of Statistics University of Toronto Collaborators: Mariana Craiu, University Politehnica, Bucharest Vienna, July 2008 Outline 1 Brief
More informationCopula methods in Finance
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 http://ise.wiwi.hu-berlin.de Motivation
More information2 (U 2 ), (0.1) 1 + u θ. where θ > 0 on the right. You can easily convince yourself that (0.3) is valid for both.
Introducing copulas Introduction Let U 1 and U 2 be uniform, dependent random variables and introduce X 1 = F 1 1 (U 1 ) and X 2 = F 1 2 (U 2 ), (.1) where F1 1 (u 1 ) and F2 1 (u 2 ) are the percentiles
More informationUsing copulas to model time dependence in stochastic frontier models
Using copulas to model time dependence in stochastic frontier models Christine Amsler Michigan State University Artem Prokhorov Concordia University November 2008 Peter Schmidt Michigan State University
More informationProbabilistic Engineering Mechanics. An innovating analysis of the Nataf transformation from the copula viewpoint
Probabilistic Engineering Mechanics 4 9 3 3 Contents lists available at ScienceDirect Probabilistic Engineering Mechanics journal homepage: www.elsevier.com/locate/probengmech An innovating analysis of
More informationConditional Least Squares and Copulae in Claims Reserving for a Single Line of Business
Conditional Least Squares and Copulae in Claims Reserving for a Single Line of Business Michal Pešta Charles University in Prague Faculty of Mathematics and Physics Ostap Okhrin Dresden University of Technology
More informationClearly, if F is strictly increasing it has a single quasi-inverse, which equals the (ordinary) inverse function F 1 (or, sometimes, F 1 ).
APPENDIX A SIMLATION OF COPLAS Copulas have primary and direct applications in the simulation of dependent variables. We now present general procedures to simulate bivariate, as well as multivariate, dependent
More informationNonparameteric Regression:
Nonparameteric Regression: Nadaraya-Watson Kernel Regression & Gaussian Process Regression Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro,
More informationBivariate extension of the Pickands Balkema de Haan theorem
Ann. I. H. Poincaré PR 40 (004) 33 4 www.elsevier.com/locate/anihpb Bivariate extension of the Pickands Balkema de Haan theorem Mario V. Wüthrich Winterthur Insurance, Römerstrasse 7, P.O. Box 357, CH-840
More informationSome new properties of Quasi-copulas
Some new properties of Quasi-copulas Roger B. Nelsen Department of Mathematical Sciences, Lewis & Clark College, Portland, Oregon, USA. José Juan Quesada Molina Departamento de Matemática Aplicada, Universidad
More informationModeling of Dependence Structures in Risk Management and Solvency
Moeling of Depenence Structures in Risk Management an Solvency University of California, Santa Barbara 0. August 007 Doreen Straßburger Structure. Risk Measurement uner Solvency II. Copulas 3. Depenent
More informationRisk Aggregation and Model Uncertainty
Risk Aggregation and Model Uncertainty Paul Embrechts RiskLab, Department of Mathematics, ETH Zurich Senior SFI Professor www.math.ethz.ch/ embrechts/ Joint work with A. Beleraj, G. Puccetti and L. Rüschendorf
More informationReducing Model Risk With Goodness-of-fit Victory Idowu London School of Economics
Reducing Model Risk With Goodness-of-fit Victory Idowu London School of Economics Agenda I. An overview of Copula Theory II. Copulas and Model Risk III. Goodness-of-fit methods for copulas IV. Presentation
More information8 Copulas. 8.1 Introduction
8 Copulas 8.1 Introduction Copulas are a popular method for modeling multivariate distributions. A copula models the dependence and only the dependence between the variates in a multivariate distribution
More informationCopula Regression RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES CASUALTY ACTUARIAL SOCIETY MAY 18,2011
Copula Regression RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES CASUALTY ACTUARIAL SOCIETY MAY 18,2011 Outline Ordinary Least Squares (OLS) Regression Generalized Linear Models
More informationConvexity of chance constraints with dependent random variables: the use of copulae
Convexity of chance constraints with dependent random variables: the use of copulae René Henrion 1 and Cyrille Strugarek 2 1 Weierstrass Institute for Applied Analysis and Stochastics, 10117 Berlin, Germany.
More informationSongklanakarin Journal of Science and Technology SJST R1 Sukparungsee
Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Bivariate copulas on the exponentially weighted moving average control chart Journal: Songklanakarin Journal of Science and Technology
More informationSemi-parametric predictive inference for bivariate data using copulas
Semi-parametric predictive inference for bivariate data using copulas Tahani Coolen-Maturi a, Frank P.A. Coolen b,, Noryanti Muhammad b a Durham University Business School, Durham University, Durham, DH1
More informationConvolution Based Unit Root Processes: a Simulation Approach
International Journal of Statistics and Probability; Vol., No. 6; November 26 ISSN 927-732 E-ISSN 927-74 Published by Canadian Center of Science and Education Convolution Based Unit Root Processes: a Simulation
More informationCorrelation & Dependency Structures
Correlation & Dependency Structures GIRO - October 1999 Andrzej Czernuszewicz Dimitris Papachristou Why are we interested in correlation/dependency? Risk management Portfolio management Reinsurance purchase
More informationA New Family of Bivariate Copulas Generated by Univariate Distributions 1
Journal of Data Science 1(212), 1-17 A New Family of Bivariate Copulas Generated by Univariate Distributions 1 Xiaohu Li and Rui Fang Xiamen University Abstract: A new family of copulas generated by a
More informationTail comonotonicity: properties, constructions, and asymptotic additivity of risk measures
Tail comonotonicity: properties, constructions, and asymptotic additivity of risk measures Lei Hua Harry Joe June 5, 2012 Abstract. We investigate properties of a version of tail comonotonicity that can
More informationEstimating Bivariate Tail: a copula based approach
Estimating Bivariate Tail: a copula based approach Elena Di Bernardino, Université Lyon 1 - ISFA, Institut de Science Financiere et d'assurances - AST&Risk (ANR Project) Joint work with Véronique Maume-Deschamps
More informationThe multivariate probability integral transform
The multivariate probability integral transform Fabrizio Durante Faculty of Economics and Management Free University of Bozen-Bolzano (Italy) fabrizio.durante@unibz.it http://sites.google.com/site/fbdurante
More informationEfficient rare-event simulation for sums of dependent random varia
Efficient rare-event simulation for sums of dependent random variables Leonardo Rojas-Nandayapa joint work with José Blanchet February 13, 2012 MCQMC UNSW, Sydney, Australia Contents Introduction 1 Introduction
More informationElements of Financial Engineering Course
Elements of Financial Engineering Course Baruch-NSD Summer Camp 0 Lecture Tai-Ho Wang Agenda Methods of simulation: inverse transformation method, acceptance-rejection method Variance reduction techniques
More informationOn Parameter-Mixing of Dependence Parameters
On Parameter-Mixing of Dependence Parameters by Murray D Smith and Xiangyuan Tommy Chen 2 Econometrics and Business Statistics The University of Sydney Incomplete Preliminary Draft May 9, 2006 (NOT FOR
More informationREMARKS ON TWO PRODUCT LIKE CONSTRUCTIONS FOR COPULAS
K Y B E R N E T I K A V O L U M E 4 3 2 0 0 7 ), N U M B E R 2, P A G E S 2 3 5 2 4 4 REMARKS ON TWO PRODUCT LIKE CONSTRUCTIONS FOR COPULAS Fabrizio Durante, Erich Peter Klement, José Juan Quesada-Molina
More informationElements of Financial Engineering Course
Elements of Financial Engineering Course NSD Baruch MFE Summer Camp 206 Lecture 6 Tai Ho Wang Agenda Methods of simulation: inverse transformation method, acceptance rejection method Variance reduction
More informationConstruction and estimation of high dimensional copulas
Construction and estimation of high dimensional copulas Gildas Mazo PhD work supervised by S. Girard and F. Forbes Mistis, Inria and laboratoire Jean Kuntzmann, Grenoble, France Séminaire Statistiques,
More informationMath Refresher - Calculus
Math Refresher - Calculus David Sovich Washington University in St. Louis TODAY S AGENDA Today we will refresh Calculus for FIN 451 I will focus less on rigor and more on application We will cover both
More informationCopulas. MOU Lili. December, 2014
Copulas MOU Lili December, 2014 Outline Preliminary Introduction Formal Definition Copula Functions Estimating the Parameters Example Conclusion and Discussion Preliminary MOU Lili SEKE Team 3/30 Probability
More informationStudy Guide on Dependency Modeling for the Casualty Actuarial Society (CAS) Exam 7 (Based on Sholom Feldblum's Paper, Dependency Modeling)
Study Guide on Dependency Modeling for the Casualty Actuarial Society Exam 7 - G. Stolyarov II Study Guide on Dependency Modeling for the Casualty Actuarial Society (CAS) Exam 7 (Based on Sholom Feldblum's
More informationMarginal Specifications and a Gaussian Copula Estimation
Marginal Specifications and a Gaussian Copula Estimation Kazim Azam Abstract Multivariate analysis involving random variables of different type like count, continuous or mixture of both is frequently required
More informationGeneralized quantiles as risk measures
Generalized quantiles as risk measures Bellini, Klar, Muller, Rosazza Gianin December 1, 2014 Vorisek Jan Introduction Quantiles q α of a random variable X can be defined as the minimizers of a piecewise
More informationarxiv: v1 [math.st] 31 Oct 2014
A two-component copula with links to insurance arxiv:1410.8740v1 [math.st] 31 Oct 2014 S. Ismail 1 3, G. Yu 2 4, G. Reinert 1, 5 and T. Maynard 2 5 1 Department of Statistics, 1 South Parks Road, Oxford
More informationLecture 7 and 8: Markov Chain Monte Carlo
Lecture 7 and 8: Markov Chain Monte Carlo 4F13: Machine Learning Zoubin Ghahramani and Carl Edward Rasmussen Department of Engineering University of Cambridge http://mlg.eng.cam.ac.uk/teaching/4f13/ Ghahramani
More informationModelling Dependence of Interest, Inflation and Stock Market returns. Hans Waszink Waszink Actuarial Advisory Ltd.
Modelling Dependence of Interest, Inflation and Stock Market returns Hans Waszink Waszink Actuarial Advisory Ltd. Contents Background Theory Article Conclusion Questions & Discussion Background Started
More information