Lecture Quantitative Finance Spring Term 2015

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