Generalized normal mean variance mixtures and subordinated Brownian motions in Finance. Elisa Luciano and Patrizia Semeraro

Transcription

1 Generalized normal mean variance mixtures and subordinated Brownian motions in Finance. Elisa Luciano and Patrizia Semeraro Università degli Studi di Torino Dipartimento di Statistica e Matematica Applicata D. De Castro 1

2 Outline Motivation Aim General model One factor subordination GH distribution and process (case 1 and case 2) Calibrations 2

3 Motivation: Time changed Brownian motion Empirical: calendar time may not be appropriate to represent financial market time (Clark (1973)): it depends on the arrival of information and can often be proxied by trade Theoretical: any arbitrage free return process (semi-martingale) can be written as a time changed Brownian motion (Monroe (1978); Ané and Geman (2000)). Lévy environment: change of time are univariate subordinators, i.e. increasing Lévy processes. 3

4 Motivation: Multivariate extension Time-changed Lévy processes have been extended to the multivariate setting: common time change. The financial meaning is that the business time is the same for all assets. Drawbacks of a common subordinator 1. A common business time to all stocks seems to be restrictive; 2. the marginal processes have common parameters joint calibration is necessary; 3. independence may not be represented. 4

5 Aim Introduce a class of multivariate Lévy processes using multivariate subordination in order to Obtain time changed BM marginal processes, capture both high dependence and independence, propose a multivariate GH distribution and process; Show that the generalization permits to easily fit the linear correlation and to include independence. 5

6 Multivariate subordinators Generalization: each stock has its own change of time multidimensional subordinator. General model A random vector Y has generalized normal mean variance mixture distribution (shortly Y Gnmv) if Y = AGµ + Q GW, (1) where W N(0, I n ), A, Q M n, QQ T is positive-definite, G = diag(g) G positive, independent from W. Notice Gnmv since for G 1 = G 2 =... = G n we have the usual N mv mixture. 6

7 Theorem If the mixing distribution G is infinitely divisible, the vector Y is i.d. with characteristic function ψ Y (z) = exp(ψ G (iµz T A 1 2 (Q z)), (2) where Q z := (( l z l q l1 ) 2,..., ( l z l q ln ) 2 ) T = (Q T z)(z T Q) and Q j = (q 1j,..., q nj ) T. Under the condition of the previous theorem the vector Y Gnmv uniquely determines a Lévy process in law. 7

8 R n + -parameter process X = {(X 1 (s),..., X n (s)) T, s R n + } is an Rn + -parameter process (see Barndorff-Nielsen et al.) if X(0) = 0 almost surely and: 1. for any m 3 and for any choice of s 1... s m, the increments X(s j ) X(s j 1 ), j = 1,..., m, are independent; 2. for any s 1 s 2 and s 3 s 4 satisfying s 2 s 1 = s 4 s 3, X(s 2 ) X(s 1 ) = d X(s 4 ) X(s 3 ) (increments are stationary); 3. X(s) is almost surely right continuous with left limits in s in the partial ordering of R n +. 8

9 Proposition A random vector Y is in Gnmv if and only if Y = d Y (1), where Y is a Lévy process obtained by subordination of a R n + -parameter Brownian motion B(s). Moreover the subordinator G is the Lévy process such that L(G(1)) = L(G). Marginal processes: Y j (t) = B j (G(t)) = B j (G 1 (t),..., G n (t)). 9

10 One factor subordinator Economic intuition: both a common and an idiosyncratic time change exist, so that independent returns can be generated. Formally: We adopt the random-additive-effect distributions proposed in Barndorff-Nielsen et al. Let G be G = (X 1 + γ 1 Z,..., X n + γ n Z), (3) where γ 1,..., γ n are positive real parameters and X i, i = 1,..., n and Z are independent and infinitely divisible. Define G = {G(t), t 0} the Lévy process such that L(G(1)) = L(W ). 10

11 Summary: one factor model Y = AGµ + Q GW, (4) where G is the one factor mixing variable. If Y (t) is the corresponding Lévy process we proved that: Y (t) = B(G(t)) = (B 1 (G 1 (t),..., G n (t)),..., B n (G 1 (t),..., G n (t))) where B be a R n + -parameter Brownian motion. We will discuss for the GH process two particular cases in order to attach to each marginal asset its own change of time. 11

12 Case 1 In Semeraro and Luciano and Semeraro we considered for the VG and NIG specifications the particular case: If Q = Σ diagonal and A = I then Y T = ( G 1 σ 1 W 1 + µ 1 G 1,..., G n σ n W n + µ n G n ). (5) The corresponding process Y becomes Y (t) = (B 1 (G 1 (t)),..., B n (G n (t))), where B 1,..., B n are n independent Brownian motions. The intuition on business time is verified, but we do not always reach high correlation. 12

13 Multivariate GH process We specify the distribution of Y in order to obtain: Y with GH margins; easy calibration of both the margins and the dependence; it answers our economic requirement to attach to each Brownian motion its own change of time. 13

14 Multivariate GH process: GIG-subordinator X i GIG( λ, δ i, b γ i ), V i Γ(λ a, b2 2γ 2 i ), Z Γ(a, b2 2 ), where λ > 0, b 0, γ i > 0, 0 < a λ and δ i, b both nonnegative and not simultaneously zero. Then G: G = (X 1 + V 1 + γ1 2 Z,..., X n + V n + γn 2 Z). (6) has GIG margins: G i GIG. The subordinator G so that L(G(1)) = L(G) has GIG margins. 14

15 Multivariate GH process Case1: independent Brownian motions B 1,..., B n independent Brownian motions with parameters (µ j, 1). Define Y (t) = (B 1 (G 1 (t)),..., B n (G n (t))), where the subordinator G is defined in the previous slide to be GIG. The process Y is infinitely divisible and it has GH margins with parameters α j, β j, δ j, λ. 15

16 Multidimensional GH distribution: Case2: correlated Brownian motions We add correlation leaving GH margins. Let Y GM GH. Y is the sum of two independent multivariate processes: Y = Y T + ΓY Z,, where Γ = diag(γ1 2,..., γ2 n) and Y T has independent GH margins (independent subordinators T j = X j + V j ) and Y Z, has a common gamma subordinator Z. Define the process Ỹ (Q GMGH) Ỹ = Y T + QY Z, where Y Z is a multivariate VG process with a common subordinator Z(t) Γ(a, b2 2 ). 16

17 Proposition Under the condition: ( i q ji ) 2 = γ 2 j, the process Ỹ has GH(α j, β j, δ j, λ) margins. 17

18 Calibrations: case1 The parameters involved in the GMGH model are: The marginal parameters of the returns: α j, β j, δ j, λ. The common marginal parameter specify the marginal distributions, for example λ = 1, the case we consider in calibration correspond to Hyperbolic margins. The parameters of the subordinator, involved in the dependence structure of the model: a, b (we have constraint on the parameters that make correlations depend only on a, therefore we fix b = 1). 18

19 Calibrations:case2 The parameters involved in the Q-GMGH model are: The marginal parameters of the returns: α j, β j, δ j, λ; The parameters of the subordinator, involved in the dependence structure of the model: a, b. The correlation matrix Q. Some constraints have to be verified. 19

20 Calibrations: Hyperbolic case marginal parameters we consider the marginal parameters calibrated in Eberlein and Keller (1995) for the hyperbolic distribution (GH with λ = 1). We consider three firms taken from their sample, namely BASF, BMW, Daimler Benz. The estimated parameters for BASF, BMW and Daimler Benz are given in the following table: α j β j δ j BASF BMW DA-BE

21 Calibrations Case 1: Correlation with independent BM The theoretical correlation matrix for a = max = 1 follows: ρ BASF BMW BMW DA-BE

22 Case 2: Q-GMGH correlation We provide an example of choice of (a, b, Q) that gives rise to significant correlations, with the same marginal parameters as above. a = 1 (maximal), b = 10 and Q = With this choice for the dependence parameters we get the following correlations: ρỹ BASF BMW BMW DA-BE

23 Conclusions We developed a multivariate model that: is based on a one factor business clock, allows to capture both independence and high dependence, permits to fit linear correlation. 23

24 References 1) Ané, T., Geman, H. (2000). Order flow, transaction clock, and normality of asset returns The Journal of Finance, 55, )Barndorff-Nielsen, O.E., Pedersen, J. Sato, K.I. (2001). Self-Decomposability and Stability. Multivariate Subordination, 3) Clark, P.K. (1973) A subordinated stochastic process model with finite variance for speculative prices Econometrica 41, ) Eberlein, E. Keller, D. (1995) Hyperbolic distributions in Finance. Bernoulli, 1(3), ) Eberlein, Ernst; Prause, Karsten (2002) The generalized hyperbolic model: financial derivatives and risk measures. Mathematical finance Bachelier Congress,(Paris), , Springer Finance, Springer, Berlin. 6) Luciano, E., Semeraro, P (2007). Extending Time-Changed Lévy Asset Models Through Multivariate Subordinators, working paper, Collegio Carlo Alberto. 7) Monroe, I. (1978) Process that can be embadded in Brownian motion Annals of Probability 6, ) Semeraro, P. (2008) A multivariate Variance Gamma model for financial application Journal of Theoretical and Applied Finance, 11,