Estimating the Wishart Affine Stochastic Correlation Model using the empirical characteristic function

José Da Fonseca; Martino Grasselli; Florian Ielpo

doi:10.1515/snde-2012-0009

Published by De Gruyter October 22, 2013

Estimating the Wishart Affine Stochastic Correlation Model using the empirical characteristic function

José Da Fonseca , Martino Grasselli and Florian Ielpo

From the journal Studies in Nonlinear Dynamics & Econometrics

https://doi.org/10.1515/snde-2012-0009

Showing a limited preview of this publication:

Abstract

This paper provides the first estimation strategy for the Wishart Affine Stochastic Correlation (WASC) model. We provide elements showing that the use of empirical characteristic function-based estimates is advisable as this function is exponential affine in the WASC case. We use a GMM estimation strategy with a continuum of moment conditions based on the characteristic function. We present the estimation results obtained using a dataset of equity indexes. The WASC model captures most of the known stylized facts associated with financial markets, including leverage and asymmetric correlation effects.

Keywords: Wishart Process; Empirical Characteristic Function; Stochastic Correlation; Generalized Method of Moments

Corresponding author: José Da Fonseca, Department of Finance, Auckland University of Technology, Private Bag 92006, 1142 Auckland, New Zealand, Phone: +64 9 9219999 extn 5063, e-mail: jose.dafonseca@aut.ac.nz

¹
A similar exercise could also be performed using options on individual indices. It remains however an exciting task that is beyond the scope of this article.
²
If instead of the characteristic function we focus on the moment generating function of the logarithm of the assets’ prices, then the matrix of equation (13) is Hamiltonian. Therefore its exponential is a symplectic matrix for which we have A11Τ(τ, ω)A22(τ, ω)−A21Τ(τ, ω)A12(τ, ω)=I. Hence, the solution depends on A₁₁(τ, ω), though indirectly.
³
Note that B₁₁ is involved in the computation of the characteristic function of the volatility.
⁴
These simulations results are not provided in this article, but available upon request.
⁵
This approximation is available upon request.
⁶
Another strategy to build an approximation scheme which remains positive in the matrix sense is through the use of the square function. This strategy was used by Gouriéroux, Jasiak, and Sufana (2009, 172). It however leads to analytical difficulties in the case of the WASC precisely because of the vector ρ that controls the leverage effect of the model.
⁷
We thank Marine Carrasco for pointing out this fact.
⁸
K−12 does not exist on the whole space of functions which are square integrable with respect to the weight π. We thus use this notational abuse to simplify the presentation and refer to Carrasco et al. (2007) for the precise mathematical aspects.
⁹
The scientific language used to implement the model and the estimation is R and is available at http://www.r-project.org/. The integration was performed using the R function “trapz” from the “caTools” package that computes a trapezoidal numerical integration.
¹⁰
We use the word “support” of a function f to denote the interval I such that ∫If∼∫f. For example, for the standardized Gaussian pdf we can qualify the interval [–3, 3] as the “support” but obviously the function is defined on ℝ.
¹¹
This remark extends to the functions (19) and (37).
¹²
In fact, the computational cost of a numerical integration mainly depends on the oscillatory behavior of the integrand.
¹³
O(n) stands for the orthogonal group i.e., O(n)={g∈GL(n, ℝ)|g^⊤g=I_n} with GL(n, ℝ) the linear group, the set of invertible matrices. We refer to Faraut (2006) for basic results on matrices.
¹⁴
Given Q∈GL(n, ℝ) there exists a unique pair (K,Q˜)∈O(n)×n, with ℘_n is the set of symmetric positive definite matrices, such that Q=KQ˜.
¹⁵
Obviously, we exclude the case i=j=0.
¹⁶
From a computational point of view it is simpler to estimate the matrix Q and then to take its polar decomposition because a matrix in ℘(n) implies more constraints on the elements of the matrix than a matrix in GL(n, ℝ). Note that all the aggregated quantities involving Q or Q and ρ are invariant by rotation.
¹⁷
On asymmetric correlation effects, see Roll (1988) and Ang and Chen (2002).
¹⁸
Gouriéroux, Jasiak, and Sufana (2009) used intraday data to estimate a Wishart based stock covariance model. This is however a different approach from what is done here.
¹⁹
We compared the results to those obtained with an Epanechnikov kernel without finding significantly different conclusions.
²⁰
If p_i is the i^th eigenvector of X and q_i is the i^th row of P^–1 then the projection operator on L_i is given by Pi=piqiΤ.
²¹
Given two matrix of same size X^kl and Y^kl then X◦Y=X^klY^kl.
²²
A difficulty with adaptive integration numerical schemes is that apart from the matrix H defined by equation (72) we need to keep track of the points {ωⁱ} and {ω^j} selected by the algorithm in order to evaluate the term g(θ)Th^(θ). Most of the algorithms implemented in software packages can not return this information.

Acknowledgements

We are particularly indebted to Marine Carrasco for remarkable insights and helpful comments. We are also grateful to Christian Gouriéroux, Fulvio Pegoraro, François-Xavier Vialard and the CREST seminar participants for useful remarks. We are thankful to the seminar participants of the 14th International Conference on Computing in Economics and Finance, Paris, France (2008), the 11th conference of the Swiss Society for Financial Market Research, Zürich (2008), Mathematical and Statistical Methods for Insurance and Finance, Venice, Italy (2008), the 2nd International Workshop on Computational and Financial Econometrics, Neuchâtel, Switzerland (2008), the First PhD Quantitative Finance Day, Swiss Banking Institute, Zürich (2008), Inference and tests in Econometrics, in honor of Russel Davidson, Marseille, France (2008), the Inaugural conference of the Society for Financial Econometrics (SoFie), New York, USA (2008), the 28th International Symposium on Forecasting, Nice, France (2008), the ESEM annual meeting, Milano, Italy (2008), the Oxford-Man Institute of Quantitative Finance Vast Data Conference, Oxford, UK (2008), the Courant Institute Mathematical Finance seminar, New York, USA (2008) and the Bloomberg Seminar, New York, USA (2008) for their comments and remarks. Any errors remain ours.

A Appendix

A.1 Computing the gradient

The gradient of the characteristic function is needed to study the asymptotic distribution of the estimates but also in the optimization process underlying the estimation procedure. Therefore we turn our attention to the differentiation of matrix function depending on a parameter. We illustrate the theoretical framework with the characteristic function of the assets’ log returns and we give without technical details the results for the forward characteristic function needed in our empirical study. We mainly rely on the work of Daleckii (1974) for the general case (i.e., in the non-Hermitian matrix case) and to Daleckii and Krein (1974), Donoghue (1974) and Bathia (2005) for the Hermitian matrix case.

Let us first state some basic results on linear algebra. Denote by {λ_i; i=1 … n} the set of eigenvalues of a matrix X∈M_n and m_i the multiplicity of λ_i as a root of the characteristic polynomial of X. Define L_i=Ker(X–λ_iI) and P_i the projection operator from ℂⁿ onto L_i, then we have ∑i=1nPi=I. Define also J_i such that (X–λ_iI)P_i=J_i. The Jordan normal form of X if given by the well-known decomposition X=∑i=1n(λiPi+Ji).

Let f be a function from M_n into M_n: the derivative of f at X in direction H, denoted D_f_,_X(H), is by definition ||f(X+tH)–f(X)–D_f,X(H)||= to(||p||) and can be computed using the following formula Daleckii (1974):

(62)

Df, X(H)=∑j1, j2n∑r1=0mj1−1∑r2=0mj2−11r1r2∂r1+r2∂λj1r1∂λj2r2[f(λj1)−f(λj2)λj1−λj2]Pj1Jj1r1HPj2Jj2r2. (62)

Remark 7.1 Whenever λj1=λj2 the term within the bracket should be replaced by f′(λj1).

Remark 7.2 When X can be diagonalized then m_j=1 for each j and we are lead to the very simple form

(63)

Df, X(H)=∑j1, j2n[f(λj1)−f(λj2)λj1−λj2]Pj1HPj2. (63)

If X is Hermitian then P^–1=P* and we recover the result presented for example in Daleckii and Krein (1974).

Simple algebra leads to D_f,X(H)=PM_f◦(P^–1HP)P^–1 where P is the matrix of the eigenvectors of X,²⁰ ◦ is the Schur product²¹ and M_f=(M_f(λ_k, λ_l))_{_k₌₁_…_n_,_l₌₁_…_n_} is the Pick matrix associated to the function f, which is defined by

(64)

Mf(λ, μ)={f(λ)−f(μ)λ−μifλ≠μf′(λ)ifλ=μ. (64)

This formulation is well known and can be found for example in Donoghue (1974, p. 79) and Bathia (2005, p. 123–124).

The gradient of the characteristic function involves ∂_αA where A is given by (12). In fact for any parameter value α which may be equal to M_kl, Q_kl, R_kl or β the gradient is given by

∂αΦYt,Σt(τ, z)=(Tr(∂αAΣt)+∂αc(τ))ΦYt, Σt(τ, z).

From (12) and ∂_α(L^–1)=–L^–1(∂_αL)L^–1 implied by the derivation of L^–1L=I we conclude that

∂αA(τ)=−A22(τ)−1∂α(A22(τ))A(τ)+A22(τ)−1∂αA21(τ),

Therefore we are lead to the computation of

(∂αA11(τ)∂αA12(τ)∂αA21(τ)∂αA22(τ)),

that is the derivative with respect to a parameter of a function of a matrix (in this case the exponential function). In order to use formula (62) we specify in the following table for each parameter of the WASC model the choice of the matrice X and H. As usual {e_l;l=1 … n} resp. {e_kl; k, l=1 … n} stands for the canonical basis of ℝⁿ resp. M_n(ℝ) (the function f being f(x)=e^x).

Parameter	X	H
M_kl	τG	τ(ekl00−eklT)
Q_ij	τG	τ((ekl)TρiωT−2(QTekl+(ekl)TQ)0−iωρTekl)
ρ_l	τG	τ(QTeliωT00−iωelTQ)

To fulfill the analytical computation of the gradient we need the derivative of c(τ) with respect to a model parameter and particularly the term log A₂₂(τ), because

(65)

∂αc(τ)=−β2[Tr(∂α(logA22(τ)))+τTr(∂αMT+iω∂α(ρTQ))]. (65)

In order to apply (62) we just need to define P₂₂ from M₂_n into M_n such that P₂₂L=L₂₂ with

(66)

L=(L11L12L21L22). (66)

Then it is easy to see that

(67)

∂αlogA22(τ)=Dlog,A22(τ)(P22Dexp,rG(H)). (67)

Once again using (62) with the log function gives the result.

Remark 7.3 If f is the exponential function then we can also compute the derivative of the exponential of a matrix using the Baker-Hausdorff’s formula [see e.g., Hall (2003, 71) formula (3.10) for the details].

(68)

∂αeτG=DexpτG∂α(τG), (68)

where DexpX=eXI−e−adXadX and ad_XY=[X, Y]=XY–YX is the Lie bracket.

The empirical study is based on the forward characteristic function of assets’ log returns defined by

(69)

ΦΣ0(t, −iA(τ))ec(τ)=exp{Tr[B(t)Σ0]+C(t)+c(τ)} (69)

with

B(t)=(A(τ)B12(t)+B22(t))−1(A(τ)B11(t)+B21(t)),C(t)=−β2Tr[log(A(τ)B12(t)+B22(t))+tMT],c(τ)=−β2Tr[log(A22(τ))+τMT+τiγ(ρTQ)].

As for the characteristic function of assets’ log returns it is straightforward to show that for any given model parameter α we have:

∂αΦΣ0(t, −iA(τ))ec(τ)=ΦΣ0(t,−iA(τ))ec(τ)(Tr(∂αB(t)Σ0)+∂αC(t)+∂αc(τ))),

where the matrix derivatives are given by

∂αB(t)=−(A(τ)B12(t)+B22(t))(∂αA(τ)B12(t)+A(τ)∂αB12(t)+∂αB22(t))−1B(t)+(A(τ)B12(t)+B22(t))−1(∂αA(τ)B11(t)+B21(t)),∂αC(t)=−β2Tr(Dlog, A(τ)B12(t)+B22(t)(∂αA(τ)B12(t)+A(τ)∂αB12(t)+∂αB22(t))).

This completes the analytical computation of the gradient.

A.2 Dynamics of the correlation process

In this Appendix we compute in the 2-dimensional case the drift and the diffusion coefficients of the correlation process ρt12 defined by

(70)

ρt12=Σt12Σt11Σt22. (70)

We differentiate both sides of the equality (ρt12)2=(Σt12)2Σt11Σt22. We refer to Da Fonseca, Grasselli, and Tebaldi (2008) for the explicit computation of all covariations involved in the below formulas. We obtain:

2ρt12dρt12=2Σt12Σt11Σt22dΣt12+(Σt12)2(1Σt22d(1Σt11)+1Σt11d(1Σt22))+(·)dt,

so that

dρt12=1Σt11Σt22(dΣt12−Σt122Σt11dΣt11−Σt122Σt22dΣt22)+(·)dt.

By using the covariations among the Wishart elements we have

d〈ρ12〉t=1Σt11Σt22[Σt11(Q122+Q222)+2Σt12(Q11Q12+Q21Q22)+Σt22(Q112+Q212)+(Σt12)2(Q112+Q212Σt11+Q122+Q222Σt22+2Σt12Σt11Σt22(Q11Q12+Q21Q22))−2Σt12Σt11(Σt11(Q11Q12+Q21Q22)+Σt12(Q112+Q212))−2Σt12Σt22(Σt12(Q122+Q222)+Σt22(Q11Q12+Q21Q22))]dt,

which leads to:

d〈ρ12〉t=(1−(ρt12)2)(Q122+Q222Σt22+Q112+Q212Σt11−2ρt12(Q11Q12+Q21Q22)Σt11Σt22)dt.

Now let us compute the drift of the process ρt12.

We differentiate both sides of the equality ρt12=Σt12Σt11Σt22 and we consider the finite variation terms:

dρt12=1Σt11Σt22dΣt12+Σt12d(1Σt11Σt22)+d〈Σ12, 1Σ11Σ22〉t=1Σt11Σt22(Ω11Ω21+Ω12Ω22+M21Σt11+M12Σt22+(M11+M22)Σt12)dt+Σt12[1Σt22(−12(Σt11)3)(Ω112+Ω122+2M11Σt11+2M12Σt12)+1Σt11(−12(Σt22)3)(Ω212+Ω222+2M21Σt12+2M22Σt22)+38Σt11Σt22(Σt11)2d〈Σ11〉t+38Σt11Σt22(Σt22)2d〈Σ22〉t+14(Σt11Σt22)3d〈Σ11, Σ22〉t]dt+1Σt22(−12(Σt11)3)d〈Σ11, Σ12〉t+1Σt11(−12(Σt22)3)d〈Σ12,Σ22〉t+Diffusions.

Now we use the formulas of the covariations of the Wishart elements and we arrive to an expression which can be written as follows:

dρt12=(At(ρt12)2+Btρt12+Ct)dt+Diffusions,

where:

At=1Σt11Σt22(Q11Q12+Q21Q22)−Σt22Σt11M12−Σt11Σt22M21,Bt=−Ω112+Ω1222Σt11−Ω212+Ω2222Σt22+Q112+Q2122Σt11+Q122+Q2222Σt22<0,Ct=1Σt11Σt22(Ω11Ω21+Ω12Ω22−2(Q11Q12+Q21Q22))+Σt22Σt11M12+Σt11Σt22M21.

From the definition of Ω=βQT and the Gindikin condition we deduce that B_t is negative. As a by-product, we easily deduce the instantaneous covariation between the Wishart element Σt11 and the correlation process:

d〈ρ12, Σ11〉t=1Σt11Σt22(d〈Σ11, Σ12〉t−Σt122Σt11d〈Σ11, Σ11〉t−Σt122Σt22d〈Σ22, Σ22〉t)=2Σt11Σt22(1−(ρt12)2)(Q11Q12+Q21Q22)dt.

Using the fact that Q∈GL(n, ℝ) there exists a unique pair (K, Q˜)∈O(n)×n such that Q=KQ˜. The law of the Σ_t being invariant by rotation of Q we rewrite this covariation as

d〈ρ12,Σ11〉t=2Σt11Σt22(1−(ρt12)2)Q˜12(Q˜11+Q˜22)dt.

Proof of Proposition 3.2.

We have to solve θ^T=arg minθ||(KTαT)−1/2h^(θ)|| which is equivalent to

θ^T=argminθ〈(KTαT)−1h^(θ), h^(θ)〉.

As in proposition 3.4 of Carrasco et al. (2007) we first compute the function g(θ) which solves

(71)

(αTI+KT2)g(θ)=KTh^(θ) (71)

with K_T an integral operator similar to (25) with the kernel given by (37). The left hand side of the above equation leads to

αTg(θ, ω)+1(T−q)2∑t, l=1T∫∫hl(ω,θ^1)Uhl(ω1, θ^1)ht(ω1, θ^1)Uht(ω2, θ^1)g(ω2)π(ω2)π(ω1)dω1dω2.

If we reorganize the integrals we can obtain the left hand side of equation B.15 of Carrasco et al. (2007). At this stage if we approximate the integrals using N equally spaced points {ωⁱ; i=1 … N} with ω_max>0, ω¹=–ω_max and ω^N=ω_max (δ_ω=ωⁱ–ωⁱ^–1) we obtain

(72)

αTg(θ,ω)+1(T−q)2∑t,l=1T ∑i,j=1Nhl(ω,θ^1)Uhl(ω1i,θ^1)ht(ω1i,θ^1)Uht(ω2j,θ^1)g(ω2j)π(ω2j)π(ω1i)δωδω. (72)

Denoting H the N×N matrix with element Hij=1(T−q)∑t=1Tht(ωi,θ^1)Uht(ωj, θ^1)π(ωj)δω and evaluating 72) for {ωⁱ; i=1 … N} this equation rewrites as (α_TI_N+H²)g(θ) with g(θ) a N-dimensional vector (which is equal to the function g(θ) of (71) evaluated at the points {ωⁱ; i=1 … N}, as such we use the same symbol). The right hand side of (71) can be handled in a similar way to obtain

(73)

Hh^(θ) (73)

with h^(θ)=(h^(ω1,θ), …, h^(ωN, θ))T. Once g(θ) is computed then after discretization of the integral involved in 〈(KTα)−1h^(θ), h^(θ)〉 we obtain g(θ)Th^(θ).²²

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Published Online: 2013-10-22

Published in Print: 2014-5-1

Estimating the Wishart Affine Stochastic Correlation Model using the empirical characteristic function

Abstract

Acknowledgements

A Appendix

A.1 Computing the gradient

A.2 Dynamics of the correlation process

References

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