Computational Statistics and Data Analysis 52 (2008) 3389–3407 www.elsevier.com/locate/csda
Approximation of multiple integrals over hyperboloids with application to a quadratic portfolio with options J. Sadefo Kamdem a,∗ , A. Genz b a Laboratoire de Math´ematiques, CNRS UMR 6056, BP 1039 Moulin de la housse 51687 Reims cedex 2, France b Department of Mathematics, Washington State University, Pullman, WA 99164-3113, USA
Received 19 November 2007; received in revised form 14 December 2007; accepted 15 December 2007 Available online 23 December 2007
Abstract An application involving a financial quadratic portfolio, where the joint underlying log-returns follow a multivariate elliptic distribution, is considered. This motivates the need for methods for the approximation of multiple integrals over hyperboloids. Transformations are used to reduce the hyperboloid integrals to products of integrals which can be approximated with appropriate numerical methods. The application of these methods is demonstrated using some financial applications examples. c 2007 Elsevier B.V. All rights reserved.
Contents 1. 2. 3. 4.
Introduction ..................................................................................................................................................3390 Quadratic portfolio of options application.........................................................................................................3391 Transformation to a hyperboloid integration region............................................................................................3393 Integration over hyperboloids..........................................................................................................................3393 4.1. The general case .....................................................................................................................................3393 4.2. The normal case .....................................................................................................................................3395 4.3. The Student-t case ..................................................................................................................................3395 4.3.1. Student-t case with n + = 0 and v = 0 ........................................................................................3396 5. Application examples.....................................................................................................................................3397 5.1. A Normal distribution example with n − > 0, n + > 0 and v = 0 ..................................................................3397 5.2. Student-t example with n+ = 0 and v = 0 ................................................................................................3397 6. Conclusions ..................................................................................................................................................3401 Appendix A. Algorithm of Sheil and O’Muircheartaigh...................................................................................3401 Appendix B. Algorithms of Genz and Monahan .............................................................................................3402 Appendix C. Algorithm of Genz for spherical-surface integrals ........................................................................3402 Appendix D. Modified Genz and Bretz method ..............................................................................................3403 Appendix E. Limit of Taylor’s quadratic portfolio...........................................................................................3404 Appendix F. Parameter estimation using MLE and EWMA..............................................................................3405
∗ Corresponding address: JSK RisK Consulting, 124 Boulevard Vasco de Gama, 51100 Reims, France. Tel.: +33 3 26 87 52 12.
E-mail address:
[email protected] (J. Sadefo Kamdem). c 2007 Elsevier B.V. All rights reserved. 0167-9473/$ - see front matter doi:10.1016/j.csda.2007.12.006
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F.1. Example of Student-t distribution .............................................................................................................3406 References....................................................................................................................................................3406
1. Introduction Value-at-Risk (VaR) is considered to be one of the standard measures of market risk. VaR measures the maximum loss that a portfolio can experience with a certain probability over a certain horizon, for example, one day. Mathematically, if the profit or loss is given by Π (t, S(t)) − Π (0, S(0)), for which Π (t, S(t)) is the price of the portfolio at t, VaRα for a confidence level 1 − α is determined by the following equation: Prob{Π (t, S(t)) − Π (0, S(0)) ≥ −VaRα } = 1 − α,
(1)
where S(t) = (S1 (t), . . . , Sn (t)) is a vector of asset prices that govern the risk factors at time t. In this paper, we consider numerical methods for the estimation of integrals over hyperboloids and their application to VaR computations for a complex portfolio that contains options depending on the market fluctuations that create risk. We reduce the problem of VaR computations to multiple integrals over hyperboloids, and show how these integrals can be approximated using techniques described by Genz (1992, 2003), Genz and Monahan (1998), Genz and Bretz (2002), and Sheil and O’Muircheartaigh (1977). Related work for probability density computations has been described by Lu (2006). One of the most important analytic methods for VaR computation, which is called ∆-normal VaR, was introduced in the RiskMetrics Technical Document (1996). The method is based on the assumptions that the distribution is normal and the portfolio is linear. Sadefo Kamdem (2004b, 2005, 2007), generalized the ∆-normal VaR by introducing the ∆-elliptic VaR for a linear portfolio, with the ∆-Student VaR given as an example. An advantage of the ∆-elliptic VaR (for example, ∆-normal or ∆-Student) is that the formula is still fairly simple to calculate. But in practice, if we deal with a ∆-hedged portfolio or a nonlinear portfolio, the ∆-elliptic VaR does not provide a realistic model, and that is why alternatives were proposed in the papers of Brummelhuis et al. (2002), Brummelhuis and Sadefo Kamdem (2004) and Sadefo Kamdem (2004b). Also, the valuation of complex derivatives exposures can be a computationally intensive task. For risk management purposes, and particularly for VaR, we need hundreds of valuations to obtain the P & L distribution of a complex instrument. As the number of Monte Carlo simulation increases, the time savings from using quadratic approximation increases. In addition, if the pricing function is smooth enough, the loss in accuracy from using quadratic approximation is very small. We assume that the approximation for the price of the portfolio is given by 1 Π (t, S(t)) − Π (0, S(0)) ≈ Θt + ∆Xtt + Xt 0Xtt , 2 and we also assume that the joint log-returns Xt is elliptically distributed. For further details about elliptic distributions, see Embrechts et al. (2002). 0, Θ and ∆ are functions of some sensitivities of the portfolio (see Taleb (1997), for a discussion concerning sensitivities). We also suppose that t = 1/252, because the time horizon for VaR is generally taken to be one day. If the log-returns vector Xt is normally distributed, Albanese and Seco (2001) have shown how to reduce the analysis of a quadratic VaR to the computation of the integral of a Gaussian over a quadric in a space of possibly very high dimension. We will use the more general assumption of an elliptic distribution for the risk factors. An alternative rigorous analytical approach to quadratic VaR was proposed by Cardenas et al. (1997) and by Rouvinez (1997). They observed that, assuming Gaussian risk factors, the portfolio’s characteristic function can be explicitly computed. Numerical Fourier inversion will then yield the portfolio’s distribution function and, consequently, its quantiles or VaR. This method was extended to jump-diffusions in Duffie and Pan (2001). Note that it is only semi-explicit, in that it still requires the numerically nontrivial step of Fourier inversion (although good algorithms are available for this). This would be a disadvantage for analyzing parameter dependence. Moreover, explicit computation of the characteristic function is possible when Xt is normally distributed (and jumps may be included by first conditioning on the number of jumps (Duffie and Pan, 2001)), but the method does not generalize to certain non-Gaussian risk factors.
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Two further papers dealing with non-Gaussian quadratic VaR are Jahel et al. (1999), who assume Xt follows a stochastic volatility process, and Glasserman et al. (2002), who consider Student-t distributed Xt . Both papers are characteristic function based, (Glasserman et al., 2002) exploiting the relation of t-distributions with quotients of independent Gaussians, and (Jahel et al., 1999) employing the characteristic function of the process to compute the moments of the portfolio distribution function, and subsequently fitting a parametric distribution from the Pearson or Johnson family to these moments (but this last step introduces an uncontrolled approximation). This paper proposes numerical methods for the approximation of integrals over hyperboloids, with application to estimate the VaR. We combine some techniques described in Genz (2003, 1992), Genz and Bretz (2002), Genz and Monahan (1998) and Sheil and O’Muircheartaigh (1977) to approximate the integrals over hyperboloids for VaR, with the generalized assumption that the underlying joint log-returns changes with an elliptic distribution. To illustrate our method, we will take the cases of normal distributions, Student-t and we consider the test examples for two ∆-hedged portfolios from the French CAC 40 market. One of the most common methods for quadratic perturbations of the linear VaR uses the Cornish–Fisher expansion for the quantile function of non-Gaussian variables. There are also some quadratic approximations in Hull (1999) and Dowd (1998). Many papers in the literature have proposed numerical methods for the quadratic approximation (for example, Sadefo Kamdem (2003) proposed the use of some numerical methods of Genz (2003), and the use of hypergeometric functions for a portfolios of equities VaR with mixture of multivariate Student-t distribution). The rest of our paper is organized as follows. In Sections 2 and 3, following Albanese and Seco (2001), we show how portfolio volatility can be used to reduce the calculation of VaR to the approximation of an integral over a hyperboloid, assuming elliptic distributions that admit a density function. In Section 4, we propose a numerical method for the approximation of integrals over hyperboloids using some methods of Genz (2003), Genz and Monahan (1998) and Sheil and O’Muircheartaigh (1977). To illustrate our method we use examples where the density function is Normal or Student-t. In Section 5, we consider two examples of financial portfolios, and we show that our method is applicable to estimate the VaR for the portfolio. In Section 6, we provide some conclusions. 2. Quadratic portfolio of options application In this section, we will define a quadratic portfolio of options. We will use the following notational conventions for vectors and matrices: single letters x, y, . . . will denote row vectors (x1 , . . . , xn ), (y1 , . . . yn ), and the corresponding column vectors will be denoted by x t , y t . Matrices A = (Ai j )i, j , B, etc. will be multiplied in the usual way. In particular, A will act on vectors by left-multiplication on column vectors, Ay t , and by right-multiplication on row vectors, x A; x · y = x y t = x1 y1 + · · · + xn yn will stand for the Euclidean inner product, with kxk2 = x x t . We first define Xt = (X1t , . . . , Xnt ), with Xit = log(Si (t)/Si (0)), i i 1 n and we define ∆1 = (∆11 , . . . , ∆n1 ), with ∆1= Si (0).∆ , and ∆ = (∆ , . . . , ∆ ), the gradient vector of the portfolio i, j
Π at time t = 0. We also define 01 = 01 i, j
01 =
Si2 (0)0 i,i + ∆i1 Si (0)S j (0) · 0 i, j
i, j=1,...,n
by
if i = j if i 6= j ,
with
0= 0
i, j
i, j=1,...,n
=
∂ 2Π (0) ∂ Si ∂ S j
,
i, j=1,...,n
the Hessian of the portfolio at time t = 0. If we use a second-order Taylor series approximation e X it −1 ≈ X it + X it2 /2 for i = 1, . . . , n, then Π (t, S(t)) − Π (0, S(0)) ≈ tΘ + ∆1 Xtt + Xt 01 Xtt /2, where Θ = ∂∂tΠ (0). This approximation contains all O(t) terms. If the random vector Xt follows an elliptic distribution √ with covariance matrix proportional to t, the expectation of its norm will be proportional to t.
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We assume that the X jt are elliptically distributed with mean µ and correlation matrix Vt = C Ct : (X1t , . . . , Xnt ) ∼ N (µ, Vt , φ). Then the pdf of Xt is of the form t f Xt (x) = |Vt |−1/2 g((x − µ)V−1 t (x − µ) ),
where |Vt | denotes the determinant of Vt , and where g : R≥0 → 0 is such that the Fourier transform of g(kxk2 ), as a generalized function on Rn , is equal to φ(kξ k2 ). Assuming that g is continuous, and nonzero everywhere, the Value-at-Risk denoted by VaRα at a confidence level of 1 − α is given by the solution to following equation: t t 1 − α = G∆ 0 (−VaRα ) = Prob(Θ t + ∆1 Xt + Xt 01 Xt /2 ≥ −VaRα ) Z dx = g (x − µ)V−1 (x − µ)t √ . t 1 t t det(Vt ) {Θ t+∆1 x + 2 x01 x ≥−VaRα }
(2)
Using the Cholesky decomposition Vt = Ct C, and changing variables in the integral to y = (x − µ)C−1 , Z G∆ (−VaR ) = g(kyk2 )dy, α 0 ˘ t +y ∆ ˘ t + 1 y 0˘ y t ≥−VaRα } {Θ 2
˘ t = Θt + µ∆t + 1 µ01 µt , ∆ ˘ = (∆1 + µ01 )Ct , and 0˘ = C01 Ct . Now, diagonalize 0˘ into its principal where Θ 2 t ˘ components, 0 = PDP , and change variables to z = yP, so that Z ∆ G 0 (−VaRα ) = g(kzk2 )dz. ˘ t +z Pt ∆ ˘ t + 1 z Dz t ≥−VaRα } {Θ 2
˘ D˘ −1 , so that zPt ∆ ˘ t = zDv t , then If we let v = ∆P ˘ t + zPt ∆ ˘ t + 1 zDz t = Θ ˘ t − 1 vDv t + 1 (z + v)D(z + v)t , Θ 2 2 2 and Z
G∆ 0 (−VaR)
= ˘ t +v Dv t } {(z+v)D(z+v)t ≥−2VaR−2Θ
g(kzk2 )dz,
or G∆ 0 (R)
Z = {−(z+v)D(z+v)t ≤R 2 }
g(kzk2 )dz,
(3)
with ˘ t − vDv t . R 2 = 2VaR + 2Θ
(4)
If we determine Rα as the solution to the equation 1 − α = G ∆ 0 (R), then the Value-at-Risk is given by VaRα = ˘ t + vDv t /2. Rα2 /2 − Θ The typical way in which such a quadratic portfolio arises in practice as a 0 − ∆ approximation of a more complicated nonlinear portfolio with value Π (X1t , . . . , X(n+1)t , t), where t denotes time. If we make the additional assumption that Π is ∆-hedged at today’s values of X, then we can, without loss of generality, assume to be ∆ = 0. A typical example would be a hedged portfolio of derivatives plus underlying assets, with X j being the log-return of the jth underlying asset S j over the time-interval [0, t]: X jt = log(S j (t)/S j (0)). The restriction to ∆-hedged portfolios simplifies the computations, but this is also an important special case, covering the hedged portfolios constructed for various kinds of options. All results in this paper can in principle be extended to general quadratic portfolios; in Brummelhuis et al. (2002) this was already done for Gaussian distributed X. If we consider a ∆-hedged Portfolio, ˘ = 0 and v = 0, then the solution Rα to the where we have ∆ = 0 and ∆1 = 0, and we assume µ = 0, so that ∆ equation Z 1 − α = G 00 (R) = g(kzk2 )dz, (5) {−z Dz t ≤R 2 }
˘ t. provides the Value-at-Risk quantity VaRα , with confidence level 1 − α, given by VaRα = 21 Rα2 − Θ
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3. Transformation to a hyperboloid integration region We start with the general equation (3), and assume that the diagonalization of 0˘ = PDPt has been constructed with D+ 0 D= , 0 −D− where d1 D = 0 0
... .. . ...
0
0 , dn
− − − + − + for = ±1, and where all d + j , d j ≥ 0, and −d1 ≤ −d2 ≤ · · · ≤ −dn − ≤ d1 ≤ · · · ≤ dn + . If z = (z − , z + ) and v = (v− , v+ ) are partitioned into subvectors associated with the respective negative and positive eigenvalues of D, then (3) becomes Z G∆ (R) = g(kzk2 )dz, (6) 1 1 0 {k(z − +v− )D−2 k2 −k(z + +v+ )D+2 k2 ≤R 2 }
The integration region is the hyperboloid in Rn defined by 1
1
{z = (z − , z + ) ∈ Rn − × Rn + : k(z − + v− )D−2 k2 − k(z + + v+ )D+2 k2 ≤ R 2 }. If D is defined to be the absolute value of D, then an alternate form of Eq. (6) is determined using the transformation 1 w = (w− , w+ ) = (z + v)D 2 , so that Z 1 dw G∆ (R) = . (7) g(kw D − 2 − vk2 ) 0 1 2 2 2 kw− k ≤R +kw+ k |D| 2 Our goal is to determine a solution Rα to G(R) = 1 − α. For a given confidence level 1 − α, once we find Rα , we have the approximate quadratic Value-at-Risk given by 1 2 1 R − Θt + vDv t . 2 α 2
VaRα =
(8)
4. Integration over hyperboloids 4.1. The general case We need to determine approximations to integrals in the form Z G(R) = ϕ(x, y)dydx, kxk2 ≤R 2 +kyk2
where x ∈ and y ∈ Rn 2 . If we use the change of variables: y = r2 ξ2 , x = r1 ξ1 , with r2 = kyk and r1 = kxk, then G(R) becomes Rn 1
∞
Z G(R) = 0
q
r2n 2 −1
Z
Z kξ2 k=1 0
R 2 +r22
r1n 1 −1
Z
ϕ(r1 ξ1 , r2 ξ2 )dσ (ξ1 )dr1 dσ (ξ2 )dr2 .
kξ1 k=1
We now have G(R) defined in terms of a product of two integrals over hyper-spherical surfaces, defined by kξ2 k = 1 and kξ1 k = 1, and two radial integrals, with dσ denoting the standard spherical-surface content measure. The hypersphere surface integrals can be approximated using methods described in the paper by Genz (2003). In this paper we assume that ϕ is an elliptic density function, so 1
1
ϕ(r− ξ− , r+ ξ+ ) = g(k(r− ξ− , r+ ξ+ )D − 2 − vk2 )/|D| 2 ,
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and, therefore Z
1
G(R) = kw− k2 ≤R 2 +kw+ k2 ∞
Z = 0
Z n −1 r++
q
g(kw|D|− 2 − vk2 ) R 2 +r22
dw 1
|D| 2
.
n −1
r−−
0
Jg (r− , r+ )dr− dr+ ,
(9)
where Jg (r− , r+ ) =
1
"Z
Z kξ+ k=1
g(k(r− ξ− , r+ ξ+ )D − 2 − vk2 ) 1
|D| 2
kξ− k=1
# dσ (ξ− ) dσ (ξ+ ).
In this general case we have the following theorem. Theorem 4.1. Z
ϕ (x, y) dxdy X w p1 w p2
G(R) =
kx−v1 k2 −ky−v2 k2 ≤R 2 X −c(u)−c(v)
≈2
| p 1 |=m 1 ∞
Z 0
r2n 2 −1
| p 2 |=m 2
Z √ R 2 +r 2 0
r1n 1 −1
XX s1
ϕ(r1 (s 1 .u p1 ) + v1 ), (r2 (s 2 .u p2 ) + v2 )dr1 dr2 ,
(10)
s2
where s · u = (s1 u 1 , . . . sn u n ), s 1 = (s11 , s21 , . . . , sn11 ), s 2 = (s12 , s22 , . . . , sn22 ), s ij = ±1, i = 1, 2 and j = 1, 2, . . . , n i . c(u p1 ) and c(u p2 ) are the respective numbers of nonzero entries in u p1 and u p2 , m 1 and m 2 determine the order of accuracy for the spherical-integration rules, and the w p ’s and u p ’s are integration rule parameters. Proof. Starting with Z G(R) = kx−v1 k2 −ky−v2 k2 ≤R 2
ϕ(x, y)dxdy,
where x ∈ Rn 1 and y ∈ Rn 2 , we change variables to y = r2 ξ2 , x = r1 ξ1 , where r2 = kyk, r1 = kxk, with ξi ∈ Sn i −1 = {x ∈ Rni : kxk = 1} for i = 1, 2 and n 1 + n 2 = n. Then Z G(R) = ϕ(x, y)dxdy kx−v1 k2 −ky−v2 k2 ≤R 2 Z Z ∞ r2n 2 −1 ϕ(x + v1 , r2 ξ2 0 Sn2−1 |x|2 ≤R 2 +r 2 Z Z ∞ J (R, r2 , r, ξ2 )dσ (ξ2 )dr2 , r2n 2 −1 Sn2−1 0
Z = =
+ v2 )dxdσ (ξ2 )dr2
with J (R, r2 , r, ξ2 ) =
R 2 +r 2
Z 0
r1n 1 −1
Z
ϕ(r1 ξ1 + v1 , r2 ξ2 + v2 )dσ (ξ1 )dr1 . Sn 1 −1
So G(R) is given by ∞
Z G(R) = 0
r2n 2 −1
Z √ R 2 +r 2 0
r1n 1 −1
Z
Z Sn 2 −1
ϕ(r1 ξ1 + v1 , r2 ξ2 + v2 )dσ (ξ1 )dσ (ξ2 )dr1 dr2 . Sn 1 −1
(11)
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The application of Lemma C.1 to the hyper-spheres Sn 1 −1 and Sn 2 −1 provides the approximation Z ∞ Z √ R 2 +r 2 X X r2n 2 −1 r1n 1 −1 G(R) ≈ 1 − 2−c(u)−c(v) w p1 w p2 | p 1 |=m 1
×
XX s1
0
| p 2 |=m 2
0
ϕ(r1 (s 1 .u p1 )t + v1 ), (r2 (s 2 .v p2 )t + v2 )dr1 dr2 ,
s2
given by the theorem.
Remark 4.2. Efficient numerical approximation of the radial integrals will depend on information about the rate of decrease of the integrand ϕ for large values of r1 and r2 . Remark 4.3. Because of limitations of the quadratic Taylor approximation for a large variation of log-returns, we can consider the quadratic method described by Studer (1999) and then use our method directly (see Appendix E). 4.2. The normal case For many applications, the distribution g is a normal distribution. In this case, Z −1 dw 2 2 . G(R) = e−kw D −vk /2 √ 2 2 2 (2π )n |D| kw− k ≤R +kw+ k Separating the w and v variables, we find −1 2
Z G(R) = kw+ k2 ≥0
e−kw+ D+ −v+ k /2 p (2π )n + |D+ | 2
−1 2
Z kw− k2 ≤R 2 +kw+ k2
e−kw− D− −v− k /2 p dw− dw+ . (2π )n − |D− | 2
The inner integral for w− can be computed using the algorithm described by Sheil and O’Muircheartaigh (1977), so we define H (R, r ) by H (R, r ) =
−1 2
e−kw− D− −v− k /2 p dw− , (2π )n − |D− |
Z kw− k2 ≤R 2 +r 2
2
(12)
1
and let w+ = (z + + v+ )D+2 . Then G(R) can rewritten as Z 1 t 1 G(R) = e−z + z + /2 H (R, k(z + + v+ )D+2 k)dz + 1 n /2 + (2π ) k(z + +v+ )D+2 k2 ≥0 Z ∞ 1 t /2 1 −z + z + 2 = e k)dz + , H (R, k(z + v )D + + + (2π )n + /2 −∞ where ∞ = (∞, . . . , ∞). Integrals in this form can be approximated using methods described by Genz (1992) and Genz and Monahan (1998) (see Appendix B). 4.3. The Student-t case In this case, g is given by x − n+ν 2 g(x) = Cν,n 1 + , ν 0((n+ν)/2) √ where x > 0, n = n − + n + , and Cν,n = 0(ν/2) . (νπ)n Substituting the Student-t g into Eq. (6), we have
Z G(R) = Cν,n
1
1
k(z − +v− )D−2 k2 ≤R 2 +k(z + +v+ )D+2 k2
kzk2 1+ ν
− n+ν 2 dz.
(13)
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J. Sadefo Kamdem, A. Genz / Computational Statistics and Data Analysis 52 (2008) 3389–3407
We now transform this problem to a χ -Normal problem using the transformations first described by Cornish (1954). First multiply G(R) by a χ integral term (with value 1), so that
Z G(R) = Cν,n
1 1 k(z − +v− )D−2 k2 ≤R 2 +k(z + +v+ )D+2 k2
Now change variables using r = s(1 + give n+ν
G(R) = Cν,n
21− 2 0( n+ν 2 )
Z
kzk2 ν
∞
kzk2 1+ ν
− n+ν 2
n+ν
dz
21− 2 0( n+ν 2 )
∞
Z
r2
r n+ν−1 e− 2 dr. 0
1
) 2 , change the order of integration, and separate the exponential terms to
s2
s n+ν−1 e− 2
Z 1 1 k(z − +v− )D−2 k2 ≤R 2 +k(z + +v+ )D+2 k2
0
e−
s 2 kzk2 2ν
dzds.
1
After the final transformation y = sz/ν 2 , and some cancellations in the constant terms ν Z Z ∞ 2 kyk2 21− 2 ν−1 − s2 G(R) = s e e− 2 dyds. 1 1 ν 2 2 n/2 sv sv (2π ) 0( 2 ) 0 k(y− + √− )D−2 k2 ≤ s νR +k(y+ + √+ )D+2 k2 ν
ν
Separating the y− and y+ variables, we obtain ν
G(R) =
21− 2 0( ν2 )
∞
Z
s2
s ν−1 e− 2
Z
0
∞
e−
ky+ k2 2
−∞
Z 1 1 2 2 sv sv )D−2 k2 ≤ s νR +k(y+ + √+ )D+2 k2 k(y− + √− ν ν
e−
ky− k2 2
dy− dy+ ds. √ (2π )n
(14)
An appropriate numerical method for the s integral can be combined with numerical methods described in previous sections to provide numerical approximations for G(R); methods similar to those described by Genz and Bretz (2002) can also be used (see Appendix D). 4.3.1. Student-t case with n + = 0 and v = 0 In this case n = n − and we can use a simplified Eq. (14) "Z # ν Z ∞ 2 21− 2 −yy t /2 ν−1 − s2 G(R) = e dy ds. s e 1 n− 2 2 ky D−2 k2 ≤ s νR 0( ν2 )(2π ) 2 0
(15)
We can also work directly with the Student-t form of G(R) and the w variables. Sadefo Kamdem (2004b) showed how this form of G(R) can be transformed into a spherical-surface integral of a hypergeometric function: #− n + ν 2 −1 t w− D− w− G(R) = 1 + dw− 1 ν |D− | 2 kw− k2 ≤R 2 " # n+ ν Z ∞ 2 ξ D −1 ξ t − 2 r− Cν,n − − − n − −1 = 1− r− 1+ dr− 1 ν |D− | 2 R Z Cν,n R −ν = 1− Φ R,ν (ξ− ) dσ (ξ− ), 1 ν|D− | 2 kξ− k=1 Cν,n
Z
"
(16)
where Φ R,ν (ξ− ) =
−1 t ξ− D − ξ− ν
!− n + ν
"
2
2 F1
# ν n− + ν ν 2 + ν , ; ;− . −1 t 2 2 2 2 ξ− D − ξ− R
If Theorem 4.1 is applied in this case, only one hyper-sphere surface integral is needed for the approximate computation of G(R).
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J. Sadefo Kamdem, A. Genz / Computational Statistics and Data Analysis 52 (2008) 3389–3407 Table 1 Data for ten CAC 40 European call and put options
Call-BNPPARIBAS Call-BOUYGUES Call-CAP GEMINI Call-CREDIT AGRICOLE Call-DEXIA Put-LOREALL Put-SOCIETEGENERALE Put-TF1 Put-THOMSON Put-VIVENDI
Exercise price
Underlying price
30.00 19.00 20.00 10.50 9.00 40.00 50 18.00 9.00 9.00
39.75 27.30 24.00 14.80 9.38 62.90 64.00 22.02 17.13 17.00
5. Application examples 5.1. A Normal distribution example with n − > 0, n + > 0 and v = 0 In this case Z
∞
G(R) =
t e−z + z + /2 H (R, z + D+ z + )√ t
−∞
dz + . (2π )n +
Integrals in this form can be approximated using methods that are a combination of the Sheil and O’Muircheartaigh (1977) and Genz and Monahan (1998) algorithms (see Appendices A and B). We consider a portfolio that contains call options and put options on equities, so that the price of the portfolio at time t, is given by: " # 5 10 X X Π (t) = [Ci (t, Si (t)) − δi Si (t)] − P j (t, S j (t)) − (δ j − 1)S j (t) . (17) i=1
j=6
The prices of each of the options are taken from data from the French CAC 40 Market, and are given in Table 1: In this example, using the three month historical data for the ten CAC 40 equities, with the exponential moving weighted average (EMWA) λ = 0.94, interest rate r = 0.05 and the time maturity is 3 month, we obtained the following V. The matrix 0 is a diagonal matrix with diagonal d = (24.186, 8.6269, 21.7320, 4.3111, 15.4949, −4.5815, −82.2915, −22.2079, −1.2957, −1.2822), and Θ = −3.8596. The eigenvalues of the D matrix are given by the vector e = (−0.1251, −0.0115, −0.0030, −0.0014, −0.0006, 0.0014, 0.0092, 0.0124, 0.0290, 0.1271). The following table provides some R and V values that were found as numerical solutions to the equation G(R) = 1 − α, for selected α’s. α R
0.05 0.025 0.01 0.6069 0.7176 0.8455
5.2. Student-t example with n+ = 0 and v = 0 We constructed a ∆-hedged portfolio that contains n = 45 equities and n = 45 European call options on these equities from the French CAC 40 Market (January 05, 2005–October 17, 2005), with data plotted in Figs. 1 and 2. The price of the portfolio is given by Π (t, S(t)) = Z
45 X i=1
[−Ci (t, Si (t)) + ∆i · Si (t)],
(18)
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Fig. 1. Histogram of 199 daily log-returns of 45 stocks (CAC 40).
Fig. 2. Normalplot of 200 daily prices of 45 stocks (CAC 40).
where Si is an equity price i, with S(t) = (S1 (t), . . . , Sn (t)), and Ci (t, Si (t)) is the price of European call option i on equity i. ∆ is known in the literature as a gradient portfolio sensitivity vector. Our portfolio has been chosen so 1 n i that ∆ = 0, with ∆i = ∂C ∂ Si (Si (0)), and ∆ = (∆ , . . . , ∆ ). We defined the volatility σi of the underlying stock i as the sample standard deviation of the log-return of stock i. We set the maturity time T = 1/4 years, the interest rate r = 0.1, and used E i = (1−c cos(k i π ) σi ) Si (0) for the exercise price of call i with k = 1, 2 and c a parameter to be chosen. The degrees-of-freedom parameter ν for the Student- t distribution can be estimated via maximum likelihood, using the EWMA covariance matrix and the sample log-returns (see Appendix F). Computations were first attempted using spherical-surface integration methods with Eq. (16) but, because of the large variation in the eigenvalues for this problem, accurate R values could not be determined. However, using the method described in Appendix D and applied to Eq. (15) we were able to compute R values with accuracy levels consistent with the data values. The R’s were determined by solving the Eq. (15) for α using a bisection method, with integral approximations that had estimated errors ≈ 5 × 10−4 , based on 20 000 quasi-Monte Carlo simulations
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for each integral approximation. Each integral approximation takes approximately one minute in Matlab. The time execution of Monte Carlo for 1000 simulations was 132 s. For 10 000 MC simulations the time was 2833 s. In the following Tables: VaRΠ MC,NSIM denotes the VaR of the full portfolios (without approximation) obtained −0 with NSIM Monte Carlo simulation using Student-t random with ν; VaRΘ MC,NSIM denotes the VaR of the quadratic approximation portfolios obtained with NSIM Monte Carlo simulations. The following Tables provide computed R and V a Rα,ν values for ν = 3, 5, 8, for selected α’s and two c = 1, 0, −1 values. NSIM denotes the number of simulations for Monte Carlo.
• In-the-money (ITM) portfolio: for c = 1, Z = 1, k = 2 ν = 3, Π (0) = 85.6680838, E i /Si (0) = 1 − cσi < ˘ t = −0.818343258, we obtained 1, Θ α R −0,ν=3 VaRΘ SG Θ −0,ν=3 VaR MC,1000 Θ−0,ν=3 VaRSG Θ−0,ν=3 − 1 VaRMC,1000
0.01 0.00329 0.81834867 0.81726040
0.001 0.00726 0.818369612 0.81726167
0.001331607 0.001355676
• At-the-money (ATM) portfolio: for Z = 1, c = 0; ν = 3, Π (0) = 8.56680838, E i /Si (0) = 1, and ˘ t = −0.828093011, we obtained Θ α R −0,ν=3 VaRΘ SG Θ −0,ν=3 VaRMC,1000
0.01 0.0794 0.831245191 0.828049899
0.001 0.173 0.843057511 0.828060544
• Out-the-money (OTM) portfolio: For Z = 1, c = −1; ν = 3, Π (0) = 85.6680838, E i /Si (0) = 1 − cσ (i) > 1, ˘ t = −0.821139785, we obtained Θ α R −0,ν=3 VaRΘ SG Θ −0,ν=3 VaRMC,1000 VaRFull,ν=3 MC,1000
0.01 0.664 1.041587785 0.816849144 0.755008872
0.001 1.45 1.872389785 0.818709816 0.817770347
• For α = 0.01, Z = 1, ν = 5, r = 0.1, T = 0.25, E i /Si (0) = (1 − c σi ) and N S I M = 1000 trials, we obtained Portfolio: c Π (0) ˘t Θ R −0,ν=5 VaRΘ SG
ITM ATM OTM 1 0 −1 85.409275235 86.535733454 85.668083823 −0.818343258 −0.828093011 −0.821139785 0.00213 0.0493 0.461 0.818345527 0.829308256 0.927400285
−0,ν=5 VaRΘ MC,1000
0.81726019
0.817437745
0.818903003
VaRFull,ν=5 MC,1000
0.717463451
0.738764720
0.757069804
VaRFull,ν=5 MC,1000 − 1 Θ−0,ν=5 VaRMC,1000
0.1371238411
0.105022419
0.113984433
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˘ t = −0.817260053, we obtained • For Z = 1, ν = 5, Π (0) = 85.6680838, E i /Si (0) = 1 − c cos(i π ) σi and Θ c R0.01 −0,ν=5 VaRΘ SG
1 0.331 0.872040553
−1 0.261 0.856283491
−0,ν=5 VaRΘ MC,1000
0.817260194
0.817260485
VaRFull,ν=5 MC,1000
0.717463454
0.738764726
VaRFull,ν=5 MC,1000 Θ−0,ν=5 − 1 VaRMC,1000
0.137123841
0.105022419
• For α = 0.01, Z = 1, ν = 8, r = 0.1, T = 0.25, E i /Si (0) = 1 − c σi and N S I M = 1000 trials, we obtained Portfolio: c Π (0) ˘t Θ R0.01 −0,ν=8 VaRΘ SG −0,ν=8 VaRΘ MC,1000
ITM ATM OTM 1 0 −1 85.409275235 86.535733454 85.668083823 −0.818343258 −0.828068306 −0.821139785 0.00171 0.0383 0.317 0.81834472 0.828826456 0.871384285 0.818343247
0.828068306
0.818967012
˘ t = −0.817260053, we obtained • For Z = 1, ν = 8, Π (0) = 85.6680838, E i /Si (0) = 1 − c cos(i π ) σi and Θ c R0.01 −0,ν=5 VaRΘ SG
1 0.202 0.842624991
−1 0.261 0.856283491
−0,ν=5 VaRΘ MC,1000
0.817260195
0.817260486
VaRFull,ν=5 MC,1000
0.717463451
0.738764721
0.137123841
0.105022419
VaRFull,ν=5 MC,1000 Θ−0,ν=5 − 1 VaRMC,1000
The following remarks summarize the results from our tables: • For near-term ATM portfolios the quadratic effect is more important. • As can be seen from the preceding tables, our method has comparable accuracy to Monte Carlo methods because −0,α,ν −0,α,ν VaR∆ ' VaR∆ , for ATM portfolio and ITM portfolio. Options at-the-money have the most nonlinear MC SG patterns. In fact, the sensitivity Θ is the greatest for short-term at-the-money options when measured in absolute value. Note also that ATM portfolios have the highest 0, which indicates that ∆ changes very fast as S changes. • The large errors in the case where we consider OTM portfolio reflect the fact that the European options in the portfolio are valueless. • The differences seen in the tables between the full MC results and the quadratic approximation results demonstrate the limits of the quadratic approximation for the VAR computations. In order to assess the quality of the quadratic approximation, one can use the Kullback–Leibler information measure as in Lu (2006). • Overall, the usefulness of the Delta–Gamma–Theta method depends on how users view the trade-off between computational speed and accuracy. Our method is more efficient than the Monte Carlo method, because the −0,α,ν −0,α,ν computation of VaR∆ takes approximately 60 s, while the computation of the VaR∆ SG MC,10000 takes 2833 s. • For a risk manager seeking a quick efficient means of computing VaR that measures Gamma–Theta risk, ∆−0 −Θ approximation offers an attractive method for doing so.
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• We have used the EWMA covariance matrix. But, the dynamic conditional correlation (DCC) of Engle (2002) could also have been used. Sadefo Kamdem (2004a), for example, has proposed the use of DCC with a Generalized Laplace distribution. Another approach is the multivariate switching regime approach for the covariance, which could be applied to other elliptic distributions (see Pelletier (2006) and Carvalho and Lopes (2007)). • Summary: In the following table we provide the R values as a function of ITM, ATM or OTM portfolio, the degree of freedom ν and the value α given by the confidence level. ν ITM,ν R0.01
3 5 8 0.00329 0.00213 0.00171
ATM,ν R0.01 0.0794
0.0493
0.0383
OTM,ν R0.01
0.461
0.317
0.664
We note that the R’s values increase as the degree of freedom ν increases. 6. Conclusions In this paper we have considered the problem of the approximate computation of Value-at-Risk. We have shown how the price of a portfolio can be approximated using a quadratic Taylor approximation. Given a specified confidence level α, and assuming an elliptic distribution for the joint log-returns, we have shown how Value-at-Risk computation problems can be converted to multivariate integration problems, where the integration region is an hyperboloid. We have described several methods for approximating the resulting integrals, focusing on details for applications where the distribution is multivariate Normal or Student-t. We illustrated the use of these methods with several examples based on real data taken from the French CAC 40 Market. We have shown that appropriately chosen numerical methods can efficiently provide accurate results for these problems. We expect that the type of results in this paper can be generalized for mixture of elliptic distributions risk factors (see Sadefo Kamdem (2004b, 2003)). An important result of this paper is the approximation of the distribution function of a nonlinear (e.g. quadratic) of elliptic distribution random vectors. Our methods were derived assuming a quadratic Taylor approximation for the portfolio price, but our methods could also be used for other quadratic approximations (those developed by Studer (1999), for example. See Appendix E). Note that Studer (1999) and Mina (2001) describe procedures by which the quadratic approximation is estimated by least-squares methods. These methods produce fairly accurate and fast delta–gamma approximations to “true” VaR. Our method is applicable to portfolios of bonds and also to a portfolio of mortgage backed securities. Even though we have used the RiskMetrics EWMA for our computation, an improvement is possible with our method by using DCC (Engle, 2002) or a regime switching volatility approach (see Pelletier (2006) and Carvalho and Lopes (2007)). An application to assess the quality of an approximate distribution by using the Kullback–Leibler information measure is possible as in Lu (2006). Appendix A. Algorithm of Sheil and O’Muircheartaigh Sheil and O’Muircheartaigh (1977) describe an algorithm for computing the distribution function of the quadratic form (z + a)t C(z + a), where the n-dimensional random vector z has a multivariate normal distribution with the expected value vector µ, non-singular covariance matrix V and a fixed value a. In fact, Sheil and O’Muircheartaigh (1977) expresses the distribution function of the quadratic form (z + a)t C(z + a) as an infinite series in central χ 2 distribution functions: both the distribution functions and the series coefficients are evaluated recursively. A brief description of the algorithm follows. Using the linear transformations z − µ = Ct Rx, and a + µ = Ct Rb, Ruben (1962) has shown that P{(z + a)t C(z + a) ≤ t} = P{(x + b)t A(x + b) ≤ t}, where the components of the vector x are uncorrelated standard normal variables, L is the upper triangular matrix defined by V = Lt L, R is the matrix whose columns are the eigenvectors of LCLt , and A = diag(αi ) is the diagonal matrix whose elements αi are the eigenvalues of LCLt .
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If f (m, y) is the density function of a central χ 2 with m degrees of freedom and F(m, y) is the corresponding distribution function, then combining the results from Ruben (1962) and Kotz et al. (1967) ∞ X c F(n 0 + 2k, t/β), t > 1 k P{(z + a)t C(z + a) ≤ t} = k=1 (A.1) 0, t ≤ 0, where β is a constant and n 0 is the rank of C. If 0 < P β < 2 mini (αi ), the above series is uniformly absolutely convergent for all t > 0. If β < αmin then ck ≥ 0, and ∞ j=0 c j = 1, and the series is truncated after N terms, then ∞ X
ck F(n 0 + 2k, t/β) ≤ F(n 0 + 2N + 2, t/β)
k=N +1
∞ X
ck
k=N +1 0
= F(n + 2N + 2, y/β) 1 −
N X
! ck .
k=1
Pk−1 P 0 Q 0 p The ck ’s are computed using c0 = Ae−λ/2 and ck = k −1 l=0 gk−l cr , with A = nj=1 β/α j , λ = nj=1 b2j , gm = Pn 0 m Pn 0 2 m−1 + 1 2 m j=1 b j γ j j=1 (1 − mb j )γ j , and γ j = 1 − β/α j . 2 2 Appendix B. Algorithms of Genz and Monahan An integral of a function f (x) over Rn , with weight function ω(x x t ), is first transformed to a spherical-radial coordinate system x = r ξ , with kξ k = 1. Z ∞ I( f ) = f (x)ω(x)dx −∞ ∞
Z =
r n−1 ω(r 2 )
Z
0
f (r ξ )dξ dr.
(B.1)
kξ k=1
The integrals in this form can then be approximated using a product of radial and spherical integration rules to provide approximations in the form I ( f ) ≈ S R Q,ρ ( f ) =
m X i=1
wi
p X
w˜ j f (ρi z j Q),
(B.2)
j=1
where the w˜ j ’s and z j ’s are weights and points for a spherical-surface integration rule and Q is a random (Haar distributed) orthogonal matrix. The ρ parameters and the wi ’s are radial integration rule parameters, randomly chosen with distribution dependent on ω and m. Averages of the S R Q,ρ ( f ) rules provide estimates for I ( f ), and the associated standard errors for these averages provide robust error estimates for the approximations. Genz and Monahan (1998) provide details for the selection of the randomized ρ and the wi parameters when ω is a Normal or Student-t density function. Appendix C. Algorithm of Genz for spherical-surface integrals In a paper by Genz (2003), the following method is derived. Suppose that we need to estimate the following integral Z J( f ) = f (z)dσ (z), Sn−1
where dσ (z) is an element of surface on Sn−1 = {z|z ∈ Rn , kzk = 1}. Let be the n −P 1 simplex by Tn−1 = {x|x ∈ Rn−1 , 0 ≤ x1 + x2 + · · · + xn−1 ≤ 1} and for any x ∈ Tn−1 , n−1 define xn = 1 − i=1 xi . Also t p = (t p1 , . . . , t pn−1 ) if points t0 , t1 , . . . , tm are given, satisfying the condition :
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Pn Pn |t p | = i=1 t pi = 1 whenever i=1 pi = m, for nonnegative integers p1 , . . . , pn , then the Lagrange interpolation formula (Sylvester, 1970) for a function g(x) on Tn−1 is given by L (m,n−1) (g, x) =
n pY i −1 X Y xi − t j g(t p ), t − tj | p|=m i=1 j=0 pi
m L (m,n−1) (g, x) is the unique polynomial of degree m which interpolates g(x) at all of the Cm+n−1 points in the set {x|x = (t p1 , . . . , t pn−1 ), | p| = m}. Silvester provided families of points, satisfying the condition |t p | = 1 when i+µ | p| = m, in the form ti = m+θn for i = 0, 1, . . . , m, and µ real. If 0 ≤ θ ≤ 1, all interpolation points for (m,n−1) L (g, x) are in Tn−1 . Sylvester derived families of Genz (2003) rules for integration over Tn−1 by integrating L (m,n−1) (g, x). Fully symmetric interpolatory integration rules can be obtained by substituting xi = z i2 , and ti = u i2 in L (m,n−1) (g, x), and defining
M (m,n) ( f, z) =
n pY i −1 z 2 − u 2 X Y i j | p|=m i=1 j=0
u 2pi − u 2j
f {u p },
where f {u} is a symmetric sum defined by X f {u} = 2−c(u) f (s1 u 1 , s2 u 2 , . . . , sn u n ) s
with c(u) the number of nonzero entries in (u 1 , . . . , u n ), and when si = ±1 for those i with u i different to zero.
P
s
is taken over all of the signs combinations that occur
Lemma C.1. If wp = J
n pY i −1 z 2 − u 2 Y i j i=1 j=0
u 2pi − u 2j
! ,
then J ( f ) ≈ Rm ( f ) =
X
w p f {u p },
| p|=m
where f {u} = 2−c(u)
X
f (s1 u 1 , s2 u 2 , . . . , sn u n ),
s
P with c(u) the number of nonzero entries in (u 1 , . . . , u n ), and s is taken over all of the signs combinations that occur when si = ±1 for those i with u i different to zero. The approximation is exact whenever f (z) is a polynomial of degree 2m + 1. The proof is given by Genz (2003). Genz (2003) also describes how the integration rules Rm ( f ) can be randomized. If an orthogonal matrix Q is P randomly chosen with Haar distribution, then rules of the form Rm,Q ( f ) = | p|=m w p f Q {u p }, where f Q {u} = P −c(u) 2 s f (s · u Q), are also degree 2m + 1 approximations to J ( f ). Averages of the Rm,Q ( f ) rules are estimates for J ( f ), and the standard errors for these averages provide robust error estimates for the averages. Appendix D. Modified Genz and Bretz method A modification of the Genz and Bretz (2002) method for problems in this article begins with Eq. (14), rewritten in the form 2 2 12 ν Z Z ∞ ∞ 1 2 ky k2 1− 2 1 sv s R sv − + − +2 2 ν−1 − s2 2 dsdy+ , s e L D− , √ , G(R) = e + k(y+ + √ )D+2 k n+ 0( ν2 ) 0 ν ν ν (2π ) 2 −∞ (D.1)
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where L(D, v, u) =
1 (2π)n/2
R
in the form L(D, v, u) =
1
k(y+v)D 2 k2 0 for all sufficiently small t, t ≤ t0 . Then for any ε > 0, lim sup t→0
t VaRΠ p+ε
VaR0pt
≤ 1 ≤ lim inf t→0
t VaRΠ p−ε
VaR0pt
.
(E.2)
Proof. The proof is given by Brummelhuis and Sadefo Kamdem (2004) and Sadefo Kamdem (2004b).
Theorem E.2. With the same hypotheses as in Theorem E.1, suppose moreover that FΠt is strictly increasing and continuous, and that F0t is continuously differentiable and strictly increasing also. Then VaRΠt √ p − 1 (E.3) ≤ C t. VaR0pt Proof. The proof of this theorem will be based on the following elementary lemma: Lemma E.3. Let X and Y be two real-valued random variables, with cumulative distribution functions FX and FY respectively, and let F Y := 1 − FY . Then, for any λ with 0 < λ < 1, we have: FX (x/λ) − F (−(1 − λ)x/λ) ≤ FX +Y (x) ≤ FX (λx) + FY ((1 − λ)x) . For more details of the proof, see Brummelhuis and Sadefo Kamdem (2004) and Sadefo Kamdem (2004b).
(E.4)
Appendix F. Parameter estimation using MLE and EWMA An important problem in a wide range of financial risk management is the modeling of the variance-covariance matrix. In this paper, it is assumed that at any period t, the return vector Xt (m × 1) follows a multivariate elliptic distribution with mean µ, covariance Σ , and the probability density function of Xt is Cν,n f X (xt ) = g (xt − µ)Σ −1 (xt − µ)> , | det(Σ )| where Cν,n is a constant normalization. If we assume this model for m time periods, t = T, T − 1, . . . , T − m + 1, we assume statistical independence between time periods, we can write the joint density function f X given the mean, and Σ (computed using the Riskmetrics EWMA (Morgan and Reuters, 1996)), as follows: f X (x T , . . . , x T −m+1 |µ, g) =
T Y t=T −m+1
Cν,n −1 > g (xt − µ)Σ (xt − µ) . | det(Σ )|
(F.1)
The function f X (x T , . . . , x T −m+1 |g) is the probability density for the data given a set of parameter values θ = (µ, g). Since Σ as given by the EWMA, we need to estimate θ given the data matrix. The likelihood function of θ given the data, L(µ, g|Σ , XT , . . . , XT −m+1 ) is the same as f X (x T , . . . , x T −m+1 |µ, g), except that it considers the parameters as random variables and takes the data as given. For a realized sample of stocks, we need to find the parameter values that most likely generated the observed data matrix. The solution lies in maximum likelihood estimation (MLE), so we determine θ M L E parameters that maximize the likelihood.
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F.1. Example of Student-t distribution For a given Σ = ν−2 V, computed using the RiskMetrics EWMA, we also need the parameter ν, of the ν ST D(µ, Σ , ν) (Student-t) distribution, from the historical data for the given portfolio. We estimate the parameter ν = ν0 which maximizes the likelihood. Suppose that we have the risk factors of the m+1 days (t = T, . . . , T −m+1), where T denotes today’s date. Using n = m + 1 and X1 = (X 11 , . . . , X 1q ), X2 = (X 21 , . . . , X 2q ), . . . , Xn = (X n,1 , . . . , X nq ), where X i j is the jth risk factor at day i and q represent the number of risk factors. We determine ν > 0 that maximizes the function L V (ν) =
n Y
G ν,V (Xi ),
(F.2)
i=1
where Xi V−1 Xi> Cν,n G ν,V (Xi ) = √ 1+ ν−2 det(V)
!− n+ν 2
,
(F.3)
and Cν,n =
0
n+ν 2
. √ 0(ν/2) (νπ )n
(F.4)
It suffices to find the value of ν that maximize the function ! n Y log(L V (ν)) = log G ν,V (Xi ) i=1
=
n X
log(G ν,V (Xi ))
i=1
=
n X i=1
Since
"
1 n+ν log(Cν,n ) − log(det(|V|)) − 2 2
Xi V−1 Xi> 1+ ν−2
!# .
(F.5)
log(det(V)) does not depend on ν, we find the ν value that maximize !# " m X Xi V−1 Xi> n+ν 1 1+ . Φ(ν) = log(Cν,n ) − log(det(|V|)) − 2 2 ν−2 i=1 1 2
In our application cases m = 199 and q = 45, so the matrix Vλ is defined with the EWMA parameter λ = 0.96. References Albanese, C., Seco, L., 2001. Harmonic analysis in value-at-risk calculations. Revista Matem´atica Iberoamericano 17, 195–219. Brummelhuis, R., Cordoba, A., Quintanilla, M., Seco, L., 2002. Principal component value-at-risk. Mathematical Finance 12, 23–43. Brummelhuis, R., Sadefo Kamdem, J., 2004. VaR for quadratic portfolio’s with generalized Laplace distributed returns. University of Reims working paper. Cardenas, J., Fruchard, E., Koehler, E., Michel, C., Thomazeau, I., 1997. VAR: One step beyond. Risk 10, 72–75. Carvalho, M.C., Lopes, H.F., 2007. Simulation-based sequential analysis of Markov switching stochastic volatility models. Computational Statistics and Data Analysis 51, 4256–4542. Cornish, E.A., 1954. The multivariate t-distribution associated with a set of normal sample deviates. Australian Journal of Physics 7, 531–542. Dowd, K., 1998. Beyond Value-at-Risk. Wiley, New York. Duffie, D., Pan, J., 2001. Analytical value-at-risk with jumps and credit risk. Finance and Stochastics 5, 155–180. Embrechts, P., McNeil, A., Straumann, D., 2002. Correlation and dependence in risk management: Properties and pitfalls. In: Dempster, M.A.H. (Ed.), Risk Management: Value at Risk and Beyond. Cambridge University Press, Cambridge, pp. 176–223. Engle, R., 2002. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business and Economic Statistics 20, 339–350. Genz, A., 1992. Numerical computation of multivariate normal probabilities. Journal of Computational and Graphical Statistics 1, 141–150.
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