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28) holds. FFT-based Fourier inversion yields approximations for the cdf F on equidistant ∆x -spaced grids. Depending on the smoothness of F , linear or higher-order interpolations may be used. Any monotone interpolation of {F (x0 + ∆x j)}j yields a quantile approximation whose interpolation error can be bounded by ∆x . This bound can be improved if an upper bound on the density f in a suitable neighborhood of the true quantile is known. 2 Approximating Value at Risk in Conditional Gaussian Models Tail Behavior 2 2 If λj = 0 for some j, then |φ(t)| = O(e−δj t /2 ).

For every u ∈ [0, 1] C(u, 1) = u and C(1, u) = u . 2) 3. For every (u1 , u2 ), (v1 , v2 ) ∈ [0, 1] × [0, 1] with u1 ≤ v1 and u2 ≤ v2 : C(v1 , v2 ) − C(v1 , u2 ) − C(u1 , v2 ) + C(u1 , u2 ) ≥ 0 . 3) A function that fulfills property 1 is also said to be grounded. Property 3 is the two-dimensional analogue of a nondecreasing one-dimensional function. A function with this feature is therefore called 2-increasing. The usage of the name ”copula” for the function C is explained by the following theorem.

Npar is the same list, containing additionally B, delta, and lambda on output. The characteristic function of a non-central χ21 variate ((Z + a)2 , with standard normal Z) is known analytically: 2 Eeit(Z+a) = (1 − 2it)−1/2 exp a2 it 1 − 2it . 4) 2 or in terms of Γ and Σ 1 Eeit∆V = det(I − itΓΣ)−1/2 exp{− t2 ∆ Σ(I − itΓΣ)−1 ∆}. 3) can be used to compute an approximation to the cumulative distribution function (cdf) F of ∆V . ) The cost of the Fourier-inversion is O(N log N ), the cost of the function evaluations is O(mN ), and the cost of the eigenvalue decomposition is O(m3 ).

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