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As we want to concentrate our attention on the modelling of the dependency structure, rather than on the modelling of the marginal distributions, we restrict our analysis to normal marginal densities. On the basis of our backtesting results, we find that the copula method produces more accurate results than “correlation dependence”. 1 Copulas In this section we summarize the basic results without proof that are necessary to understand the concept of copulas. Then, we present the most important properties of copulas that are needed for applications in finance.
The steps of the algorithm are following: 1. 57) 2. 58) where Σ(ω) = (I − ωΛ)−1 and µ(ω) = ωΣ(ω)δ. 58). The loss exceeding scenarios (L > l), which were previously rare under N(0, I), are typical under N(µ, Σ), since the expected value of the approximate value L is now l. According to Glasserman et al. (2000), the effectiveness of this importance sampling procedure is not very sensitive to the choice of ω. After we get N(µ(ω), Σ(ω)), we can follow the same steps in the basic partial Monte-Carlo simulation to calculate the VaR.
Is the normal cdf with mean µ and standard deviation 2 2 σ and eiµt−σ t /2 its characteristic function. t. ) Applying the Fourier inversion to F (x) − Φ(x; µ, σ) instead of F (x) solves the (numerical) problem that ti φ(t) has a pole at 0. Alternative distributions with known Fourier transform may be chosen if they better approximate the distribution F under consideration. 5): 1 1 µ= (θi + λi ) = θ 11 + tr(ΓΣ) 2 2 i and σ2 = i 1 1 (δi2 + λ2i ) = ∆ Σ∆ + tr((ΓΣ)2 ). 2 2 def 2 2 Let ψ(t) = ti (φ(t)−eiµt−σ t /2 ).