By F. J. W. Hahne

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IV is the empirically determined parameter that ‘makes the BS formula ﬁt market prices of options’. 30), there exists a unique solution σ > 0. g. by a Newton-Raphson algorithm as discussed in Manaster and Koehler (1982). 26), put and call IV are equal. In the derivation of the BS model it is presumed that the diﬀusion coeﬃcient of the Brownian motion is a constant. IV σ, however, displays a pronounced curvature across option strikes K and, albeit to a lesser extent, across diﬀerent expiry days T .

11 for the years 1997 and 2000. Similar results are reported by Tompkins (2001) for a variety of stochastic and jump models. Summing up, it appears that only a combination of jump and stochastic volatility models is suﬃciently capable of capturing the stylized facts of the IVS, Bakshi et al. (1997). g. Breidt et al. (1998). There is evidence that in particular the upwardsloping term structure of the IVS can strongly be inﬂuenced by long memory in volatility, Taylor (2000). g. Ederington and Guan (2002) for an analysis that is based on delta-hedging an option portfolio, processes with a marginal distribution tails heavier than the Gaussian ones are considered.

This point is further discussed in the context of the principal component analysis in Sect. 5. 6. IV appears to be mean-reverting, Cont and da Fonseca (2002). For ATM IV (three months) we ﬁnd a mean reversion of approximately 60 days, see also Sect. 4. 7. Shocks across the IVS are highly correlated. Thus, IVS dynamics can be decomposed into a small number of driving factors, Chap. 5. 25 Fig. 12. 2 DAX Index IV between 1995 and 2001 An overview on the three-month ATM IV time series between 1995 and May 2001 is given in Fig.