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Without loss of generality we assume that σ is strictly positive. 28) ~ To simplify notation we introduce f(t) = ln F(t) and f ( t ) = f ( t ) + ωt . 29) is well-defined for u ∈ Λx, its strip of regularity. As the VG model is fully time-homogeneous, the maximum and minimum allowed moments do not depend on the maturity T, in contrast with the situation in the Heston model. 29) we are using the complex power function, again a multivalued function. The complex discontinuities in the Heston model were in fact also caused by the branch switching of the complex power function.

26) since we use the convention that the real part of the square root is nonnegative and τ ≥ 0. If the inequality is strict, we can immediately conclude that whenever the imaginary parts of c(u ) or c(u )e D ( u ) τ are zero their real parts are not in the interval [0,1]. When the inequality is an equality, we need only worry about the cases where G = -1 or G = 1. 1. 9 Note that the principal argument of zero is undefined. 14). 4. 4. Related issues in other models Having analysed the Heston characteristic function in great detail, it is time to turn to other models.

Assume that z never passes through the origin. Adding y ∈  where y ≠ 0 to z(x) does not add any discontinuities to the principal argument of z(x) + y when compared to the principal argument of z(x), if and only if: • • Re(z(x)) ∉ (-y,0) for y > 0 whenever Im(z(x)) changes sign; Re(z(x)) ∉ (0,-y) for y < 0 whenever Im(z(x)) changes sign. 42) Let a trajectory from quadrant i to quadrant j, without crossing any quadrants inbetween, be denoted as a tuple (i,j). The direction in which the trajectory is traversed does not matter here.

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