By Roger Lord
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Extra resources for Efficient pricing algorithms for exotic derivatives
The crucial step of the algorithm is the simulation of the integrated square root process conditional upon its start and endpoint. 39) with D( u ) = κ 2 − 2ω 2 iu , the degrees of freedom ν = 4κθ/ω2 and Iν representing the modified Bessel function of the first kind. Finally, τ equals t – s. 39) depends non-trivially on the two realisations v(s) and v(t), it is not an easy task to precompute major parts of the calculations. As a result this step of the algorithm will be highly timeconsuming. It is therefore not surprising that one of our findings in Chapter 6 is that several biased simulation schemes outperform the exact simulation scheme in terms of both speed and accuracy, even when the asset value is only required at one time instance.
If we write u = x + yi, one can check there are values of y where lim D( x + yi) = lim D( x + yi) . Nevertheless this is not a problem either, as for u ∈ Λx the x ↑0 x ↓0 characteristic function is real on the imaginary axis, so that this discontinuity in D(u) does not have an impact on the characteristic function. Under the premise of the rotation count algorithm the assertion is thus true. The remaining problem is now to check that the premise of the rotation count algorithm is true. 25) where x ∈ .
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. Trajectories of z that do not cause any discontinuities are (1,2), (1,4), (3,4).
Efficient pricing algorithms for exotic derivatives by Roger Lord