European Options in BS Markets - Computational Methods for Quantitative Finance - page 57

Fig. 4.2 L 2 (J ; L 2 (G))
convergence rate for θ = 1 / 2

using uniform time steps
( top ) and graded time steps
( bottom )

both with uniform time steps and with graded time steps. The exact values are ob-
tained using the analytic formulas [21]. We use the algebraic grading factor β = 3

for the call option and the extreme grading factor β 25 for the digital option.

It can be seen in Fig. 4.2 that for the call and the digital option we only obtain
the optimal convergence rate using a graded time mesh.

=

Remark 4.4.2 Note that the theory from [146, Chap. 3.3] suggests that choosing
β 10 is sufficient in the case of a digital option in the Black-Scholes market to

obtain full convergence order of the given discretization. But the analysis in [146,
Chap. 3.3] is performed in a semi-discrete setting, i.e. there is no discretization error
in the space domain. This explains why in our situation, discretizing in space and
time, a larger grading factor has to be chosen.

=