Partial Differential Equations - Modelling and ... - ResearchGate

288 S. Ikonen and J. Toivanen
1.6
1.4
1.2
Black-Scholes
Kou
Heston
1
v
0.8
0.6
0.4
0.2
0
8.5 9 9.5 10 10.5 11 11.5 12 12.5
Fig. 1. The price of the option with respect to the value of the underlying asset for
the three different models.
Table 2. Results for different grids with Black–Scholes model
l m error ratio time
10 20 0.01056 0.02
18 40 0.00208 5.1 0.06
34 80 0.00058 3.6 0.21
66 160 0.00022 2.7 0.79
130 320 0.00007 3.3 3.08
x
where we have chosen β =6andγ =1/2 which leads to some refinement near
the strike price K. For the temporal discretization, we choose the approximation
times to be
( a −k/(l−2) )
− 1
t k =
a −1 T, k =0, 1,...,l− 4, (23)
− 1
and
( a −(k+l−4)/(2l−4) )
− 1
t k =
a −1 T, k = l − 3,...,l. (24)
− 1
The parameter a in (23) and (24) has been chosen to be a = 2 which leads to
a mild refinement near the expiry.
Table 2 reports the l 2 errors computed using the reference prices in Table
1atx = 9, 10, and 11 for five different space-time grids. The ratio column
in the table gives the ratios between two successive l 2 errors. The time is the
CPU time in milliseconds needed to price the options.
7.2 Kou’s Jump-Diffusion Model
The parameters defining the jump probability and its distribution in Kou’s
model are chosen to be