The Scientific World Journal

Research Article

Finite Difference Methods for Option Pricing under Lévy Processes: Wiener-Hopf Factorization Approach

Table 2

Convergence of the down-and-out put prices in KoBoL model, : FDS & WH versus CV.
(a)  
Parameters CV FDS & WH
Option price CPU time, sec. Option price ϵ CPU time, sec.
0.001 95 0.372 −28.9% 1 0.551 5.3% 1
0.0005 218 0.402 −23.3% 2 0.543 3.6% 3
0.00025 501 0.431 −17.7% 10 0.536 2.3% 15
0.0001 1501 0.464 −11.4% 188 0.529 1.0% 98
0.00005 3440 0.481 −8.2% 1045 0.526 0.4% 482
0.000025 7882 0.492 −6.1% 5520 0.524   2697
(b)  
Parameters CV FDS & WH
Option price CPU time, sec. Option price CPU time, sec.
0.001 95 2.314 −5.7% 1 2.565 4.5% 1
0.0005 218 2.344 −4.5% 2 2.522 2.7% 3
0.00025 501 2.369 −3.5% 10 2.494 1.6% 15
0.0001 1501 2.394 −2.5% 188 2.471 0.7% 98
0.00005 3440 2.407 −1.9% 1045 2.461 0.3% 482
0.000025 7882 2.418 −1.5% 5520 2.455   2697
(c)  
Parameters CV FDS & WH
Option price CPU time, sec. Option price CPU time, sec.
0.001 95 2.304 −3.4% 1 2.428 1.8% 1
0.0005 218 2.320 −2.7% 2 2.414 1.2% 3
0.00025 501 2.335 −2.1% 10 2.403 0.7% 15
0.0001 1501 2.349 −1.5% 188 2.393 0.3% 98
0.00005 3440 2.357 −1.2% 1045 2.389 0.1% 482
0.000025 7882 2.364 −0.9% 5520 2.386   2697
(d)  
Parameters CV FDS & WH
Option price CPU time, sec. Option price CPU time, sec.
0.001 95 1.547 −2.2% 1 1.567 −0.9% 1
0.0005 218 1.554 −1.8% 2 1.575 −0.4% 3
0.00025 501 1.560 −1.4% 10 1.579 −0.2% 15
0.0001 1501 1.566 −1.0% 188 1.581 0.0% 98
0.00005 3440 1.570 −0.7% 1045 1.581 0.0% 482
0.000025 7882 1.573 −0.5% 5520 1.581   2697
(e)  
Parameters CV FDS & WH
Option price CPU time, sec. Option price CPU time, sec.
0.001 95 0.862 −1.5% 1 0.849 −3.0% 1
0.0005 218 0.865 −1.2% 2 0.861 −1.7% 3
0.00025 501 0.867 −0.9% 10 0.868 −0.9% 15
0.0001 1501 0.870 −0.6% 188 0.872 −0.3% 98
0.00005 3440 0.872 −0.4% 1045 0.874 −0.1% 482
0.000025 7882 0.873 −0.3% 5520 0.875   2697
KoBoL parameters: , , , , .
, , , ; : number of time steps; : the relative difference between the current option price and the price computed by FDS & WH method for space step .
(a): ; (b): ; (c): ; (d): ; (e): .