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

Table 1

Convergence of the down-and-out put prices in KoBoL model, 𝜈<1: FDS & WH versus CV.

(a)  

Parameters		CV			FDS & WH
Δ𝑥	𝑁𝜏	Option price	𝜖	CPU time, sec.	Option price	𝜖	CPU time, sec.
0.001	93	0.0577	−59.1%	1	0.1380	−2.2%	1
0.0005	152	0.0716	−49.2%	3	0.1400	−0.8%	2
0.00025	253	0.0873	−38.1%	19	0.1408	−0.2%	8
0.0001	520	0.1073	−24.0%	78	0.1411	0.0%	33
0.00005	926	0.1197	−15.2%	324	0.1411	0.0%	126
0.000025	1688	0.1281	−9.2%	1348	0.1411		568
0.00001	4000	0.1330	−5.7%	14655

(b)  

Parameters		CV			FDS & WH
Δ𝑥	𝑁𝜏	Option price	𝜖	CPU time, sec.	Option price	𝜖	CPU time, sec.
0.001	93	0.2344	−19.8%	1	0.2899	−0.8%	1
0.0005	152	0.2464	−15.7%	3	0.2915	−0.2%	2
0.00025	253	0.2571	−12.0%	19	0.2920	0.0%	8
0.0001	520	0.2679	−8.3%	78	0.2922	0.0%	33
0.00005	926	0.2740	−6.2%	324	0.2922	0.0%	126
0.000025	1688	0.2787	−4.6%	1348	0.2922		568
0.00001	4000	0.2832	−3.1%	14655

(c)  

Parameters		CV			FDS & WH
Δ𝑥	𝑁𝜏	Option price	𝜖	CPU time, sec.	Option price	𝜖	CPU time, sec.
0.001	93	0.2182	−16.7%	1	0.2583	−1.4%	1
0.0005	152	0.2273	−13.3%	3	0.2604	−0.6%	2
0.00025	253	0.2353	−10.2%	19	0.2614	−0.3%	8
0.0001	520	0.2434	−7.1%	78	0.2618	−0.1%	33
0.00005	926	0.2481	−5.3%	324	0.2620	0.0%	126
0.000025	1688	0.2517	−4.0%	1348	0.2621		568
0.00001	4000	0.2552	−2.6%	14655

(d)  

Parameters		CV			FDS & WH
Δ𝑥	𝑁𝜏	Option price	𝜖	CPU time, sec.	Option price	𝜖	CPU time, sec.
0.001	93	0.1718	−15.4%	1	0.1995	−1.7%	1
0.0005	152	0.1781	−12.2%	3	0.2012	−0.9%	2
0.00025	253	0.1838	−9.4%	19	0.2022	−0.4%	8
0.0001	520	0.1896	−6.6%	78	0.2027	−0.2%	33
0.00005	926	0.1930	−4.9%	324	0.2029	−0.1%	126
0.000025	1688	0.1956	−3.6%	1348	0.2030		568
0.00001	4000	0.1981	−2.4%	14655

KoBoL parameters: 𝜎=0, 𝜈=0.5, 𝜆₊=4, 𝜆₋=−6, 𝑐=1.
𝐾=100, 𝐻=90, 𝑟=0.04879, 𝑇=0.5, 𝜖: the relative difference between the current option price and the price computed by FDS & WH method for space step Δ𝑥=0.000025.
(a): 𝑆=91; (b): 𝑆=101; (c): 𝑆=111; (d): 𝑆=121.