A novel stochastic modeling framework for coal production and logistics through options pricing analysis

Financial Innovation

Table 4 Semi-analytical pricing compared with Monte Carlo

T	In-the-money			At-the-money			Out-of-the-money
	Analytics	Monte Carlo		Analytics	Monte Carlo		Analytics	Monte Carlo
	Semi-analytical	Price	Std errors	Semi-analytical	Price	Std errors	Semi-analytic	Price	Std errors
0.5	9.655	9.680	0.00005	9.682	9.690	0.00006	10.050	10.054	0.00007
1.0	13.887	13.923	0.00014	14.160	14.145	0.00014	15.827	15.827	0.00019
1.5	17.263	17.285	0.00022	17.731	17.689	0.00023	20.425	20.458	0.00031
2.0	20.124	20.129	0.00030	20.758	20.715	0.00032	24.342	24.393	0.00041
2.5	22.601	22.629	0.00037	23.381	23.410	0.00039	27.849	27.809	0.00051
3.0	24.770	24.788	0.00043	25.676	25.644	0.00045	30.822	30.805	0.00059
3.5	26.679	26.632	0.00047	27.698	27.694	0.00050	33.456	33.445	0.00065
4.0	28.365	28.367	0.00051	29.483	29.526	0.00054	35.831	35.781	0.00070
4.5	29.857	29.897	0.00055	31.064	31.059	0.00058	37.857	37.851	0.00075
5.0	31.177	31.155	0.00057	32.465	32.416	0.00060	39.661	39.687	0.00078
CPU	35	126		34	124		36	128

For semi-analytical method [refer to Eq. (30)], we chose \(\alpha _1=0.65, \alpha _2=1.4, \alpha _3=0.25\), CPU times in secs