An analytical approximation of option prices via TGARCH model

Figures & data

Table 1. The average of standardized mean and variance of the cumulative asset return from the simulation results.

			Time to maturity (days)
			10	30	90	150
h1	μT	Analytical	3.2998 × 10^–5	4.7529 × 10^–5	5.2318 × 10^–5	5.3275 × 10^–5
		Monte Carlo	3.4322 × 10^–5	3.7050 × 10^–5	3.8005 × 10^–5	3.8161 × 10^–5
	${\sigma }_T^2$	Analytical	4.1145 × 10^–5	1.3715 × 10^–5	4.7168 × 10^–6	3.0016 × 10^–6
		Monte Carlo	2.4299 × 10^–5	2.6147 × 10^–5	2.6743 × 10^–5	2.4428 × 10^–5
$0.8h_1$	μT	Analytical	3.3186 × 10^–5	4.7584 × 10^–5	5.2336 × 10^–5	5.3286 × 10^–5
		Monte Carlo	3.4382 × 10^–5	3.7072 × 10^–5	3.7904 × 10^–5	3.8153 × 10^–5
	${\sigma }_T^2$	Analytical	6.5576 × 10^–5	1.3607 × 10^–5	4.6815 × 10^–6	2.6819 × 10^–6
		Monte Carlo	2.4301 × 10^–5	2.6142 × 10^–5	2.6732 × 10^–5	2.4421 × 10^–5
$1.2h_1$	μT	Analytical	3.2835 × 10^–5	4.7475 × 10^–5	5.2299 × 10^–5	5.3264 × 10^–5
		Monte Carlo	3.4323 × 10^–5	3.7219 × 10^–5	3.8023 × 10^–5	3.8179 × 10^–5
	${\sigma }_T^2$	Analytical	4.1434 × 10^–5	1.3823 × 10^–5	4.7281 × 10^–6	2.7659 × 10^–6
		Monte Carlo	2.4326 × 10^–5	2.6137 × 10^–5	2.6744 × 10^–5	2.4415 × 10^–5

Source: Created by the authors.

Table 2. The average of standardized skewness and kurtosis of the cumulative asset return from the simulation results.

			Time to maturity (days)
			10	30	90	150
h1	κ3,T	Analytical	–7.1456 × 10^–1	–2.4385 × 10^–1	–4.9193 × 10^–1	–5.4059 × 10^–5
		Monte Carlo	5.1928 × 10^–5	–1.5990 × 10^–5	8.6571$\times {10}^{-7}$	9.5133$\times {10}^{-9}$
	κ4,T	Analytical	1.8721 × 10^2	2.0573	1.9667 × 10^2	3.1578 × 10^2
		Monte Carlo	3.0017 × 10^–1	3.2986 × 10^–3	3.3332$\times {10}^{-2}$	2.0006$\times {10}^{-2}$
0.8h1	κ3,T	Analytical	–18.1490	–1.8091 × 10^–1	–5.1764 × 10^–1	–5.6884 × 10^–3
		Monte Carlo	5.2657 × 10^–5	–5.3836 × 10^–5	7.3248 × 10^–6	8.0493$\times {10}^{-8}$
	κ4,T	Analytical	3.7421 × 10^2	4.1122	1.9910 × 10^2	3.2032 × 10^2
		Monte Carlo	2.9995 × 10^–1	3.2961 × 10^–3	3.3321$\times {10}^{-2}$	1.9992$\times {10}^{-2}$
$1.2h_1$	κ3,T	Analytical	–3.5495 × 10^–1	–2.6187 × 10^–1	–1.6556 × 10^–1	–1.8193 × 10^–3
		Monte Carlo	2.9480 × 10^–5	–3.9989 × 10^–6	–2.0195 × 10^–6	–2.2192$\times {10}^{-8}$
	κ4,T	Analytical	1.8494 × 10^2	2.0323	1.9570 × 10^2	3.1551 × 10^2
		Monte Carlo	3.0005 × 10^–1	3.2972 × 10^–3	3.3319$\times {10}^{-2}$	2.0001$\times {10}^{-2}$

Source: Created by the authors.

Figure 1. Average of simulation results of an analytical approximation for call option prices. Source: Created by the authors.

Figure 2. Average of simulation results of an analytical approximation for call option prices: ‘ITM Cases’. Source: Created by the authors.

Figure 3. Impact of skewness and kurtosis correspondence on call option prices. Source: Created by the authors.

Figure 4. CORR, MAPE, RMSE and MaxAPE of call option prices. Source: Created by the authors based on SET50 data.

Table 3. The performance of option prices with $K/S_t\le 0.95.$

T–t	St	K	$K/S_t$	Observed	GARCH model (1995)	TGARCH model (2016)	GJR-GARCH model (1993)	Analytical approximation via GARCH (1999)	Monte Carlo simulation	Proposed method
30	1002.83	925	0.9224	77.1	80.9976	80.6944	80.6840	80.0163	81.2703	80.0169
		950	0.9473	53.6	57.3141	57.4602	57.7509	55.0754	59.1362	55.0768
	814.10	725	0.8906	87.0	91.6920	93.2941	91.3098	90.8136	90.5869	90.8142
		775	0.9520	44.8	50.4312	49.6412	46.7440	40.9319	44.7024	40.9375
	1025.18	925	0.9511	47.6	55.4279	52.6966	55.0622	59.1919	55.1542	52.4853
60	1002.57	950	0.9476	57.4	64.9077	64.4133	65.1244	73.2825	66.6853	57.0486
	1079.50	1000	0.9264	76.9	85.9039	84.1349	86.0368	90.3446	85.8393	84.2234
		1025	0.9495	56.3	63.9742	59.2604	63.8736	68.7463	63.8847	59.3415
90	1057.56	950	0.8983	111.1	120.4084	118.6975	120.2725	128.9308	119.4781	114.2647
	1085.81	1025	0.9440	64.5	75.4322	67.9778	75.3993	82.9388	75.4068	68.0643
				CORR	0.9935	0.9977	0.9915	0.9538	0.9886	0.9922
				MAPE	11.0139%	7.9727%	10.1307%	15.5802%	10.1672%	5.3720%
				RMSE	7.3924	5.4494	7.1523	11.9181	7.2855	3.8697
				MaxAPE	16.9492%	12.2183%	16.8981%	28.5873%	16.9097%	10.2632%

Source: Created by the authors based on SET50 data.

Figure 5. Impact of $K/S_t$ correspondence on call option prices. Source: Created by the authors based on SET50 data.