A methodology for temperature option pricing in the equatorial regions

Figures & data

Figure 1. General methodology for temperature option pricing.

Figure 2. Monthly mean temperature at Bogotá, Colombia (1961–2018). Data source: American National Oceanic and Atmospheric Administration (NOAA, 2018).

Figure 3. Yearly mean temperature at Bogotá, Colombia (1961–2018). Data source: American National Oceanic and Atmospheric Administration (NOAA, 2018).

Table 1. Regression results for the temperature models with independent errors ($T_t=f(t)+{\epsilon }_t,$ where ${\epsilon }_t$ are i.i.d. normal random variables with mean zero and variance ${\sigma }_{\epsilon }^2$).

Parameters	Model 1	Model 2	Model 3	Model 4	Model 5
α	53.903***	55.093***	55.099***	55.101***	55.100***
	(0.0684)	(0.0378)	(0.0367)	(0.0367)	(0.0366)
ω	0.045***	0.045***	0.045***	0.045***	0.045***
	(0.0012)	(0.0012)	(0.0012)	(0.0012)	(0.0012)
β2	1.097***
	(0.0877)
β3	1.847***
	(0.0861)
β4	2.519***
	(0.0868)
β5	2.267***
	(0.0861)
β6	1.334***
	(0.0868)
β7	0.641***
	(0.0861)
β8	0.530***
	(0.0861)
β9	0.585***
	(0.0868)
β10	1.315***
	(0.0861)
β11	1.604***
	(0.0868)
β12	0.653***
	(0.0861)
γ1		0.545***	0.547***	0.547***	0.547***
		(0.0260)	(0.0252)	(0.0251)	(0.0251)
δ1		−0.224***	−0.220***	−0.220***	−0.220***
		(0.0260)	(0.0251)	(0.0251)	(0.0250)
γ2			−0.633***	−0.633***	−0.633***
			(0.0252)	(0.0251)	(0.0251)
δ2			−0.561***	−0.560***	−0.560***
			(0.0252)	(0.0251)	(0.0251)
γ3				−0.158***	−0.158***
				(0.0251)	(0.0251)
δ3				−0.162***	−0.161***
				(0.0251)	(0.0251)
γ4					−0.018***
					(0.0251)
δ4					−0.211***
					(0.0251)
Observations	17955	17955	17955	17955	17955
AIC	82178.02	83272.24	82180.28	82103.18	82035.55
BIC	82287.16	83311.22	82234.85	82173.34	82121.30

Standard errors are in parentheses. ***p < 0.01, **p < 0.05, *p < 0.10.

Table 2. Regression results for the temperature models with mean reversion.

Parameters	Model 1	Model 2	Model 3	Model 4	Model 5
$\varphi$	0.310***	0.353***	0.312***	0.309***	0.306***
	(0.0071)	(0.0070)	(0.0071)	(0.0070)	(0.0071)
α	53.937***	55.092***	55.100***	55.100***	55.100***
	(0.0929)	(0.0547)	(0.0507)	(0.0504)	(0.0502)
ω	0.045***	0.045***	0.045***	0.045***	0.045***
	(0.0017)	(0.0018)	(0.0017)	(0.0017)	(0.0016)
β2	1.083***
	(0.1171)
β3	1.794***
	(0.1163)
β4	2.459***
	(0.1173)
β5	2.211***
	(0.1163)
β6	1.313***
	(0.1173)
β7	0.633***
	(0.1163)
β8	0.490***
	(0.1163)
β9	0.547***
	(0.1173)
β10	1.266***
	(0.1163)
β11	1.551***
	(0.1172)
β12	0.625***
	(0.1152)
γ1		0.547***	0.548***	0.548***	0.548***
		(0.0376)	(0.0348)	(0.0346)	(0.0344)
δ1		−0.219***	−0.215***	−0.215***	−0.215***
		(0.0375)	(0.0347)	(0.0345)	(0.0344)
γ2			−0.625***	−0.624***	−0.625***
			(0.0348)	(0.0346)	(0.0344)
δ2			−0.570***	−0.569***	−0.569***
			(0.0348)	(0.0346)	(0.0344)
γ3				−0.155***	−0.155***
				(0.0346)	(0.0344)
δ3				−0.165***	−0.164***
				(0.0346)	(0.0344)
γ4					−0.012**
					(0.0344)
δ4					−0.212***
					(0.0344)
Observations	17955	17955	17955	17955	17955
AIC	80360.68	80878.79	80333.91	80295.26	80261.29
BIC	80477.62	80925.56	80396.27	80373.22	80354.84

Standard errors are in parentheses. ***p < 0.01, **p < 0.05, *p < 0.10.

Table 3. Regression results for the temperature models with the mean reversion and jump process.

Parameters	Model 1	Model 2	Model 3	Model 4	Model 5
$\varphi$	0.311***	0.353***	0.313***	0.310***	0.307***
	(0.0070)	(0.0069)	(0.0070)	(0.0070)	(0.0070)
λ	0.0245***	0.0255***	0.0242***	0.0235***	0.0238***
	(0.0050)	(0.0053)	(0.0049)	(0.0048)	(0.0049)
μJ	−0.4499	−0.7625**	−0.5513	−0.3657	−0.4211
	(0.3553)	(0.3468)	(0.3529)	(0.3674)	(0.3604)
σJ	4.1525***	4.0366***	4.1670***	4.2059***	4.1657***
	(0.3381)	(0.3434)	(0.3403)	(0.3444)	(0.3409)
α	53.981***	55.150***	55.133***	55.135***	55.136***
	(0.0943)	(0.0572)	(0.0525)	(0.0521)	(0.0519)
ω	0.044***	0.044***	0.044***	0.044***	0.044***
	(0.0016)	(0.0018)	(0.0016)	(0.0016)	(0.0016)
β2	1.055***
	(0.1165)
β3	1.791***
	(0.1152)
β4	2.447***
	(0.1162)
β5	2.211***
	(0.1151)
β6	1.318***
	(0.1160)
β7	0.597***
	(0.1152)
β8	0.484***
	(0.1152)
β9	0.546***
	(0.1161)
β10	1.280***
	(0.1151)
β11	1.551***
	(0.1159)
β12	0.641***
	(0.1145)
γ1		0.541***	0.542***	0.541***	0.541***
		(0.0370)	(0.0343)	(0.0341)	(0.0340)
δ1		−0.203***	−0.206***	−0.207***	−0.208***
		(0.0372)	(0.0343)	(0.0341)	(0.0340)
γ2			−0.628***	−0.632***	−0.631***
			(0.0343)	(0.0341)	(0.0340)
δ2			−0.569***	−0.566***	−0.566***
			(0.0343)	(0.0341)	(0.0340)
γ3				−0.150***	−0.150***
				(0.0341)	(0.0340)
δ3				−0.163***	−0.163***
				(0.0341)	(0.0340)
γ4					−0.015
					(0.0340)
δ4					−0.203***
					(0.0340)
Observations	17955	17955	17955	17955	17955
AIC	80179.32	80716.05	80154.85	80116.28	80084.40
BIC	80311.85	80778.41	80232.80	80209.82	80193.53

Standard errors are in parentheses. ***p < 0.01, **p < 0.05, *p < 0.10.

Table 4. RMSE of the temperature simulation using five deterministic seasonality functions and two stochastic processes.

Stochastic processes (Xt)	Model 1	Model 2	Model 3	Model 4	Model 5
Independent	3.0456	3.1205	3.0466	3.0398	3.0341
Mean reversion	2.9491	2.9987	2.9512	2.9478	2.9461
Mean reversion and Jump process	3.0321	3.0830	3.0336	3.0296	3.0230

Table 5 HDD option pricing on temperature using Model 5 and mean reversion with a strike price (K) of 5 ${}^{°}$ F in Bogotá, Colombia.

		$\mathit{Call}(\mathit{USD}$$)			$\mathit{Put}(\mathit{USD}$$)
Month		θ = 0	$\theta =-0.1$	$\theta =+0.1$	θ =  0	$\theta =-0.1$	$\theta =+0.1$
1		272.47	432.77	144.65	0.02	0.01	1.36
2		100.91	202.84	38.49	4.70	0.63	15.34
3		33.48	97.04	7.85	18.79	4.96	42.03
4		13.07	47.56	2.79	32.94	13.84	55.09
5		16.72	62.14	4.16	30.34	10.63	51.62
6		84.11	189.90	29.91	7.21	0.80	23.54
7		197.09	351.03	84.47	0.96	0.00	7.43
8		206.39	384.14	96.72	0.85	0.04	6.47
9		194.58	342.31	83.32	1.30	0.08	8.04
10		105.35	211.58	34.05	5.92	0.87	23.16
11		58.95	146.81	15.84	15.03	2.65	37.65
12	211.26		381.26	94.18	0.72	0.05	8.01

Table 6. CDD option pricing on temperature using Model 5 and mean reversion with a strike price (K) of 5 ${}^{°}$ F in Bogotá, Colombia.

Month	$\mathit{Call}(\mathit{USD}$$)			$\mathit{Put}(\mathit{USD}$$)
	θ = 0	$\theta =-0.1$	$\theta =+0.1$	θ =  0	$\theta =-0.1$	$\theta =+0.1$
1	542.24	347.62	775.02	0.00	0.03	0.00
2	797.88	553.73	1048.16	0.00	0.00	0.00
3	1262.42	962.15	1614.10	0.00	0.00	0.00
4	1471.84	1137.83	1808.48	0.00	0.00	0.00
5	1496.92	1159.81	1860.86	0.00	0.00	0.00
6	1035.45	750.02	1347.61	0.00	0.00	0.00
7	793.97	541.57	1105.24	0.00	0.00	0.00
8	793.16	521.71	1088.14	0.00	0.04	0.00
9	801.25	545.01	1093.21	0.00	0.00	0.00
10	1135.08	845.59	1497.61	0.00	0.00	0.00
11	1287.61	963.61	1657.30	0.00	0.00	0.00
12	849.56	589.23	1191.87	0.00	0.00	0.00

Figure 4. HDD and CDD option pricing with strike price (K) of 5 ${}^{°}$ F for Bogotá, Colombia (θ = 0).

NOAA. (2018). National oceanic and atmospheric administration. Retrieved September 23, 2018, from https://www.ncdc.noaa.gov

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