Upside and downside correlated jump risk premia of currency options and expected returns

He, Jie-Cao; Chang, Hsing-Hua; Chen, Ting-Fu; Lin, Shih-Kuei

doi:10.1186/s40854-023-00493-3

Financial Innovation

Table 2 Parametric estimation results for EUR/USD

From: Upside and downside correlated jump risk premia of currency options and expected returns

	1-Month				3-Month				6-Month				12-Month
	CB	CB–ISJ	CB–CSJ	CB–CAJ	CB	CB–ISJ	CB–CSJ	CB–CAJ	CB	CB–ISJ	CB–CSJ	CB–CAJ	CB	CB–ISJ	CB–CSJ	CB–CAJ
Panel (a): Continuous Part
\({\mu }_{D}\)	− 0.000921	0.002124	− 0.000727	− 0.000775	− 0.000135	0.003679	0.001183	− 0.000083	0.000055	0.006230	0.000152	− 0.001298	0.005301	0.003183	− 0.000608	− 0.000815
	(0.0006)	(0.0008)	(0.0004)	(0.0003)	(0.0003)	(0.0013)	(0.0007)	(0.0004)	(0.0000)	(0.0014)	(0.0010)	(0.0003)	(0.0009)	(0.0010)	(0.0006)	(0.0004)
\({\mu }_{F}\)	− 0.001714	− 0.004993	− 0.004543	− 0.004944	− 0.003632	− 0.005032	− 0.006675	− 0.006107	− 0.006200	− 0.005742	− 0.006905	− 0.006749	− 0.006774	− 0.008479	− 0.011742	− 0.011111
	(0.0005)	(0.0014)	(0.0010)	(0.0009)	(0.0007)	(0.0013)	(0.0008)	(0.0008)	(0.0009)	(0.0013)	(0.0010)	(0.0009)	(0.0010)	(0.0013)	(0.0011)	(0.0011)
\({\mu }_{X}\)	0.009624	0.066988	0.043084	0.036623	0.011472	0.059233	0.035017	0.032122	0.008565	0.058256	0.030661	0.032705	0.017788	0.033536	0.007475	0.010433
	(0.0017)	(0.0055)	(0.0041)	(0.0032)	(0.0017)	(0.0067)	(0.0056)	(0.0053)	(0.0010)	(0.0082)	(0.0058)	(0.0052)	(0.0020)	(0.0074)	(0.0059)	(0.0055)
\({\sigma }_{D}\)	0.000217	0.037925	0.013870	0.006984	0.000383	0.051679	0.020141	0.008219	0.000458	0.060723	0.019860	0.001230	0.051970	0.048572	0.016423	0.005987
	(0.0001)	(0.0052)	(0.0032)	(0.0022)	(0.0002)	(0.0069)	(0.0043)	(0.0025)	(0.0002)	(0.0072)	(0.0044)	(0.0005)	(0.0088)	(0.0060)	(0.0034)	(0.0020)
\({\sigma }_{F}\)	0.000233	0.012128	0.009276	0.004120	0.000799	0.017671	0.001741	0.000439	0.001475	0.015408	0.003728	0.000174	0.026883	0.014897	0.002050	0.000528
	(0.0001)	(0.0040)	(0.0034)	(0.0024)	(0.0002)	(0.0043)	(0.0008)	(0.0004)	(0.0002)	(0.0042)	(0.0017)	(0.0000)	(0.0042)	(0.0036)	(0.0006)	(0.0002)
\({\sigma }_{X}\)	0.005820	0.214761	0.141936	0.118903	0.005567	0.224697	0.083906	0.061079	0.006091	0.172552	0.084800	0.046469	0.057822	0.154126	0.065259	0.033368
	(0.0001)	(0.0119)	(0.0099)	(0.0080)	(0.0002)	(0.0128)	(0.0078)	(0.0065)	(0.0002)	(0.0136)	(0.0088)	(0.0050)	(0.0087)	(0.0124)	(0.0068)	(0.0037)
\({\rho }_{DF}\)	0.042803	− 0.084923	0.176238	0.155151	0.045667	− 0.117856	0.082364	0.055962	0.030065	− 0.054998	0.064719	0.097790	0.023745	0.013141	0.059020	0.094724
	(0.0220)	(0.0437)	(0.0364)	(0.0362)	(0.0222)	(0.0403)	(0.0332)	(0.0327)	(0.0227)	(0.0408)	(0.0338)	(0.0317)	(0.0204)	(0.0400)	(0.0324)	(0.0312)
\({\rho }_{DX}\)	− 0.076668	0.234528	0.347905	0.369591	− 0.119865	0.272173	0.277339	0.245178	− 0.114798	0.247017	0.318158	0.331476	− 0.274496	0.265379	0.281603	0.221663
	(0.0175)	(0.0307)	(0.0249)	(0.0248)	(0.0198)	(0.0293)	(0.0283)	(0.0287)	(0.0202)	(0.0303)	(0.0289)	(0.0263)	(0.0261)	(0.0299)	(0.0298)	(0.0292)
\({\rho }_{FX}\)	0.046973	− 0.213778	− 0.403137	− 0.403660	− 0.054756	− 0.159177	− 0.328638	− 0.317723	0.007442	− 0.143966	− 0.374716	− 0.319880	0.073694	− 0.134918	− 0.253538	− 0.264014
	(0.0161)	(0.0293)	(0.0241)	(0.0232)	(0.0229)	(0.0307)	(0.0277)	(0.0275)	(0.0266)	(0.0322)	(0.0293)	(0.0286)	(0.0272)	(0.0308)	(0.0316)	(0.0292)
\({h}_{1,D}\)	0.036477	− 0.138793	− 0.080802	− 0.076683	0.036340	− 0.125473	− 0.042385	− 0.022603	0.020723	− 0.097229	− 0.056506	− 0.011766	0.077461	− 0.062419	− 0.042461	− 0.015872
	(0.0061)	(0.0113)	(0.0124)	(0.0124)	(0.0105)	(0.0104)	(0.0130)	(0.0136)	(0.0124)	(0.0116)	(0.0138)	(0.0143)	(0.0141)	(0.0122)	(0.0126)	(0.0127)
\({h}_{1,F}\)	0.018294	− 0.122076	− 0.022409	− 0.017339	0.001183	− 0.053533	− 0.021341	− 0.016073	− 0.035795	− 0.053031	− 0.002279	0.011101	− 0.052295	− 0.025456	0.033677	0.026347
	(0.0057)	(0.0121)	(0.0129)	(0.0126)	(0.0113)	(0.0132)	(0.0124)	(0.0129)	(0.0117)	(0.0140)	(0.0136)	(0.0140)	(0.0146)	(0.0139)	(0.0133)	(0.0132)
\({h}_{1,X}\)	0.025029	0.005214	0.011537	0.015560	0.037336	0.003009	0.004603	− 0.003558	0.041968	− 0.004925	− 0.007003	− 0.006280	0.071737	− 0.000818	− 0.023318	− 0.025890
	(0.0052)	(0.0114)	(0.0109)	(0.0109)	(0.0050)	(0.0122)	(0.0119)	(0.0120)	(0.0063)	(0.0126)	(0.0128)	(0.0129)	(0.0115)	(0.0126)	(0.0129)	(0.0128)
Panel (b): Jump Part
\({\theta }_{D,u}\left({\theta }_{D}\right)\)		− 0.100688	− 0.079268	− 0.041040		− 0.095572	− 0.055884	− 0.024121		− 0.076280	− 0.091150	− 0.036647		− 0.102691	− 0.086787	− 0.041951
		(0.0106)	(0.0088)	(0.0060)		(0.0111)	(0.0085)	(0.0049)		(0.0102)	(0.0084)	(0.0060)		(0.0120)	(0.0097)	(0.0097)
\({\theta }_{F,u}\left({\theta }_{F}\right)\)		− 0.107118	− 0.090019	− 0.048872		− 0.096301	− 0.121035	− 0.062113		− 0.092275	− 0.090156	− 0.052071		− 0.106841	− 0.133530	− 0.053116
		(0.0122)	(0.0093)	(0.0061)		(0.0108)	(0.0089)	(0.0088)		(0.0094)	(0.0066)	(0.0063)		(0.0113)	(0.0106)	(0.0091)
\({\theta }_{X,u}\left({\theta }_{X}\right)\)		0.010272	0.008520	0.057719		− 0.011493	− 0.007382	0.093479		0.007724	− 0.020850	0.096152		− 0.043176	− 0.113097	0.077248
		(0.0165)	(0.0125)	(0.0084)		(0.0221)	(0.0179)	(0.0113)		(0.0214)	(0.0208)	(0.0130)		(0.0228)	(0.0193)	(0.0127)
\({\theta }_{T,u}\left({\theta }_{T}\right)\)		− 0.127245	− 0.047148	− 0.021196		− 0.093518	− 0.027779	− 0.013091		− 0.073321	− 0.019149	− 0.009557		− 0.064898	− 0.008095	− 0.004291
		(0.0089)	(0.0052)	(0.0028)		(0.0066)	(0.0037)	(0.0026)		(0.0041)	(0.0023)	(0.0016)		(0.0045)	(0.0012)	(0.0010)
\({\theta }_{D,d}\)				− 0.037718				− 0.040424				− 0.036153				− 0.056477
				(0.0074)				(0.0065)				(0.0056)				(0.0093)
\({\theta }_{F,d}\)				− 0.031286				− 0.053169				− 0.036759				− 0.083891
				(0.0072)				(0.0092)				(0.0070)				(0.0109)
\({\theta }_{X,d}\)				− 0.065160				− 0.115084				− 0.113685				− 0.171123
				(0.0083)				(0.0137)				(0.0120)				(0.0154)
\({\theta }_{T,d}\)				− 0.026578				− 0.021690				− 0.016061				− 0.010816
				(0.0041)				(0.0031)				(0.0017)				(0.0014)
\({\nu }_{D,u}\left({\nu }_{D}\right)\)		0.335811	0.314102	0.204786		0.337551	0.291981	0.169110		0.292838	0.356751	0.244913		0.340109	0.353738	0.200328
		(0.0192)	(0.0157)	(0.0159)		(0.0200)	(0.0140)	(0.0138)		(0.0184)	(0.0150)	(0.0166)		(0.0203)	(0.0167)	(0.0183)
\({\nu }_{F,u}\left({\nu }_{F}\right)\)		0.259790	0.268501	0.202711		0.280429	0.323837	0.196639		0.255154	0.277162	0.213445		0.279514	0.326765	0.169572
		(0.0167)	(0.0130)	(0.0148)		(0.0167)	(0.0144)	(0.0162)		(0.0161)	(0.0110)	(0.0142)		(0.0164)	(0.0128)	(0.0150)
\({\nu }_{X,u}\left({\nu }_{X}\right)\)		0.348739	0.286168	0.185695		0.374289	0.354920	0.192929		0.379360	0.328660	0.220082		0.328770	0.337435	0.167851
		(0.0145)	(0.0111)	(0.0130)		(0.0156)	(0.0140)	(0.0151)		(0.0173)	(0.0134)	(0.0153)		(0.0153)	(0.0140)	(0.0146)
\({\nu }_{T,u}\)		0.385073	0.214064	0.117978		0.348324	0.157044	0.077925		0.327029	0.153397	0.071872		0.314040	0.112876	0.046910
		(0.0115)	(0.0094)	(0.0094)		(0.0100)	(0.0076)	(0.0066)		(0.0087)	(0.0088)	(0.0071)		(0.0097)	(0.0065)	(0.0051)
\({\nu }_{D,d}\)				0.178138				0.202540				0.245294				0.261442
				(0.0158)				(0.0172)				(0.0177)				(0.0179)
\({\nu }_{F,d}\)				0.145932				0.189175				0.196922				0.265231
				(0.0136)				(0.0161)				(0.0138)				(0.0159)
\({\nu }_{X,d}\)				0.148396				0.198658				0.204077				0.243379
				(0.0120)				(0.0158)				(0.0153)				(0.0158)
\({\nu }_{T,d}\)				0.122344				0.111278				0.113663				0.091415
				(0.0104)				(0.0094)				(0.0085)				(0.0070)
\({\phi }_{DF,u}\left({\phi }_{DF}\right)\)			0.056727	0.023693			− 0.010389	− 0.029232			0.070793	0.027077			0.092833	0.045615
			(0.0340)	(0.0266)			(0.0362)	(0.0286)			(0.0379)	(0.0299)			(0.0384)	(0.0254)
\({\phi }_{DX,u}\left({\phi }_{DX}\right)\)			− 0.021682	− 0.033227			0.017727	0.001234			0.033392	− 0.015567			0.047684	− 0.022359
			(0.0330)	(0.0242)			(0.0350)	(0.0285)			(0.0367)	(0.0311)			(0.0352)	(0.0258)
\({\phi }_{FX,u}\left({\phi }_{FX}\right)\)			− 0.021208	− 0.044786			− 0.037041	− 0.079530			0.009425	− 0.025997			0.085253	− 0.023325
			(0.0324)	(0.0257)			(0.0329)	(0.0256)			(0.0383)	(0.0297)			(0.0358)	(0.0260)
\({\phi }_{DF,d}\)				− 0.016848				0.065124				− 0.008434				0.042699
				(0.0250)				(0.0279)				(0.0309)				(0.0327)
\({\phi }_{DX,d}\)				− 0.004270				0.013855				0.041967				− 0.007523
				(0.0247)				(0.0276)				(0.0280)				(0.0309)
\({\phi }_{FX,d}\)				0.032736				0.043400				0.024660				0.063930
				(0.0208)				(0.0256)				(0.0289)				(0.0301)
\({\lambda }_{u}\left(\lambda \right)\)		0.280881	0.206799	0.106294		0.295578	0.220862	0.096217		0.299466	0.255834	0.100483		0.322897	0.285843	0.076891
		(0.0101)	(0.0054)	(0.0069)		(0.0090)	(0.0076)	(0.0072)		(0.0100)	(0.0093)	(0.0070)		(0.0114)	(0.0106)	(0.0073)
\({\lambda }_{d}\)				0.105518				0.109510				0.129113				0.169682
				(0.0084)				(0.0096)				(0.0104)				(0.0123)
\({h}_{D,u}\left({h}_{D}\right)\)		0.001281	− 0.062000	− 0.034035		− 0.005769	− 0.044923	− 0.023695		0.005560	− 0.038939	− 0.031663		0.011681	− 0.041468	− 0.025064
		(0.0092)	(0.0076)	(0.0059)		(0.0087)	(0.0076)	(0.0064)		(0.0087)	(0.0092)	(0.0077)		(0.0088)	(0.0085)	(0.0076)
\({h}_{F,u}\left({h}_{F}\right)\)		− 0.005464	0.028080	0.024562		0.016891	0.032475	0.047101		0.008289	− 0.003961	0.032792		− 0.026714	− 0.090545	− 0.003138
		(0.0113)	(0.0111)	(0.0086)		(0.0116)	(0.0115)	(0.0108)		(0.0110)	(0.0119)	(0.0093)		(0.0114)	(0.0109)	(0.0087)
\({h}_{X,u}\left({h}_{X}\right)\)		0.006704	− 0.006453	− 0.011616		0.013229	0.012234	0.017305		0.015738	0.050872	0.048587		0.013443	0.026627	0.012721
		(0.0020)	(0.0039)	(0.0036)		(0.0024)	(0.0074)	(0.0058)		(0.0037)	(0.0094)	(0.0070)		(0.0034)	(0.0092)	(0.0063)
\({h}_{D,d}\)				− 0.022063				− 0.014611				0.014805				− 0.008055
				(0.0060)				(0.0073)				(0.0081)				(0.0077)
\({h}_{F,d}\)				0.003424				− 0.003981				− 0.019386				− 0.057708
				(0.0082)				(0.0095)				(0.0100)				(0.0102)
\({h}_{X,d}\)				0.004931				0.005036				0.027671				0.033705
				(0.0029)				(0.0054)				(0.0061)				(0.0072)
Panel (c): Log- Likelihood
LL	2600.65	2784.55	3061.16	3123.21	2477.30	2518.48	3024.49	3118.30	2226.84	2385.18	2751.96	2903.53	2028.43	2278.30	2572.47	2675.10
LRT		367.81	553.22	124.10		82.37	1012.01	187.64		316.69	733.56	303.14		499.74	588.35	205.26

The results are calibrated from the sample shown in Table 1. This table lists the average and standard error of the mean of parametric estimation results for EUR/USD. The results are divided into the continuous part (Panel (a)) and the jump part (Panel (b)). For the jump part’s parameters in CB–ISJ and CB–CSJ models, the notations are shown in the brackets. For example, the mean of jump amplitude for the domestic interest rate in CB–ISJ and CB–CSJ models is \({\uptheta }_{\mathrm{D}}\) in the bracket

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