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 4 Parametric estimation results for JPY/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
\({\mu }_{D}\)	− 0.000724	0.002790	0.000202	− 0.000345	− 0.000140	0.003769	0.000163	− 0.000201	− 0.000025	0.006647	0.001585	− 0.000752	0.002471	0.004365	0.003167	− 0.001568
	(0.0006)	(0.0011)	(0.0004)	(0.0002)	(0.0003)	(0.0012)	(0.0005)	(0.0004)	(0.0001)	(0.0015)	(0.0011)	(0.0005)	(0.0006)	(0.0014)	(0.0017)	(0.0004)
\({\mu }_{F}\)	− 0.002540	− 0.013534	− 0.013198	− 0.013227	− 0.003591	− 0.014064	− 0.014798	− 0.014493	− 0.005207	− 0.014572	− 0.016581	− 0.016151	− 0.005799	− 0.017797	− 0.018464	− 0.018679
	(0.0007)	(0.0015)	(0.0011)	(0.0011)	(0.0007)	(0.0013)	(0.0013)	(0.0012)	(0.0009)	(0.0015)	(0.0014)	(0.0013)	(0.0009)	(0.0016)	(0.0016)	(0.0016)
\({\mu }_{X}\)	0.012065	0.074566	0.037860	0.037697	0.012294	0.062645	0.037402	0.034098	0.008886	0.058460	0.037401	0.032931	0.011999	0.020883	0.009416	0.014147
	(0.0019)	(0.0064)	(0.0032)	(0.0032)	(0.0018)	(0.0067)	(0.0037)	(0.0035)	(0.0011)	(0.0079)	(0.0051)	(0.0046)	(0.0017)	(0.0091)	(0.0047)	(0.0034)
\({\sigma }_{D}\)	0.001283	0.045123	0.010410	0.004763	0.000387	0.050116	0.012271	0.006201	0.000643	0.062721	0.027526	0.007445	0.027741	0.050773	0.032656	0.007195
	(0.0010)	(0.0061)	(0.0028)	(0.0017)	(0.0002)	(0.0066)	(0.0032)	(0.0022)	(0.0002)	(0.0074)	(0.0051)	(0.0024)	(0.0067)	(0.0070)	(0.0067)	(0.0021)
\({\sigma }_{F}\)	0.001508	0.016329	0.001445	0.001331	0.000761	0.014872	0.004829	0.001694	0.001221	0.018566	0.004784	0.000251	0.014728	0.012520	0.004372	0.001172
	(0.0011)	(0.0049)	(0.0006)	(0.0006)	(0.0002)	(0.0042)	(0.0019)	(0.0009)	(0.0002)	(0.0047)	(0.0019)	(0.0001)	(0.0032)	(0.0038)	(0.0017)	(0.0007)
\({\sigma }_{X}\)	0.006340	0.249889	0.117325	0.109459	0.005559	0.213643	0.090310	0.074485	0.006006	0.181977	0.112538	0.059714	0.033455	0.156570	0.076310	0.039318
	(0.0005)	(0.0128)	(0.0066)	(0.0066)	(0.0001)	(0.0132)	(0.0070)	(0.0061)	(0.0002)	(0.0138)	(0.0098)	(0.0063)	(0.0067)	(0.0142)	(0.0080)	(0.0042)
\({\rho }_{DF}\)	0.033618	− 0.046936	0.117929	0.103691	0.027450	− 0.069172	0.033432	0.070554	0.015880	− 0.163274	− 0.008536	− 0.078817	0.014507	− 0.078124	− 0.015887	− 0.015277
	(0.0234)	(0.0439)	(0.0364)	(0.0367)	(0.0209)	(0.0415)	(0.0326)	(0.0326)	(0.0232)	(0.0405)	(0.0356)	(0.0344)	(0.0222)	(0.0405)	(0.0324)	(0.0301)
\({\rho }_{DX}\)	− 0.100361	0.245872	0.319529	0.325771	− 0.135105	0.272584	0.251802	0.246466	− 0.146818	0.184796	0.221529	0.196845	− 0.254027	0.197142	0.235278	0.200726
	(0.0199)	(0.0310)	(0.0297)	(0.0297)	(0.0202)	(0.0307)	(0.0318)	(0.0313)	(0.0196)	(0.0337)	(0.0334)	(0.0337)	(0.0248)	(0.0320)	(0.0308)	(0.0311)
\({\rho }_{FX}\)	0.030153	− 0.203839	− 0.360288	− 0.387780	− 0.064767	− 0.219899	− 0.353414	− 0.368244	− 0.016641	− 0.117808	− 0.343492	− 0.348256	0.062092	− 0.215963	− 0.355031	− 0.331898
	(0.0173)	(0.0303)	(0.0258)	(0.0250)	(0.0232)	(0.0315)	(0.0289)	(0.0266)	(0.0239)	(0.0332)	(0.0312)	(0.0295)	(0.0286)	(0.0311)	(0.0284)	(0.0264)
\({h}_{1,D}\)	0.048652	− 0.144661	− 0.048426	− 0.041649	0.034390	− 0.111108	− 0.033412	− 0.024636	0.049456	− 0.091031	− 0.049862	− 0.033178	0.090390	− 0.089294	− 0.037697	− 0.030640
	(0.0049)	(0.0099)	(0.0129)	(0.0129)	(0.0107)	(0.0108)	(0.0126)	(0.0125)	(0.0123)	(0.0117)	(0.0138)	(0.0140)	(0.0137)	(0.0121)	(0.0123)	(0.0127)
\({h}_{1,F}\)	0.015571	− 0.111249	− 0.036274	− 0.028987	− 0.012991	− 0.097861	− 0.018851	− 0.015554	− 0.021189	− 0.068303	− 0.012708	− 0.016123	− 0.034405	− 0.065312	− 0.021774	− 0.014086
	(0.0060)	(0.0132)	(0.0130)	(0.0127)	(0.0117)	(0.0125)	(0.0127)	(0.0127)	(0.0112)	(0.0131)	(0.0137)	(0.0137)	(0.0151)	(0.0139)	(0.0133)	(0.0136)
\({h}_{1,X}\)	0.026452	0.015471	0.018559	0.018534	0.039785	− 0.017453	0.009041	0.007072	0.035143	0.009824	− 0.003340	0.005323	0.082119	0.033215	0.009578	0.005886
	(0.0054)	(0.0121)	(0.0111)	(0.0110)	(0.0049)	(0.0120)	(0.0109)	(0.0107)	(0.0057)	(0.0125)	(0.0120)	(0.0119)	(0.0103)	(0.0129)	(0.0124)	(0.0127)
Panel (b): Jump Part
\({\theta }_{D,u}\left({\theta }_{D}\right)\)		− 0.077217	− 0.040962	− 0.026035		− 0.096232	− 0.046493	− 0.020441		− 0.095027	− 0.092940	− 0.047002		− 0.096997	− 0.091239	− 0.039089
		(0.0113)	(0.0053)	(0.0044)		(0.0127)	(0.0046)	(0.0038)		(0.0117)	(0.0100)	(0.0079)		(0.0115)	(0.0088)	(0.0079)
\({\theta }_{F,u}\left({\theta }_{F}\right)\)		− 0.139962	− 0.118199	− 0.067298		− 0.149719	− 0.152991	− 0.081523		− 0.120970	− 0.163119	− 0.088605		− 0.130385	− 0.132831	− 0.062873
		(0.0122)	(0.0104)	(0.0093)		(0.0138)	(0.0112)	(0.0095)		(0.0119)	(0.0113)	(0.0102)		(0.0113)	(0.0110)	(0.0094)
\({\theta }_{X,u}\left({\theta }_{X}\right)\)		0.016231	0.018971	0.064418		0.003326	0.019707	0.064708		− 0.018151	− 0.015685	0.065297		− 0.088590	− 0.077671	0.040334
		(0.0178)	(0.0102)	(0.0079)		(0.0188)	(0.0123)	(0.0081)		(0.0213)	(0.0161)	(0.0101)		(0.0245)	(0.0144)	(0.0084)
\({\theta }_{T,u}\left({\theta }_{T}\right)\)		− 0.106679	− 0.031547	− 0.021507		− 0.101126	− 0.032968	− 0.019479		− 0.080597	− 0.023948	− 0.008753		− 0.065351	− 0.012870	− 0.006549
		(0.0073)	(0.0033)	(0.0032)		(0.0066)	(0.0038)	(0.0035)		(0.0056)	(0.0032)	(0.0014)		(0.0055)	(0.0016)	(0.0012)
\({\theta }_{D,d}\)				− 0.018009				− 0.021483				− 0.048310				− 0.049949
				(0.0039)				(0.0041)				(0.0080)				(0.0068)
\({\theta }_{F,d}\)				− 0.042573				− 0.060715				− 0.075859				− 0.074966
				(0.0078)				(0.0085)				(0.0093)				(0.0111)
\({\theta }_{X,d}\)				− 0.046338				− 0.061940				− 0.091672				− 0.125535
				(0.0058)				(0.0077)				(0.0095)				(0.0115)
\({\theta }_{T,d}\)				− 0.015228				− 0.018851				− 0.020360				− 0.014544
				(0.0021)				(0.0030)				(0.0024)				(0.0018)
\({\nu }_{D,u}\left({\nu }_{D}\right)\)		0.307731	0.235797	0.151557		0.365046	0.249631	0.161195		0.336846	0.354567	0.232892		0.347175	0.340592	0.186919
		(0.0203)	(0.0128)	(0.0134)		(0.0202)	(0.0119)	(0.0121)		(0.0205)	(0.0185)	(0.0181)		(0.0197)	(0.0168)	(0.0172)
\({\nu }_{F,u}\left({\nu }_{F}\right)\)		0.269488	0.231884	0.152615		0.283391	0.264876	0.180132		0.262703	0.292638	0.216453		0.283342	0.289277	0.176559
		(0.0179)	(0.0112)	(0.0127)		(0.0179)	(0.0124)	(0.0141)		(0.0158)	(0.0135)	(0.0153)		(0.0160)	(0.0137)	(0.0149)
\({\nu }_{X,u}\left({\nu }_{X}\right)\)		0.341559	0.264691	0.169215		0.350813	0.305284	0.201426		0.354049	0.318202	0.207832		0.348720	0.299622	0.153921
		(0.0149)	(0.0097)	(0.0124)		(0.0154)	(0.0115)	(0.0133)		(0.0167)	(0.0129)	(0.0146)		(0.0163)	(0.0126)	(0.0134)
\({\nu }_{T,u}\)		0.372766	0.189095	0.120676		0.368955	0.188932	0.115260		0.341030	0.154653	0.074490		0.307825	0.140148	0.056371
		(0.0110)	(0.0088)	(0.0099)		(0.0106)	(0.0106)	(0.0105)		(0.0110)	(0.0093)	(0.0063)		(0.0120)	(0.0078)	(0.0061)
\({\nu }_{D,d}\)				0.126384				0.131621				0.248069				0.240878
				(0.0125)				(0.0126)				(0.0176)				(0.0175)
\({\nu }_{F,d}\)				0.122497				0.143187				0.206913				0.218347
				(0.0117)				(0.0132)				(0.0145)				(0.0149)
\({\nu }_{X,d}\)				0.129250				0.143826				0.193444				0.199804
				(0.0111)				(0.0128)				(0.0140)				(0.0139)
\({\nu }_{T,d}\)				0.096627				0.098558				0.120820				0.113042
				(0.0089)				(0.0098)				(0.0097)				(0.0084)
\({\phi }_{DF,u}\left({\phi }_{DF}\right)\)			0.063737	0.028103			0.075084	0.015236			0.088882	0.039773			0.096012	0.018120
			(0.0317)	(0.0251)			(0.0318)	(0.0255)			(0.0357)	(0.0277)			(0.0364)	(0.0248)
\({\phi }_{DX,u}\left({\phi }_{DX}\right)\)			− 0.002817	− 0.006861			0.022876	0.014827			0.001231	− 0.037128			0.056436	− 0.020936
			(0.0321)	(0.0243)			(0.0339)	(0.0253)			(0.0359)	(0.0287)			(0.0333)	(0.0244)
\({\phi }_{FX,u}\left({\phi }_{FX}\right)\)			− 0.039602	− 0.032789			− 0.077907	− 0.053047			− 0.050654	− 0.113436			− 0.010334	− 0.083805
			(0.0304)	(0.0229)			(0.0312)	(0.0244)			(0.0333)	(0.0258)			(0.0347)	(0.0241)
\({\phi }_{DF,d}\)				0.020391				0.010796				0.020712				0.035938
				(0.0232)				(0.0247)				(0.0285)				(0.0325)
\({\phi }_{DX,d}\)				− 0.025167				− 0.020794				0.000596				0.016750
				(0.0224)				(0.0232)				(0.0282)				(0.0288)
\({\phi }_{FX,d}\)				0.003330				0.021847				0.023164				0.045226
				(0.0227)				(0.0224)				(0.0275)				(0.0291)
\({\lambda }_{u}\left(\lambda \right)\)		0.275868	0.199275	0.102557		0.284816	0.192973	0.104484		0.300577	0.246232	0.101458		0.333489	0.277182	0.088177
		(0.0083)	(0.0062)	(0.0073)		(0.0095)	(0.0066)	(0.0069)		(0.0091)	(0.0089)	(0.0076)		(0.0119)	(0.0077)	(0.0074)
\({\lambda }_{d}\)				0.096512				0.085225				0.120717				0.166303
				(0.0085)				(0.0077)				(0.0090)				(0.0108)
\({h}_{D,u}\left({h}_{D}\right)\)		− 0.006271	− 0.049757	− 0.030428		− 0.006466	− 0.052215	− 0.034054		− 0.011305	− 0.048651	− 0.022894		− 0.014017	− 0.046253	− 0.023847
		(0.0086)	(0.0070)	(0.0060)		(0.0088)	(0.0072)	(0.0064)		(0.0090)	(0.0094)	(0.0074)		(0.0092)	(0.0079)	(0.0073)
\({h}_{F,u}\left({h}_{F}\right)\)		0.009905	0.052307	0.043514		0.024171	0.024264	0.042562		0.032573	-0.009302	0.014825		− 0.013567	− 0.029517	0.019235
		(0.0108)	(0.0097)	(0.0082)		(0.0111)	(0.0104)	(0.0094)		(0.0107)	(0.0114)	(0.0090)		(0.0118)	(0.0124)	(0.0091)
\({h}_{X,u}\left({h}_{X}\right)\)		0.000244	− 0.016493	− 0.015157		0.010492	0.017190	− 0.003271		0.016237	0.047250	0.016391		0.011176	− 0.004335	0.002691
		(0.0022)	(0.0044)	(0.0030)		(0.0026)	(0.0057)	(0.0048)		(0.0035)	(0.0080)	(0.0064)		(0.0038)	(0.0091)	(0.0063)
\({h}_{D,d}\)				− 0.020347				− 0.006948				− 0.003349				− 0.016719
				(0.0044)				(0.0067)				(0.0081)				(0.0074)
\({h}_{F,d}\)				0.018355				− 0.007953				0.011487				− 0.045573
				(0.0064)				(0.0082)				(0.0092)				(0.0098)
\({h}_{X,d}\)				− 0.005902				0.015773				0.035390				0.010974
				(0.0035)				(0.0040)				(0.0057)				(0.0063)
Panel (c): Log-Likelihood
LL	2409.87	2646.89	3080.72	3131.02	2423.00	2585.50	3003.17	3058.59	2400.09	2465.06	2628.32	2809.77	2372.92	2393.51	2563.52	2679.95
LRT		474.05	867.66	100.59		325.00	835.34	110.84		129.95	326.52	362.89		41.17	340.03	232.86

The results are calibrated form the sample shown as Table 1. This table lists the average and standard error of the mean of parametric estimation results for JPY/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 domestic interest rate in CB–ISJ and CB–CSJ models is \({\uptheta }_{\mathrm{D}}\) in the bracket

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