On a robust estimation of option-implied interest rates and dividend yields

Figures & data

Table 1. Parameter estimation results on the market data of SPX options on ${27}^{\mathrm{t}\mathrm{h}}$ June 2022 with (standard) options maturing on 19 August , 16 September 2022, 21 October 2022, 18 November 2022, 16 December 2022, 20 January 2023, 17 February 2023, 17 March 2023, 21 April 2023, 19 May 2023, 16 June 2023, 21 July 2023, 15 September 2023, 15 December 2023, 21 June 2024, 20 December 2024, 19 December 2025, and 18 December 2026. The S&P 500 index level was $3,900.11$

OLS
Time to maturity	$n$	${\hat{\alpha }}_{OLS}$	${\hat{\beta }}_{OLS}$	$R^2$	B-P p-value	${\hat{\alpha }}_{TS}$	${\hat{\beta }}_{TS}$	${\hat{\alpha }}_{RM}$	${\hat{\beta }}_{RM}$
53	321	−3894.70	0.9972698	100.00%	0.0710420	−3894.67	0.9972683	−3894.65	0.9972480
81	343	−3889.87	0.9953856	100.00%	0.0811177	−3889.90	0.9953947	−3889.95	0.9953990
116	306	−3885.55	0.9926458	100.00%	0.0793824	−3885.57	0.9926606	−3885.86	0.9927213
144	224	−3879.74	0.9901449	100.00%	0.0218289	−3879.74	0.9901339	−3879.76	0.9901425
172	252	−3874.41	0.9874683	100.00%	0.1905207	−3874.44	0.9874699	−3874.37	0.9874569
207	99	−3871.74	0.9837331	100.00%	0.0051267	−3871.72	0.9837300	−3871.71	0.9837362
235	120	−3866.57	0.9808222	100.00%	0.4572098	−3866.46	0.9808000	−3866.41	0.9807697
263	125	−3860.92	0.9780119	100.00%	0.0075156	−3860.77	0.9779789	−3860.55	0.9779171
298	113	−3857.62	0.9745484	100.00%	0.8104278	−3857.61	0.9745432	−3857.60	0.9745379
326	136	−3852.33	0.9716616	99.99%	0.3714814	−3852.41	0.9716522	−3852.27	0.9716172
354	140	−3846.36	0.9685746	100.00%	0.0001904	−3846.39	0.9685882	−3846.30	0.9685990
389	104	−3843.11	0.9652180	99.99%	0.8988304	−3843.13	0.9652353	−3843.35	0.9653012
445	94	−3832.03	0.9595769	100.00%	0.9272889	−3831.95	0.9595517	−3831.86	0.9595056
536	160	−3818.72	0.9505707	99.99%	0.7729168	−3819.03	0.9505714	−3818.86	0.9505430
725	98	−3785.02	0.9320216	99.99%	$1.4\times {10}^{-8}$	−3792.91	0.9334648	−3791.08	0.9330000
907	59	−3751.37	0.9134997	99.99%	0.6911559	−3760.90	0.9145357	−3763.47	0.9173452
1271	58	−3705.27	0.8851711	99.99%	0.9202375	−3713.12	0.8871667	−3713.14	0.8872094
1635	60	−3676.35	0.8590789	99.99%	0.3278899	−3670.64	0.8578634	−3663.35	0.8569000

Notes: $n$ is the sample size and B-P p-value refers to the p-value obtained from the Breusch–Pagan test for heteroscedasticity.

Table 2. Parameter estimation results on the market data of SPX options on $6^{\mathrm{t}\mathrm{h}}$ July 2022 with (standard) options maturing on 19 August , 16 September 2022, 21 October 2022, 18 November 2022, 16 December 2022, 20 January 2023, 17 February 2023, 17 March 2023, 21 April 2023, 19 May 2023, 16 June 2023, 21 July 2023, 15 September 2023, 15 December 2023, 21 June 2024, 20 December 2024, 19 December 2025, and 18 December 2026. The S&P 500 index level was $3,845.08$

OLS
Time to maturity	$n$	${\hat{\alpha }}_{OLS}$	${\hat{\beta }}_{OLS}$	$R^2$	B-P p-value	${\hat{\alpha }}_{TS}$	${\hat{\beta }}_{TS}$	${\hat{\alpha }}_{RM}$	${\hat{\beta }}_{RM}$
44	292	−3840.19	0.9975816	99.99%	0.0040887	−3840.05	0.9975472	−3839.90	0.9974943
72	327	−3835.49	0.9956784	99.99%	0.6247568	−3835.54	0.9956911	−3835.69	0.9957143
107	298	−3831.22	0.9930040	99.99%	0.1672682	−3831.18	0.9930035	−3831.37	0.9930320
135	220	−3825.52	0.9906368	99.99%	0.5725747	−3825.66	0.9906701	−3825.80	0.9907143
163	244	−3820.10	0.9879679	99.99%	0.0117681	−3820.11	0.9879829	−3820.29	0.9880000
198	96	−3818.05	0.9843804	99.99%	0.1308988	−3818.29	0.9844444	−3818.65	0.9845000
226	118	−3812.69	0.9815543	99.99%	0.5301284	−3812.76	0.9815429	−3812.75	0.9815563
254	123	−3807.05	0.9788335	99.99%	0.0397333	−3807.07	0.9788108	−3806.84	0.9787619
289	110	−3803.68	0.9755280	99.99%	0.0147180	−3803.73	0.9755410	−3803.75	0.9755581
317	134	−3798.26	0.9727844	99.99%	0.8690491	−3798.27	0.9727872	−3798.15	0.9727940
345	139	−3792.62	0.9699243	99.99%	0.8952470	−3792.67	0.9699231	−3792.35	0.9698554
380	104	−3788.90	0.9667018	99.99%	0.8895741	−3789.03	0.9667292	−3789.22	0.9668115
436	94	−3777.58	0.9614737	99.99%	0.4516035	−3777.42	0.9614157	−3777.06	0.9613203
527	159	−3763.37	0.9531406	99.99%	0.0008067	−3763.69	0.9531579	−3763.86	0.9533307
716	104	−3737.81	0.9379104	99.98%	0.0148570	−3737.14	0.9376645	−3739.52	0.9381818
898	50	−3709.02	0.9228855	99.99%	0.9582215	−3709.33	0.9230455	−3708.74	0.9230094
1262	58	−3663.67	0.8974606	99.99%	0.2432914	−3663.82	0.8974778	−3665.04	0.8980217
1626	54	−3619.56	0.8712580	99.99%	0.1649295	−3620.87	0.8710000	−3620.98	0.8710000

Notes: $n$ is the sample size and B-P p-value refers to the p-value obtained from the Breusch–Pagan test for heteroscedasticity.

Table 3. Parameter estimation results on the market data of SPX options on $8^{\mathrm{t}\mathrm{h}}$ July 2022 with (standard) options maturing on 19 August , 16 September 2022, 21 October 2022, 18 November 2022, 16 December 2022, 20 January 2023, 17 February 2023, 17 March 2023, 21 April 2023, 19 May 2023, 16 June 2023, 21 July 2023, 15 September 2023, 15 December 2023, 21 June 2024, 20 December 2024, 19 December 2025, and 18 December 2026. The S&P 500 index level was $3,899.38$

OLS
Time to maturity	$n$	${\hat{\alpha }}_{OLS}$	${\hat{\beta }}_{OLS}$	$R^2$	B-P p-value	${\hat{\alpha }}_{TS}$	${\hat{\beta }}_{TS}$	${\hat{\alpha }}_{RM}$	${\hat{\beta }}_{RM}$
42	289	−3894.29	0.9977834	99.99%	$9.1\times {10}^{-7}$	−3894.29	0.9977778	−3894.28	0.9977778
70	328	−3889.32	0.9958429	99.99%	0.2303731	−3889.36	0.9958519	−3889.39	0.9958559
105	305	−3884.51	0.9929699	99.99%	0.1929528	−3884.46	0.9929630	−3884.43	0.9929630
133	219	−3879.06	0.9905733	99.99%	0.9340346	−3879.02	0.9905800	−3879.08	0.9905844
161	244	−3873.83	0.9878743	99.99%	$7.7\times {10}^{-8}$	−3873.84	0.9878889	−3873.92	0.9879009
196	97	−3871.63	0.9841235	99.99%	0.4185494	−3871.59	0.9841250	−3871.61	0.9841197
224	118	−3866.34	0.9812033	99.99%	0.2343355	−3866.36	0.9812000	−3866.39	0.9812093
252	124	−3860.59	0.9783467	99.99%	0.0114025	−3860.58	0.9783333	−3860.47	0.9783093
287	111	−3857.20	0.9748543	99.99%	0.0142255	−3857.20	0.9748571	−3857.25	0.9748644
315	134	−3851.93	0.9719726	99.99%	0.7221770	−3851.95	0.9719756	−3851.98	0.9719789
343	139	−3845.63	0.9688625	99.99%	0.4267937	−3845.56	0.9688750	−3845.65	0.9688761
378	104	−3841.67	0.9653789	99.99%	0.0001093	−3842.11	0.9654923	−3842.38	0.9655556
434	94	−3831.25	0.9599978	99.99%	0.1455253	−3831.47	0.9600000	−3831.59	0.9601283
525	159	−3817.48	0.9515584	99.99%	0.1125040	−3817.19	0.9515556	−3817.14	0.9514202
714	54	−3795.74	0.9361488	99.99%	0.1138336	−3795.72	0.9361429	−3795.70	0.9361379
896	20	−3771.85	0.9218466	99.99%	0.8965776	−3771.84	0.9218462	−3771.81	0.9218333
1260	26	−3724.61	0.8937922	99.99%	0.4100380	−3724.60	0.8937917	−3724.62	0.8937929
1624	32	−3679.96	0.8653655	99.99%	0.1854253	−3679.94	0.8653636	−3679.94	0.8653611

Notes: $n$ is the sample size and B-P p-value refers to the p-value obtained from the Breusch–Pagan test for heteroscedasticity.

Table 4. Parameter estimation results on the market data of SPX options on ${19}^{\mathrm{t}\mathrm{h}}$ July 2022 with (standard) options maturing on 19 August , 16 September 2022, 21 October 2022, 18 November 2022, 16 December 2022, 20 January 2023, 17 February 2023, 17 March 2023, 21 April 2023, 19 May 2023, 16 June 2023, 21 July 2023, 15 September 2023, 15 December 2023, 21 June 2024, 20 December 2024, 19 December 2025, and 18 December 2026. The S&P 500 index level was $3,936.69$

OLS
Time to maturity	$n$	${\hat{\alpha }}_{OLS}$	${\hat{\beta }}_{OLS}$	$R^2$	B-P p-value	${\hat{\alpha }}_{TS}$	${\hat{\beta }}_{TS}$	${\hat{\alpha }}_{RM}$	${\hat{\beta }}_{RM}$
31	274	−3932.42	0.9980633	99.99%	$5.2\times {10}^{-30}$	−3932.32	0.9980315	−3932.32	0.9980315
59	327	−3927.69	0.9961924	99.99%	0.0043989	−3927.70	0.9961932	−3927.68	0.9961952
94	306	−3922.58	0.9932619	99.99%	0.0436663	−3922.50	0.9932432	−3922.26	0.9932047
122	220	−3917.45	0.9907463	99.99%	0.0039935	−3917.24	0.9906977	−3916.96	0.9906344
150	246	−3912.49	0.9879869	99.99%	0.0005688	−3912.58	0.9880000	−3912.51	0.9880205
185	98	−3909.45	0.9840047	99.99%	0.9882805	−3909.39	0.9840000	−3909.35	0.9840000
213	118	−3904.66	0.9810844	99.99%	0.4409974	−3904.63	0.9810625	−3904.43	0.9810328
241	124	−3898.91	0.9781998	99.99%	0.0094427	−3898.83	0.9781739	−3898.60	0.9781237
276	111	−3896.41	0.9749143	99.99%	0.0006763	−3896.61	0.9749541	−3896.61	0.9749843
304	134	−3891.17	0.9720133	99.99%	0.0294642	−3891.20	0.9720000	−3891.15	0.9720000
332	139	−3885.48	0.9690983	99.99%	0.2806403	−3885.59	0.9691150	−3885.58	0.9691162
367	104	−3882.23	0.9657893	99.99%	$3.3\times {10}^{-6}$	−3882.46	0.9658462	−3882.51	0.9658794
395	104	−3876.58	0.9629540	99.99%	0.9878203	−3876.65	0.9629705	−3876.43	0.9629512
423	93	−3871.58	0.9602386	99.99%	0.4755883	−3871.50	0.9602240	−3871.64	0.9602252
514	160	−3859.01	0.9518376	99.99%	0.2738522	−3859.19	0.9518421	−3859.20	0.9518436
703	55	−3836.28	0.9359060	100.00%	0.6359660	−3836.31	0.9359130	−3836.33	0.9359167
885	21	−3808.71	0.9207201	99.99%	$4.1\times {10}^{-5}$	−3810.94	0.9213125	−3810.96	0.9213153
1249	17	−3766.99	0.8942335	99.99%	0.5085847	−3766.98	0.8942308	−3766.95	0.8942219
1613	31	−3721.61	0.8674378	99.99%	$1.4\times {10}^{-11}$	−3718.42	0.8668077	−3718.41	0.8668038

Notes: $n$ is the sample size and B-P p-value refers to the p-value obtained from the Breusch–Pagan test for heteroscedasticity.

Figure 1. Comparison of estimated option-implied risk-free interest rates obtained via the three estimation techniques (OLS, TS, and RM) across different maturities with treasury yields.

Figure 2. Estimated option-implied dividend yields obtained via the three estimation techniques (OLS, TS, and RM) across different maturities.

Figure 3. Comparison of estimated annualized option-implied dividend yields obtained via the three estimation techniques (OLS, TS, and RM) across different maturities with historical S&P 500 index yield.

Table 5. The US. Daily treasury yield curve rates (%) between 27 June 2022 and 19 July 2022

	Maturity
Date	1 Mo	2 Mo	3 Mo	6 Mo	1 Yr	2 Yr	3 Yr	5 Yr	7 Yr	10 Yr	20 Yr	30 Yr
27-06-22	1.16	1.57	1.79	2.56	2.89	3.08	3.21	3.24	3.27	3.20	3.56	3.31
28-06-22	1.12	1.59	1.79	2.55	2.88	3.10	3.21	3.25	3.27	3.20	3.55	3.30
29-06-22	1.12	1.53	1.78	2.55	2.88	3.06	3.13	3.15	3.17	3.10	3.46	3.22
30-06-22	1.28	1.68	1.72	2.51	2.80	2.92	2.99	3.01	3.04	2.98	3.38	3.14
01-07-22	1.27	1.68	1.73	2.52	2.79	2.84	2.85	2.88	2.92	2.88	3.35	3.11
05-07-22	1.33	1.71	1.90	2.59	2.77	2.82	2.82	2.82	2.87	2.82	3.31	3.05
06-07-22	1.36	1.70	1.90	2.62	2.82	2.97	2.99	2.96	2.99	2.93	3.42	3.14
07-07-22	1.55	1.90	1.95	2.64	2.87	3.03	3.05	3.05	3.07	3.01	3.45	3.20
08-07-22	1.57	1.92	1.98	2.68	2.96	3.12	3.14	3.13	3.16	3.09	3.53	3.27
11-07-22	1.58	1.95	2.18	2.79	2.97	3.07	3.09	3.05	3.06	2.99	3.43	3.18
12-07-22	1.63	2.01	2.22	2.78	3.07	3.03	3.07	3.01	3.01	2.96	3.37	3.13
13-07-22	1.78	2.20	2.39	2.96	3.21	3.13	3.14	3.02	3.00	2.91	3.35	3.08
14-07-22	1.99	2.32	2.40	2.93	3.16	3.15	3.16	3.06	3.05	2.96	3.36	3.11
15-07-22	1.98	2.26	2.37	2.94	3.12	3.13	3.14	3.05	3.03	2.93	3.34	3.10
18-07-22	1.96	2.28	2.50	3.02	3.13	3.15	3.15	3.06	3.05	2.96	3.39	3.14
19-07-22	1.93	2.29	2.52	3.06	3.18	3.23	3.22	3.14	3.11	3.01	3.42	3.17