Variance Premium and Implied Volatility in a Low-Liquidity Option Market*

Astorino, Eduardo Sanchez; Chague, Fernando; Giovannetti, Bruno; Silva, Marcos Eugênio da

doi:10.5935/0034-7140.20170001

Eduardo Sanchez Astorino

Faculdade de Economia, Administração e Contabilidade da Universidade de São Paulo (FEA-USP). Av. Prof. Luciano Gualberto, 908, Cidade Universitária, São Paulo, SP, Brasil. CEP 05508-010. Email: eduardo.astorino@usp.br

Fernando Chague

FEA-USP. Email: fchague@usp.br

Bruno Giovannetti

FEA-USP. Email: bcg@usp.br

Marcos Eugênio da Silva

FEA-USP. Email: medsilva@usp.br

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Figure 1
IMPLIED VOLATILITY IN BRAZIL – THE IVOL-BR
This figure shows the daily time-series of the IVol-Br in percentage points and annualized.

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Figure 2
COMPARING IVOL-BR AND VXEWZ
This figure shows the daily time-series of the IVol-Br and the VXEWZ. Both series are in percentage points and annualized. VXEWZ is the implied volatility index on the Brazilian stocks ETF EWZ and is calculated by CBOE.

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Figure 3
COMPARING IVOL-BR AND VXEWZ
This figure shows the daily time-series of the IVol-Br and the IVol-ATM. Both series are in percentage points and annualized. IVol-ATM is the Black-Scholes implied volatility calculated using only at-the-money options.

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Figure 4
IMPLIED VARIANCE AND EXPECTED VARIANCE
This figure shows the weekly time-series of the implied variance-the squared of the IVol-Br-and the estimated expected variance. The model for expected volatility is the GETS model shown on . Both series are in percentage points and annualized.

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Figure 5
THE VARIANCE PREMIUM
This figure shows the weekly time-series of the variance premium calculated by the difference of the implied variance and expected variance as predicted by the GETS model shown on , and its three month moving average.

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Figure 6
RISK AVERSION
This figure show a time-series for the risk aversion index in Brazil. It is computed by combining the weekly series for the variance premium with Bollerslev et al. (2009)Bollerslev, T., Tauchen, G., & Zhou, H. (2009). Expected stock returns and variance risk premia. Review of Financial Studies, 22(11), 4463-4492. doi: 10.1093/rfs/hhp008
https://doi.org/10.1093/rfs/hhp008... functional form for the variance premium, as explained in section 3.3.

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Figure 7
RISK AVERSION AS A FUNCTION OF ψ AND κ1
This figure shows the image of the γ_t function for reasonable values of the variance premium and for different choices of ψ and κ1.

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Table 1
NUMBER OF OPTION STRIKES USED IN THE IVOL-BR

The table shows the quarterly daily averages of the number of strikes that were used in the construction of the IVol-BR.

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Table 2
POSSIBLE VALUES OF j

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Table 3
GENERAL-TO-SPECIFIC BEST MODEL

The table shows the estimates of the best variance forecasting model following the General-to Specific selection method. The starting model, also called GUM or General Unrestricted Model, comprises of all independent variables. The standard errors reported in parenthesis are robust to heteroskedasticity. Regressions are based on daily observations. The corresponding p-values are denoted by * if p < 0:10, ** if p < 0:05 and *** if p < 0:01.

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Table 4
PREDICTABILITY REGRESSIONS

The table shows the estimates of predictability regressions. The dependent variable is the return on the market portfolio 4-weeks ahead. The explanatory variables are: i) ${\hat{σ}}_{t}^{2}$ , the expected variance on the next 8 weeks estimated by best model following the General to-Specific selection method; ii) IVar_t , the expected implied variance on the next 8 weeks estimated from prices of options contracts at time t ; iii) IVar_t - ${\hat{σ}}_{t}^{2}$ , the variance premium; iv) γ_t is the risk aversion computed using the functional form in Bollerslev et al. (2009)Bollerslev, T., Tauchen, G., & Zhou, H. (2009). Expected stock returns and variance risk premia. Review of Financial Studies, 22(11), 4463-4492. doi: 10.1093/rfs/hhp008
https://doi.org/10.1093/rfs/hhp008... and the variance premium; and v) log ( D_t =P_t), the log dividend yield. Regressions are based on weekly observations. To account for error correlation, the standard errors are computed using Newey-West lags. The standard errors are reported in parenthesis. The corresponding p-values are denoted by * if p < 0:10, ** if p < 0:05 and *** if p < 0:01.

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Table 5
PREDICTABILITY REGRESSIONS AT DIFFERENT HORIZONS WITH VARIANCE PREMIUM

The table shows regressions having future market returns 1-week ahead, 2-, 3- and 4-weeks ahead as the dependent variable. The explanatory variables are: i) IVar_t ${\hat{σ}}_{t}^{2}$ is the ex-ante volatility premium; and ii) log (D_t/P_t) is the log dividend yield. Regressions in columns (1) through (8) are based on weekly observations. Regressions in columns (9) and (10) are non-overlapping on the dependent variable and are based on monthly observations. To account for error correlation, standard errors in columns (3) through (8) are computed using Newey-West lags. The standard errors reported in parenthesis. The corresponding p-values are denoted by * if p < 0:10, ** if p < 0:05 and *** if p < 0:01.

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Table 6
PREDICTABILITY REGRESSIONS AT DIFFERENT HORIZONS WITH RISK AVERSION

The table shows regressions having future market returns 1-week ahead, 2-, 3- and 4-weeks ahead as the dependent variable. The explanatory variables are: i) γ_t is the risk aversion computed using the functional form in Bollerslev et al. (2009)Bollerslev, T., Tauchen, G., & Zhou, H. (2009). Expected stock returns and variance risk premia. Review of Financial Studies, 22(11), 4463-4492. doi: 10.1093/rfs/hhp008
https://doi.org/10.1093/rfs/hhp008... ; and ii) log (D_t/P_t) is the log dividend yield. Regressions in columns (1) through (8) are based on weekly observations. Regressions in columns (9) and (10) are non overlapping on the dependent variable and are based on monthly observations. To account for error correlation, standard errors in columns (3) through (8) are computed using Newey-West lags. The standard errors reported in parenthesis. The corresponding p-values are denoted by * if p < 0:10, ** if p < 0:05 and *** if p < 0:01.

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Table 7
ROBUSTNESS: DIFFERENT VARIANCE MODELS

The table shows the estimates of different models of expected variance. The dependent variable is the realized variance over the following 42 days, calculated from 5-minute returns on the Ibovespa portfolio. The explanatory variables are: i) IVar_t is the expected implied variance on the next 8 weeks estimated from prices of options contracts at time t - 1; ii) RV^(k)_t-1 is the realized volatility on the following k days at time t - 1, where k = 42, 21, 5, 1; computed iii) Lev^(k)_t-1 is the cumulative negative 5-minute returns continuous component of the realized variance on the following k days at time t-1, where k =42, 21, 5, 1. The standard errors reported in parenthesis. The corresponding p-values are denoted by * if p < 0:10, ** if p < 0:05 and *** if p < 0:01.

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Table 8
ROBUSTNESS: DIFFERENT VARIANCE MODELS (CONT.)

The table shows the estimates of different models of expected realized variance. The dependent variable is the realized variance over the following 8-weeks, calculated from 5-minute returns on the Ibovespa portfolio. The explanatory variables are: i) IVar_t-1 is the expected implied variance on the next 8 weeks estimated from prices of options contracts at time t - 1; ii) C^(k)_t-1 is the continuous component of the realized variance during the following k days at time t-1 , where k = 42, 21, 5, 1; iii) J^(k)_t-1 is the jump component of the realized variance during the following k days at time t-1, where k = 42, 21, 5, 1; and iv) Lev^(k)_t-1 is the absolute of the sum 5-minute negative returns during the following k days at time t-1, where k = 42, 21, 5, 1. The standard errors reported in parenthesis. The regressions are based on daily observations. The corresponding p values are denoted by * if p < 0:10, ** if p < 0:05 and *** if p < 0:01.

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Table 9
ROBUSTNESS: PREDICTABILITY REGRESSION

The table shows the estimates of predictability regressions having the 4-week ahead return as the dependent variable and using various measures of variance premium as regressors. Each measure of variance premium is calculated with a different model for expected variance. Column (1) uses a simple model for expected variance: σ̂²_t = σ²_t-42 , and was proposed by Bollerslev et al. (2009)Bollerslev, T., Tauchen, G., & Zhou, H. (2009). Expected stock returns and variance risk premia. Review of Financial Studies, 22(11), 4463-4492. doi: 10.1093/rfs/hhp008
https://doi.org/10.1093/rfs/hhp008... . Column (2) replicates our main regression that uses the GETS model to forecast variance. Columns (3) through (10) varies the expected variance model from M1 to M10. log (Dt/Pt) is the log dividend yield. Regressions are based on weekly observations. To account for error correlation, the standard errors are computed using Newey-West lags. The standard errors are reported in parenthesis. The corresponding p-values are denoted by * if p < 0:10, ** if p < 0:05 and *** if p < 0:01.

(1)

(2)

(3)

(4)

(A-1)

Period	Call Strikes	Near Term Put Strikes	Total Strikes	Call Strikes	Near Term Put Strikes	Total Strikes
2011Q3	3	5	8	2	4	6
2011Q4	5	5	10	4	4	9
2012Q1	4	6	10	4	5	9
2012Q2	5	5	10	4	4	9
2012Q3	5	6	11	4	5	9
2012Q4	5	6	11	5	4	10
2013Q1	5	6	11	5	5	10
2013Q2	6	6	12	5	5	10
2013Q3	5	6	11	4	4	9
2013Q4	3	3	7	4	3	7
2014Q1	4	5	9	3	4	7
2014Q2	4	5	8	3	4	7
2014Q3	4	5	9	5	7	12
2014Q4	4	5	9	4	5	9

	K₀ < F	K₀ > F	K + 0 = F
Ǝcall, Ǝput	1	1	1
Ǝcall, ∄put	2	0	0
∄call, Ǝput	0	2	0

IVart	0.152***
	(0.040)
C_t ⁽⁴²⁾	3.215***
	(0.267)
C_t ⁽⁵⁾	-0.656***
	(0.123)
J_t ⁽²¹⁾	-0.540***
	(0.127)
J_t ⁽⁵⁾	0.307***
	(0.082)
Lev_t ⁽⁴²⁾	-2.237***
	(0.155)
Lev_t ⁽²¹⁾	0.722***
	(0.084)
Lev_t ⁽⁵⁾	0.367***
	(0.066)
Constant	950.186***
	(73.048)
Number of Obs.	741
R ²	0.400
Adjusted R²	0.393
RMQE	176.345
BIC	9818.8

	4 weeks
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
${\hat{σ}}_{t}^{2}$	0.000				-0.002
	(0.005)				(0.004)
IVar_t		0.004*				0.002
		(0.002)				(0.002)
IVar_t - ${\hat{σ}}_{t}^{2}$			0.089***				0.066**
			(0.032)				(0.031)
γ_t				0.180***				0.135**
				(0.057)				(0.059)
log(D/P)					14.928***	12.684***	11.449***	11.303***
					(5.482)	(6.169)	(6.157)	(6.132)
Constant	0.589	-1.854	-0.314	-4.228***	47.919***	38.601*	35.594*	32.204
	(2.349)	(1.345)	(0.579)	(1.538)	(17.131)	(19.719)	(19.315)	(19.956)
Number of Obs.	175	175	175	175	175	175	175	175
R²	0.000	0.044	0.082	0.086	0.096	0.108	0.135	0.137
Adjusted R²	-0.001	0.038	0.077	0.081	0.086	0.098	0.125	0.127

	1 week		2 weeks		3 weeks		4 weeks		3 weeks N-O
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
IVar_t ${\hat{σ}}_{t}^{2}$	0.028*	0.023	0.038*	0.026	0.059*	0.043	0.089**	0.066*	0.136**	0.112*
	(0.017)	(0.018)	(0.021)	(0.020)	(0.030)	(0.028)	(0.032)	(0.031)	(0.050)	(0.063)
log(D/P)		3.067		6.102*		7.939		11.449*		6.032
		(2.399)		(3.571)		(5.385)		(6.157)		(9.099)
Constant	-0.129	9.487	-0.095	19.042*	-0.191	24.707	-0.314	35.594*	-0.431	18.508
	(0.208)	(7.559)	(0.338)	(11.188)	(0.462)	(16.933)	(0.579)	(19.315)	(0.730)	(28.585)
Number of Obs.	178	178	177	177	176	176	175	175	41	41
R²	0.033	0.047	0.028	0.056	0.047	0.079	0.082	0.135	0.135	0.149
Adjusted R²	0.028	0.036	0.023	0.045	0.041	0.067	0.077	0.125	0.113	0.104

	1 week		2 weeks		3 weeks		4 weeks		3 weeks N-O
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
γ_t	0.052	0.039	0.078*	0.054	0.128**	0.097*	0.180***	0.135**	0.238***	0.198*
	(0.033)	(0.035)	(0.043)	(0.043)	(0.051)	(0.052)	(0.057)	(0.059)	(0.087)	(0.106)
log( D/P )		3.185		6.018*		7.601		11.303*		6.631
		(2.336)		(3.587)		(5.332)		(6.132)		(8.894)
Constant	-1.233	9.026	-1.804	17.593	-3.010**	21.491	-4.228***	32.204	-5.534**	16.142
	(0.875)	(7.711)	(1.164)	(11.540)	(1.340)	(17.328)	(1.538)	(19.956)	(2.271)	(29.341)
Number of Obs.	178	178	177	177	176	176	175	175	41	41
R²	0.027	0.042	0.030	0.057	0.056	0.086	0.086	0.137	0.135	0.153
Adjusted R²	0.022	0.031	0.025	0.046	0.051	0.076	0.081	0.127	0.113	0.109

	M1	M2	M3	M4
RV _t ⁽⁴²⁾	-0.113*	-0.166***		1.273***
RV _t ⁽²¹⁾	0.256***	0.247***		-0.532***
RV ^t(5)	0.139***	0.069*		-0.009
RV _t ⁽¹⁾	0.021	0.011		0.000
IVar_t		0.214***	0.218***	0.194***
Lev_t ⁽⁴²⁾			-0.513***	-1.837**
Lev_t ⁽²¹⁾			0.484***	1.031***
Lev_t ⁽⁵⁾			0.091***	0.101
Lev_t ⁽¹⁾			0.029	0.031
Constant	324.831***	271.370***	251.331***	691.146***
Number of Obs.	762	741	741	741
R²	0.209	0.228	0.277	0.369
Adjusted R²	0.205	0.223	0.272	0.361
RMQE	201.504	199.543	193.193	180.900
BIC	10,276.7	9,985.2	9,937.2	9,862.2

	M5	M6	M7	M8
IVar _t-1	0.147***	0.142***		0.286***
C ⁽⁴²⁾ _t-1	3.282***	3.679***		-1.675***
C ⁽²¹⁾ _t-1	-0.514	-1.171***		1.545***
C ⁽⁵⁾ _t-1	-0.461***	-0.225*		0.010
C ⁽¹⁾ _t-1	-0.057	-0.025		-0.015
J ⁽⁴²⁾ _t-1	0.294		1.583***	1.308***
J ⁽²¹⁾ _t-1	-0.540**		-0.768***	-1.126***
J ⁽⁵⁾ _t-1	0.247***		0.052	0.163
J ⁽¹⁾ _t-1	0.021		0.007	0.026
Lev ⁽⁴²⁾ _t-1	-2.433***	-2.535***	-1.289***
Lev ⁽²¹⁾ _t-1	1.023***	1.178***	0.802***
Lev ⁽⁵⁾ _t-1	0.263***	0.211***	0.077
Lev ^>(1) _t-1	0.043	0.043	0.021
IVol _t-1			13.969***
Constant	930.537***	935.469***	337.173***	256.965***
Number of Obs.	741	741	741	741
R²	0.406	0.397	0.349	0.269
Adjusted R²	0.396	0.390	0.341	0.260
RMQE	175.994	176.881	183.726	194.766
BIC	9,843.8	9,828.9	9,885.2	9,971.7

		GETS
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
Constant	50.656***	35.594*	37.988**	38.108**	37.454*	33.082*	34.887*	34.788*	33.337*	37.230*
	(17.423)	(19.315)	(18.852)	(19.122)	(19.386)	(19.675)	(19.394)	(19.569)	(19.601)	(19.185)
log(D/P)	16.234***	11.449*	12.188**	12.235**	12.007*	10.675*	11.233*	11.201*	10.751*	11.964*
	(5.583)	(6.157)	(6.021)	(6.098)	(6.179)	(6.268)	(6.181)	(6.235)	(6.245)	(6.114)
IVar_t-σ²_t-1	0.005***
IVar_tσ̂²_t	(0.002)	0.007**
IVar_tσ̂²_t(M1)		(0.003)	0.005**
IVar_tσ̂²_t(M2)			(0.002)	0.005*
IVar_tσ̂²_t(M3)				(0.003)	0.004*
IVar_tσ̂²_t(M4)					(0.002)	0.006**
IVar_tσ̂²_t(M5)						(0.003)	0.006**
IVar_tσ̂²_t(M6)							(0.003)	0.006**
IVar_tσ̂²_t(M7)								(0.003)	0.006**
IVar_tσ̂²_t(M8)									(0.003)	0.005**
										(0.003)
Number of Obs.	175	175	175	175	175	175	175	175	175	175
R²	0.015	0.135	0.127	0.119	0.114	0.147	0.140	0.138	0.143	0.126
Adjusted R²	0.140	0.125	0.116	0.109	0.104	0.137	0.130	0.128	0.133	0.116

σ_{k}^{2} (t) = \frac{2}{T_{k} - t} \sum_{i} \frac{Δ K_{i}}{K_{i}^{2}} e^{r_{i} (T_{k} - t)} O_{t} (K_{i}) - \frac{j}{T_{k} - t} {[\frac{F (t, T_{k})}{K_{0}} - 1]}^{2},

{IVol}_{t} = 100 \times \sqrt{[(T_{1} - t) σ_{1}^{2} (\frac{N_{T_{2}} - N_{42}}{N_{T_{2}} - N_{T_{1}}}) + (T_{2} - t) σ_{2}^{2} (\frac{N_{42} - N_{T_{1}}}{N_{T_{2}} - N_{T_{1}}})] \times \frac{N_{252}}{N_{42}}},

\begin{matrix} {RV}_{t + 42} = c + γ_{1} {IVar}_{t} + γ_{2} C_{t}^{(42)} + γ_{3} C_{t}^{(21)} + γ_{4} C_{t}^{(5)} + γ_{5} C_{t}^{(1)} + γ_{6} J_{t}^{(42)} + γ_{7} J_{t}^{(21)} + γ_{t}^{(5)} + γ_{9} J_{t}^{(1)} \\ + γ_{10} {Lev}_{i}^{(42)} + γ_{11} {Lev}_{t}^{(21)} + γ_{12} {Lev}_{t}^{(5)} + γ_{13} {Lev}_{t}^{(1)} + ɛ_{t} . \end{matrix}

{VariancePremium}_{t} = \frac{ψ^{- 1} - γ_{t}}{1 - ψ^{- 1}} \times \frac{κ_{1} {(1 - γ_{t})}^{2}}{2 (\frac{1 - γ t}{1 - ψ^{- 1}}) (1 - κ_{1} ρ_{σ})} q,

\begin{matrix} \frac{2}{T - t} \frac{Δ K_{0}}{K_{0}^{2}} e^{r_{t} (T - t)} O (K_{0}) = 2 \frac{Δ K_{0}}{K_{0}^{2}} \frac{P (K_{0})}{T - t} e^{r_{t} (T - t)} + \frac{1}{T - t} \frac{Δ K_{0}}{K_{0}^{2}} (F - K_{0}) \\ = 2 \frac{Δ K_{0}}{K_{0}^{2}} \frac{P (K_{0})}{T - t} e^{r_{t} (T - t)} + \frac{1}{T - t} {(\frac{F}{K_{0}} - 1)}^{2}, \end{matrix}

σ^{2} (t) = \frac{2}{T - t} \sum_{i} \frac{Δ K}{K_{i}^{2}} e^{r (T - t)} O_{t} (K_{i}) + \underset{= 0}{\underset{︸}{\frac{1}{T - t} \frac{Δ K_{0}}{K_{0}^{2}} (F - K_{0}) - \frac{1}{T - t} {(\frac{F}{K_{0}} - 1)}^{2}}} .

σ^{2} (t) = \frac{2}{T - t} \sum_{i} \frac{Δ K}{K_{i}^{2}} e^{r (T - t)} O_{t} (K_{0}) + \frac{1}{T - t} \frac{Δ K_{0}}{K_{0}^{2}} (F - K_{0}) - \frac{1}{T - t} {(\frac{F}{K_{0}} - 1)}^{2},

\frac{2}{T - t} \frac{Δ K_{0}}{K_{0}^{2}} e^{r_{t}} (T - t) O (K_{0}) = \frac{2}{T - t} \frac{Δ K_{0}}{K_{0}^{2}} P (K_{0}) e^{r_{t}} (T - t) + \frac{2}{T - t} \frac{Δ K_{0}}{K_{0}^{2}} (F - K_{0}) .

σ^{2} (t) = \frac{2}{T - t} \sum_{i} \frac{Δ K}{K_{i}^{2}} e^{r (T - t)} O_{t} (K_{i}) + \underset{= 0}{\underset{︸}{\frac{2}{T - t} \frac{Δ K_{0}}{K_{0}^{2}} (F - K_{0}) - \frac{2}{T - t} {(\frac{F}{K_{0}} - 1)}^{2},}}