# Dynamic Relationship between Volatility Risk Premia of Stock and Oil Returns

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Volatility Risk Premium (VRP)

## 3. Data

## 4. Model Selection

#### 4.1. Unit Root Tests

#### 4.2. VAR Model

#### 4.3. Choice of Lag Length

## 5. Empirical Results

#### 5.1. Results for the Whole Period

#### 5.2. Results for the Sub-Periods

## 6. Robustness Analysis

## 7. Summary and Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Estimation of Realized Volatility by a Stochastic Volatility Model

Parameter | Period 1 | Period 2 | Period 3 | Period 4 | Period 5 |
---|---|---|---|---|---|

${\mathsf{\mu}}_{\mathrm{oil}}$ | 0.305 (0.192, 0.415) | −0.336 (−0.496, −0.178) | −0.017 (−0.03, −0.01) | 0.007 (0.003, 0.010) | −0.194 (−0270, −0.122) |

${\mathsf{\beta}}_{\mathrm{oil}}$ | 0.041 (−0.039, 0.123) | 0.068 (−0.017, 0.151) | 0.032 (−0.044, 0.109) | 0.028 (−0.046, 0.101) | 0.085 (0.004, 0.164) |

${\mathsf{\mu}}_{\mathrm{h}}$ | 1.232 (0.936, 1.530) | 2.472 (2.084, 2.820) | 1.278 (1.134, 1.428) | 0.292 (0.095, 0.509) | 1.675 (1.413, 1.939) |

${\mathsf{\beta}}_{\mathrm{h}}$ | 0.849 (0.774, 0.915) | 0.891 (0.835, 0.940) | 0.809 (0.733, 0.873) | 0.846 (0.771, 0.903) | 0.914 (0.876, 0.946) |

${\mathsf{\sigma}}_{\mathrm{h}}$ | 0.380 (0.336, 0.431) | 0.369 (0.325, 0.417) | 0.363 (0.325, 0.406) | 0.368 (0.328, 0.413) | 0.349 (0.311, 0.391) |

${\mathsf{\rho}}_{}$ | −0.488 (−0.630, −0.335) | −0.4649 (−0.608, −0.310) | −0.467 (−0.587, −0.331) | −0.457 (−0.595, −0.308) | −0.413 (−0.550, −0.269) |

${\mathsf{\nu}}_{}$ | 8.047 (6.455, 9.753) | 8.083 (6.520, 9.757) | 8.455 (6.972, 10.116) | 8.014 (6.476, 9.664) | 8.534 (6.994, 10.210) |

## Notes

1 | Results by extant researches about the effects of oil volatility-related variables on stock returns and volatility are mixed. For example, Ornelas and Mauad (2019) find little predictability of oil VRP on S&P 500 returns, Bams et al. (2017) find that difference of oil VRP is priced only on returns of oil-related stocks, and Christoffersen and Pan (2018) find predictability of oil implied volatility on stock returns and implied volatility. |

2 | Volatility swap and variance swap, where variance is the square of volatility, are traded in over-the-counter derivative markets. |

3 | Ornelas and Mauad (2019) explain what kind of realized volatility is used in the literature to approximate the expected future volatility. |

4 | https://www.cboe.com/us/indices/dashboard/VIX/ (3 March 2022). |

5 | http://realized.oxford-man.ox.ac.uk/ (3 March 2023). |

6 | https://www.cboe.com/us/indices/dashboard/OVX/ (3 March 2023). |

7 | Appendix A explains how we estimate the realized volatility of oil. |

8 | For Augmented Dickey–Fuller (ADF), Dickey–Fuller–GLS (DF–GLS), and Phillips–Perron (PP) tests, see Dickey and Fuller (1979); Elliott et al. (1996); and Phillips and Perron (1988), respectively. |

9 | Analyses with different lag length provide results quite similar to those in this paper. |

10 | We obtain the similar result if we reverse the order of ${\mathrm{VRP}}_{\mathrm{oil}}$ and ${\mathrm{VRP}}_{\mathrm{sp}}$. We select this ordering since the results of Granger causality tests show more persistent Granger causality from oil to stock than from stock to oil for most of the sub-periods. For more detail, see next subsection. |

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**Figure 1.**index_sp (S&P 500) and index_oil (USO multiplied by 50). Notes: Daily oil index (USO) multiplied by 50 and stock index (S&P 500) from 10 May 2007 to 16 May 2017 where the vertical axis is measured in U.S. dollars and time in the horizontal axis represents the date where time 1 corresponds to 10 May 2007, time 500 to 13 May 2009, time 1000 to 6 May 2011, time 1500 to 3 May 2013, time 2000 to 29 April 2015, and time 2500 to 24 April 2017, respectively.

**Figure 2.**${\mathrm{VRP}}_{\mathrm{oil}}$ and ${\mathrm{VRP}}_{\mathrm{sp}}$ from 10 May 2007 to 16 May 2017. Notes: Daily volatility risk premium of oil (${\mathrm{VRP}}_{\mathrm{oil}}$) and stock (${\mathrm{VRP}}_{\mathrm{sp}}$) from 10 May 2007 to 16 May 2017. The vertical axis represents their values and time in the horizontal axis represents the date where time 1 corresponds to 10 May 2007, time 500 to 13 May 2009, time 1000 to 6 May 2011, time 1500 to 3 May 2013, time 2000 to 29 April 2015, and time 2500 to 24 April 2017, respectively.

**Figure 3.**Impulse response functions (whole period). Notes: Vrp_oil (resp. vrp_sp) represents volatility risk premium of oil (resp. stock). The solid line represents orthogonalized impulse response functions and the gray area represents their 95% confidence intervals. The above-right graph shows the impulse response functions of vrp_sp to vrp_oil. The below-left graph shows the impulse response functions of vrp_oil to vrp_sp.

**Figure 8.**Impulse Response Function (Period 5). Notes: Vrp_oil (resp. vrp_sp) represents volatility risk premium of oil (resp. stock). The solid line represents orthogonalized impulse response functions and the gray area represents their 95% confidence intervals. The above-right graph shows the impulse response functions of vrp_sp to vrp_oil. The below-left graph shows the impulse response functions of vrp_oil to vrp_sp.

**Figure 9.**Impulse Response Function with Order ${\mathrm{VRP}}_{\mathrm{sp}}$ before ${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 1).

**Figure 10.**Impulse Response Function with Order ${\mathrm{VRP}}_{\mathrm{sp}}$ before ${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 2).

**Figure 11.**Impulse Response Function with Order ${\mathrm{VRP}}_{\mathrm{sp}}$ before ${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 3).

**Figure 12.**Impulse Response Function with Order ${\mathrm{VRP}}_{\mathrm{sp}}$ before ${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 4).

**Figure 13.**Impulse Response Function with Order ${\mathrm{VRP}}_{\mathrm{sp}}$ before ${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 5). Notes: Vrp_oil (resp. vrp_sp) represents volatility risk premium of oil (resp. stock). The solid line represents orthogonalized impulse response functions and the gray area represents their 95% confidence interval. The above-right graph shows the impulse response functions of vrp_sp to vrp_oil. The below-left graph shows the impulse response functions of vrp_oil to vrp_sp.

Whole Period | 10 May 2007–16 May 2017 (Time = 1–2516) |
---|---|

Period 1 (Pre-crisis period) | 10 May 2007–31 May 2008 (time = 1–266) |

Period 2 (Crisis outbreak period) | 1 June 2008–30 June 2009 (time = 267–533) |

Period 3 (Post-crisis recovery period I) | 1 July 2009–31 July 2012 (time = 534–1311) |

Period 4 (Post-crisis recovery period II) | 1 August 2012–30 September 2014 (time = 1312–1855) |

Period 5 (Plunging oil price period) | 1 October 2014–16 May 2017 (time = 1856–2516) |

**Table 2.**Descriptive statistics of ${\mathrm{VRP}}_{\mathrm{sp}}$ and ${\mathrm{VRP}}_{\mathrm{oil}}$.

Mean | St. Dev. | Skew. | Kurt. | Corr. | #Obs. | |
---|---|---|---|---|---|---|

${\mathrm{VRP}}_{\mathrm{sp}}$ (Whole) | 7.785 | 4.891 | −3.141 | 49.116 | 0.273 | 2516 |

${\mathrm{VRP}}_{\mathrm{sp}}$ (Period 1) | 6.509 | 4.794 | −1.502 | 7.724 | 0.097 | 266 |

${\mathrm{VRP}}_{\mathrm{sp}}$ (Period 2) | 9.583 | 10.148 | −2.778 | 22.344 | 0.278 | 267 |

${\mathrm{VRP}}_{\mathrm{sp}}$ (Period 3) | 9.536 | 4.168 | −4.291 | 49.595 | 0.344 | 778 |

${\mathrm{VRP}}_{\mathrm{sp}}$ (Period 4) | 6.441 | 2.115 | −0.722 | 5.156 | 0.212 | 544 |

${\mathrm{VRP}}_{\mathrm{sp}}$ (Period 5) | 6.616 | 2.801 | −2.395 | 24.814 | 0.222 | 661 |

${\mathrm{VRP}}_{\mathrm{oil}}$ (Whole) | 4.202 | 6.627 | −0.230 | 5.047 | 0.273 | 2516 |

${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 1) | 2.529 | 5.842 | 0.161 | 2.596 | 0.097 | 266 |

${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 2) | 3.623 | 9.754 | −0.061 | 3.518 | 0.278 | 267 |

${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 3) | 5.846 | 6.317 | 0.135 | 4.315 | 0.344 | 778 |

${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 4) | 4.231 | 4.619 | 0.491 | 3.134 | 0.212 | 544 |

${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 5) | 3.151 | 6.724 | −0.929 | 5.407 | 0.222 | 661 |

ADF | DF-GLS | PP | |
---|---|---|---|

${\mathrm{VRP}}_{\mathrm{sp}}$ (Whole) | −11.991 *** | −9.888 *** | −1485.051 *** |

${\mathrm{VRP}}_{\mathrm{sp}}$ (Period 1) | −4.928 *** | −4.815 *** | −130.293 *** |

${\mathrm{VRP}}_{\mathrm{sp}}$ (Period 2) | −3.623 *** | −3.938 *** | −180.399 *** |

${\mathrm{VRP}}_{\mathrm{sp}}$ (Period 3) | −8.022 *** | −7.041 *** | −517.148 *** |

${\mathrm{VRP}}_{\mathrm{sp}}$ (Period 4) | −7.744 *** | −7.827 *** | −443.083 *** |

${\mathrm{VRP}}_{\mathrm{sp}}$ (Period 5) | −8.864 *** | −8.651 *** | −450.466 *** |

${\mathrm{VRP}}_{\mathrm{oil}}$ (Whole) | −11.445 *** | −11.071 *** | −370.718 *** |

${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 1) | −4.447 *** | −4.450 *** | −43.848 *** |

${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 2) | −4.476 *** | −4.656 *** | −62.622 *** |

${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 3) | −5.895 *** | −5.917 *** | −111.345 *** |

${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 4) | −3.628 *** | −3.864 *** | −32.445 *** |

${\mathrm{VRP}}_{\mathrm{oil}}$ (Period 5) | −5.982 *** | −5.598 *** | −92.555 *** |

AIC | HQIC | SBIC | Selected Length | |
---|---|---|---|---|

Whole Period | 18 | 5 | 2 | 5 |

Period 1 | 1 | 1 | 1 | 1 |

Period 2 | 2 | 2 | 2 | 2 |

Period 3 | 3 | 3 | 3 | 3 |

Period 4 | 2 | 2 | 2 | 2 |

Period 5 | 7 | 2 | 2 | 2 |

Null Hypothesis | Period | Chi 2 | # of Lags |
---|---|---|---|

${\mathrm{VRP}}_{\mathrm{sp}}$ does not GC ${\mathrm{VRP}}_{\mathrm{oil}}$ | Whole | 21.174 *** | 5 |

${\mathrm{VRP}}_{\mathrm{oil}}$ does not GC ${\mathrm{VRP}}_{\mathrm{sp}}$ | Whole | 26.076 *** | 5 |

Impulse | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{o}\mathbf{i}\mathbf{l}}$ | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{s}\mathbf{p}}$ | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{o}\mathbf{i}\mathbf{l}}$ | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{s}\mathbf{p}}$ |
---|---|---|---|---|

Response | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{o}\mathbf{i}\mathbf{l}}$ | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{s}\mathbf{p}}$ | ||

1 | 1 | 0 | 0.005 | 0.995 |

5 | 0.996 | 0.004 | 0.020 | 0.980 |

10 | 0.986 | 0.014 | 0.037 | 0.963 |

15 | 0.977 | 0.023 | 0.048 | 0.952 |

20 | 0.973 | 0.027 | 0.052 | 0.948 |

Null Hypothesis | Period | Chi 2 | # of Lags |
---|---|---|---|

${\mathrm{VRP}}_{\mathrm{sp}}$ does not GC ${\mathrm{VRP}}_{\mathrm{oil}}$ | Period 1 | 1.214 | 1 |

Period 2 | 1.011 | 2 | |

Period 3 | 35.073 *** | 3 | |

Period 4 | 1.439 | 2 | |

Period 5 | 4.786 * | 2 | |

${\mathrm{VRP}}_{\mathrm{oil}}$ does not GC ${\mathrm{VRP}}_{\mathrm{sp}}$ | Period 1 | 0.024 | 1 |

Period 2 | 6.861 ** | 2 | |

Period 3 | 27.029 *** | 3 | |

Period 4 | 18.452 *** | 2 | |

Period 5 | 9.066 ** | 2 |

Impulse | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{o}\mathbf{i}\mathbf{l}}$ | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{s}\mathbf{p}}$ | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{o}\mathbf{i}\mathbf{l}}$ | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{s}\mathbf{p}}$ |
---|---|---|---|---|

Response | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{o}\mathbf{i}\mathbf{l}}$ | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{s}\mathbf{p}}$ | ||

Period 1 | ||||

1 | 1 | 0 | 0.004 | 0.996 |

5 | 0.992 | 0.008 | 0.005 | 0.995 |

10 | 0.989 | 0.011 | 0.005 | 0.995 |

15 | 0.989 | 0.011 | 0.005 | 0.995 |

20 | 0.989 | 0.011 | 0.005 | 0.995 |

Period 2 | ||||

1 | 1 | 0 | 0.004 | 0.996 |

5 | 0.995 | 0.005 | 0.044 | 0.956 |

10 | 0.992 | 0.008 | 0.070 | 0.930 |

15 | 0.991 | 0.009 | 0.078 | 0.922 |

20 | 0.991 | 0.009 | 0.080 | 0.920 |

Period 3 | ||||

1 | 1 | 0 | 0.015 | 0.985113 |

5 | 0.981 | 0.019 | 0.056 | 0.944 |

10 | 0.985 | 0.015 | 0.095 | 0.905 |

15 | 0.985 | 0.015 | 0.110 | 0.890 |

20 | 0.985 | 0.015 | 0.114 | 0.886 |

Period 4 | ||||

1 | 1 | 0 | 0.001 | 0.999 |

5 | 0.999 | 0.001 | 0.027 | 0.973 |

10 | 0.998 | 0.002 | 0.037 | 0.963 |

15 | 0.998 | 0.002 | 0.044 | 0.956 |

20 | 0.998 | 0.002 | 0.048 | 0.952 |

Period 5 | ||||

1 | 1 | 0 | 0.009 | 0.991 |

5 | 0.990 | 0.010 | 0.032 | 0.968 |

10 | 0.986 | 0.014 | 0.037 | 0.963 |

15 | 0.985 | 0.015 | 0.039 | 0.961 |

20 | 0.985 | 0.015 | 0.040 | 0.960 |

**Table 9.**Variance decompositions with order ${\mathrm{VRP}}_{\mathrm{sp}}$ before ${\mathrm{VRP}}_{\mathrm{oil}}$. (Effects on the 20th trading day after the shock).

Impulse | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{o}\mathbf{i}\mathbf{l}}$ | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{s}\mathbf{p}}$ | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{o}\mathbf{i}\mathbf{l}}$ | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{s}\mathbf{p}}$ |
---|---|---|---|---|

Response | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{o}\mathbf{i}\mathbf{l}}$ | $\mathbf{V}\mathbf{R}{\mathbf{P}}_{\mathbf{s}\mathbf{p}}$ | ||

Period 1 | ||||

20 | 0.975 | 0.025 | 0 | 1 |

Period 2 | ||||

20 | 0.979 | 0.021 | 0.065 | 0.935 |

Period 3 | ||||

20 | 0.961 | 0.039 | 0.084 | 0.916 |

Period 4 | ||||

20 | 0.995 | 0.005 | 0.045 | 0.955 |

Period 5 | ||||

20 | 0.956 | 0.044 | 0.023 | 0.977 |

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## Share and Cite

**MDPI and ACS Style**

Nakamura, N.; Ohashi, K.; Yokouchi, D.
Dynamic Relationship between Volatility Risk Premia of Stock and Oil Returns. *J. Risk Financial Manag.* **2023**, *16*, 173.
https://doi.org/10.3390/jrfm16030173

**AMA Style**

Nakamura N, Ohashi K, Yokouchi D.
Dynamic Relationship between Volatility Risk Premia of Stock and Oil Returns. *Journal of Risk and Financial Management*. 2023; 16(3):173.
https://doi.org/10.3390/jrfm16030173

**Chicago/Turabian Style**

Nakamura, Nobuhiro, Kazuhiko Ohashi, and Daisuke Yokouchi.
2023. "Dynamic Relationship between Volatility Risk Premia of Stock and Oil Returns" *Journal of Risk and Financial Management* 16, no. 3: 173.
https://doi.org/10.3390/jrfm16030173