# Pricing the Volatility Risk Premium with a Discrete Stochastic Volatility Model

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Stochastic Volatility and the Market Price of Risk

#### 3.1. Volatility Risk Premium

#### 3.2. NGARCH Stochastic Volatility Model

**Definition**

**1.**

- (i)
- $\frac{{S}_{k}}{{S}_{k-1}}\phantom{\rule{0.166667em}{0ex}}|{\mathcal{F}}_{k-1}$ is lognormally distributed (under $\tilde{\mathbb{P}}$),
- (ii)
- ${\mathrm{Var}}^{\tilde{\mathbb{P}}}\left[ln\left(\frac{{S}_{k}}{{S}_{k-1}}\right)\phantom{\rule{0.166667em}{0ex}}|{\mathcal{F}}_{k-1}\right]={\mathrm{Var}}^{\mathbb{P}}\left[ln\left(\frac{{S}_{k}}{{S}_{k-1}}\right)\phantom{\rule{0.166667em}{0ex}}|{\mathcal{F}}_{k-1}\right]$, almost surely with respect to $\mathbb{P}$,

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

## 4. Model Estimation and Numerical Results

`fmincon`function (Matlab code can be provided by the authors upon reasonable request). The iterative algorithm was tested for robustness from various initial values, and ultimately the initial values were adapted from [16,25] for SPX and BTC, respectively. The one-period risk-free interest rate r was calculated according to [25], using the US Treasury Bill Rate for 13 weeks bank discount on 31 May 2018, which was 1.89% for a 360-day year. Therefore, $r=0.0189/360=5.25\times {10}^{-5}$. Standard errors used for the (one-sided) test of parameter significance were calculated from an approximation of the Fisher information matrix (see Appendix A).

#### 4.1. Parameter Estimates and Interpretation

#### 4.2. Model Fit

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**(

**a**): S&P 500 returns and the asset risk premium. (

**b**): S&P 500 variance and the volatility risk premium.

**Figure 2.**(

**a**): Bitcoin returns and the asset risk premium. (

**b**): Bitcoin variance and the volatility risk premium.

**Figure 5.**Autocorrelation function for the squared returns for (

**a**) SPX and (

**b**) BTC data and squared standard residuals for (

**c**) SPX and (

**d**) BTC data.

SPX | BTC | |||||
---|---|---|---|---|---|---|

Par. | Est. Value | Std. Error | Z-Stat. | Est. Value | Std. Error | Z-Stat. |

$\widehat{{\omega}_{0}}$ | 3.131 $\times {10}^{-6}$ | $3.411\times {10}^{-7}$ | 9.1783 | $1.736\times {10}^{-4}$ | $4.928\times {10}^{-6}$ | 35.2208 |

$\widehat{\alpha}$ | 0.1106 | 0.0122 | 9.0585 | 0.1657 | 0.0057 | 29.3083 |

$\widehat{\beta}$ | 0.6768 | 0.0199 | 34.0946 | 0.7962 | 0.0037 | 217.9210 |

$\widehat{c}$ | 1.3328 | 0.1243 | 10.7268 | 0.2504 | 0.0272 | 9.1968 |

$\widehat{\mu}$ | $2.325\times {10}^{-4}$ | $1.498\times {10}^{-4}$ | 1.5521 | 0.0039 | 0.0011 | 3.6623 |

Persistence | 0.9838 | 0.9723 | ||||

Log-likelihood | −8697.73 | −7145.99 |

Min ($\widehat{\mathit{\lambda}}$) | Max ($\widehat{\mathit{\lambda}}$) | Med. ($\widehat{\mathit{\lambda}}$) | Mean ($\widehat{\mathit{\lambda}}$) | St.dev. ($\widehat{\mathit{\lambda}}$) | Skew. ($\widehat{\mathit{\lambda}}$) | Kurt. ($\widehat{\mathit{\lambda}}$) | |
---|---|---|---|---|---|---|---|

SPX | −0.0441 | −0.0039 | −0.0230 | −0.0227 | 0.0079 | 0.0173 | 2.2497 |

BTC | −0.0428 | −0.0019 | −0.0294 | −0.0279 | 0.0100 | 0.4793 | 2.2233 |

Mean ($\widehat{\mathit{\epsilon}}$) | St.dev. ($\widehat{\mathit{\epsilon}}$) | Skewness ($\widehat{\mathit{\epsilon}}$) | Kurtosis ($\widehat{\mathit{\epsilon}}$) | |
---|---|---|---|---|

SPX | 0.0145 | 0.9994 | −0.5663 | 5.0493 |

BTC | 0.0121 | 1.0008 | 3.6860 | 93.2303 |

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Posedel Šimović, P.; Tafro, A. Pricing the Volatility Risk Premium with a Discrete Stochastic Volatility Model. *Mathematics* **2021**, *9*, 2038.
https://doi.org/10.3390/math9172038

**AMA Style**

Posedel Šimović P, Tafro A. Pricing the Volatility Risk Premium with a Discrete Stochastic Volatility Model. *Mathematics*. 2021; 9(17):2038.
https://doi.org/10.3390/math9172038

**Chicago/Turabian Style**

Posedel Šimović, Petra, and Azra Tafro. 2021. "Pricing the Volatility Risk Premium with a Discrete Stochastic Volatility Model" *Mathematics* 9, no. 17: 2038.
https://doi.org/10.3390/math9172038