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

^{1}

^{2}

^{*}

## 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

- Black, F.; Scholes, M. The Pricing of Options and Corporate Liabilities. J. Political Econ.
**1973**, 81, 637–654. [Google Scholar] [CrossRef] [Green Version] - Merton, R.C. Theory of Rational Option Pricing. Bell J. Econ. Manag. Sci.
**1973**, 4, 141–183. [Google Scholar] [CrossRef] [Green Version] - Barndorff-Nielsen, O.E.; Shephard, N. Non-Gaussian Ornstein-Uhlenbeck-Based Models and Some of Their Uses in Financial Economics. J. R. Stat. Soc. Ser. (Stat. Methodol.)
**2001**, 63, 167–241. [Google Scholar] [CrossRef] - Hubalek, F.; Posedel, P. Joint analysis and estimation of stock prices and trading volume in Barndorff-Nielsen and Shephard stochastic volatility models. Quant. Financ.
**2011**, 11, 917–932. [Google Scholar] [CrossRef] [Green Version] - Pindyck, R. Risk, Inflation, and the Stock Market. Am. Econ. Rev.
**1984**, 74, 335–351. [Google Scholar] - Brunnermeier, M.K.; Sannikov, Y. A Macroeconomic Model with a Financial Sector. Am. Econ. Rev.
**2014**, 104, 379–421. [Google Scholar] [CrossRef] [Green Version] - Miyajima, K.; Shim, I. Asset managers in emerging market economies. BIS Q. Rev.
**2014**, 19–34. Available online: https://www.bis.org/publ/qtrpdf/r_qt1409e.htm (accessed on 1 June 2021). - Engle, R.F.; NG, V.K. Measuring and Testing the Impact of News on Volatility. J. Financ.
**1993**, 48, 1749–1778. [Google Scholar] [CrossRef] - Duan, J.C. The GARCH option pricing model. Math. Financ.
**1995**, 5, 13–32. [Google Scholar] [CrossRef] - Bollerslev, T.; Gibson, M.; Zhou, H. Dynamic estimation of volatility risk premia and investor risk aversion from option-implied and realized volatilities. J. Econom.
**2011**, 160, 235–245. [Google Scholar] [CrossRef] [Green Version] - Corradi, V.; Distaso, W.; Mele, A. Macroeconomic determinants of stock volatility and volatility premiums. J. Monet. Econ.
**2013**, 60, 203–220. [Google Scholar] [CrossRef] - Negrea, B. The Volatility Premium Risk: Valuation and Forecasting. J. Appl. Quant. Methods
**2009**, 4, 154–165. [Google Scholar] - Christoffersen, P.; Heston, S.; Jacobs, K. Capturing option anomalies with a variance-dependent pricing kernel. Rev. Financ. Stud.
**2013**, 26, 1963–2006. [Google Scholar] [CrossRef] - Zhang, W.; Zhang, J.E. GARCH Option Pricing Models and the Variance Risk Premium. J. Risk Financ. Manag.
**2020**, 13, 51. [Google Scholar] [CrossRef] [Green Version] - Duan, J.C. Cracking the smile. Risk
**1996**, 9, 55–59. [Google Scholar] - Christoffersen, P.; Jacobs, K. Which GARCH Model for Option Valuation? Manag. Sci.
**2004**, 50, 1204–1221. [Google Scholar] [CrossRef] [Green Version] - Fouque, J.P.; Papanicolaou, G.; Sircar, K.R. Derivatives in Financial Markets with Stochastic Volatility; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Hobson, D.G. Stochastic Volatility; School of Mathematical Sciences, University of Bath: Bath, UK, 1996. [Google Scholar]
- Javaheri, A. Inside Volatility Arbitrage: The Secrets of Skewness; John Wiley & Sons: Hoboken, NJ, USA, 2011; Volume 317. [Google Scholar]
- Bollerslev, T. Generalized autoregressive conditional heteroskedasticity. J. Econom.
**1986**, 31, 307–327. [Google Scholar] [CrossRef] [Green Version] - Posedel, P. Analysis of the exchange rate and pricing foreign currency options on the croatian market: The ngarch model as an alternative to the black-scholes model. Financ. Theory Pract.
**2006**, 30, 347–368. [Google Scholar] - Duan, J.C. Augmented GARCH (p, q) process and its diffusion limit. J. Econom.
**1997**, 79, 97–127. [Google Scholar] [CrossRef] - Peetz, D.; Mall, G. Why Bitcoin is Not a Currency But a Speculative Real Asset. 2017. Available online: https://ssrn.com/abstract=3098765 (accessed on 1 June 2021). [CrossRef]
- MATLAB; Version 2021a; The MathWorks Inc.: Natick, MA, USA, 2021.
- Siu, T.K.; Elliott, R.J. Bitcoin option pricing with a SETAR-GARCH model. Eur. J. Financ.
**2020**, 27, 564–595. [Google Scholar] [CrossRef] - Greene, W.H. Econometric Analysis, 4th ed.; International, Ed.; Prentice Hall: Hoboken, NJ, USA, 2000; pp. 201–215. [Google Scholar]

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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