# Volatility Is Log-Normal—But Not for the Reason You Think

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## Abstract

**:**

## 1. Introduction: Seeing and Believing

## 2. Theory: Three Pieces

#### 2.1. Continuous-Time Stochastic Volatility Models

#### 2.2. Estimation of Discretely Observed Diffusion Processes

#### Goodness of Fit and Uniform Residuals

#### 2.3. Measuring Volatility

#### Realized Volatility for an Asset Price with Jumps

## 3. Simulation: Model Discrimination

#### S&P 500 Data Revisited

## 4. Empirics: Horseraces between Models

# obs | IBM | MCD | CAT | MMM | MCO | XOM | AZN | GS | HPQ | FDX |
---|---|---|---|---|---|---|---|---|---|---|

price | 246,308 | 328,111 | 328,107 | 328,108 | 271,580 | 288,846 | 302,350 | 300,759 | 239,356 | 328,104 |

variance | 1282 | 1708 | 1708 | 1708 | 1414 | 1504 | 1574 | 1566 | 1246 | 1708 |

#### 4.1. Individual Stocks

#### 4.2. Jump Corrections

## 5. Conclusions and Future Work

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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1. | Often, we will use the term (instantaneous) volatility. This means taking the square root of the instantaneous variance and annualizing by a multiplying factor $\sqrt{252}$, where 252 represents the (approximate) number of trading days in a year. The annualization is a convention, not a theorem; when returns are not iid—which is the case for mean-reverting stochastic volatility models—we do not get the standard deviation of yearly returns by multiplying with $\sqrt{252}$. Notice further that log-normal models are stable to roots and squares: it is not misleading to say “volatility is log-normal” even though we model variance. The Heston and the 3-over-2 models do not posses this stability. |

2. | A further complication is that for the models we consider, not all parameters are identifiable by their stationary distribution. |

3. | A detailed derivation of this is in Cairns (2004, Appendix B.2). |

4. | Notice that stationary distribution depends on $\kappa $ and $\epsilon $ only through ${\epsilon}^{2}/\kappa $. This was what we meant when we talked about identification problems in an earlier footnote. |

5. | The first moment of the inverse-gamma distribution is defined if the shape parameter is greater than one, a condition certified by the Feller condition. In this case $\mathbb{E}\left[V(t)\right]=2\tilde{\kappa}/(2\tilde{\kappa}\tilde{\theta}-{\epsilon}^{2})=2\kappa \theta /(2\kappa +{\epsilon}^{2})$. |

6. | Older readers may remember that Chan et al. (1992), a very influential paper in the 90’ies, estimated the volatility of the short-term interest rate, r to be of the form $1.18{r}^{1.48}$, i.e., almost exactly 3/2, but their model had an affine drift, which (7) doesn’t. |

7. | The convergence holds for any partition whose mesh size goes to 0. |

8. | Although, as Christensen et al. (2014) points out, perhaps not as restrictive as believed in some areas in the literature. |

9. | The presence of the $\pi /2$-factor indicates that this is not as obvious as it might seem. |

10. | We have been trying to find at least a heuristic explanation for this small (or coarse) sample log-normality, but have not managed to come up with anything particularly convincing. |

11. | The two-day aggregation of 5-min returns may seem like a strange choice; one-day aggregation is common in the literature and appears the natural choice. However, our experiments (not reported here but see Jönsson (2016) for more details) show that one-day aggregation of 5-min returns diminishes model separation power: lower noise is indeed worth an increased bias. |

12. | Quantitative evidence: In https://tinyurl.com/y9epg6a7 we simulate a log-normal volatility model with the parameters from Table 3 in this paper at 5-min intervals and add market microstructure noise using the parameter specification in Table 2 in Christensen et al. (2014). We find that the average absolute deviation between true volatility and volatility measured by noise-free realized variance is about 75 times higher than the average absolute difference between realized variances with and without noise. |

13. | We are not allowed to make the high-frequency data public. |

**Figure 1.**Estimated empirical density of the annualized measured instantaneous S&P500 variance (4552 daily observations 1996–2013 using Equation (1)) together with fitted log-normal and gamma densities.

**Figure 2.**Fitted empirical, log-normal and gamma densities to measured variance with $\lambda =0.94$ of 4532 simulated prices with Heston’s (

**left**figure) and the log-normal model (

**right**figure).

**Figure 3.**The left figure shows the S&P 500 ETF price from 2 January 1996 to 31 December 2013 with a 5-min. frequency and a total of 368,724 observations. Right figure: variance of the S&P 500 ETF price measured by realized volatility with frequency $n=192$ and a total of 1920 observations.

**Figure 4.**Quantile plots of the uniform $[0,1]$ distribution from uniform residuals calculated with measured S&P 500 variance.

**Table 1.**Impact of stochastic volatility research measured by citations in Google Scholar on 15 February 2018.

Model Type | Source | # Citations (Yearly Rate) |
---|---|---|

Affine, basic | Heston (1993) | 7637 (310) |

Affine, fancy | Duffie et al. (2000) | 3009 (171) |

Log-normal, no mean-reversion $\kappa $ | Hagan et al. (2002) | 974 (62) |

Log-normal, fast mean-reversion $\kappa $ | Fouque et al. (2000) | 175 (10) |

3-over-2 | Carr and Sun (2007) | 94 (9) |

All of the above | Lewis (2000) | 766 (43) |

**Table 2.**Discriminatory power of Kolmogorov-Smirnov tests on uniform residuals for different methods of variance measurement: p-values from the chi-square test under the hypothesis indicated in the test alternative column.

True Model: Heston | Variance Measurement | ||
---|---|---|---|

Test alternative | true simulated | 2-day, 5-min. realized | Equation (1) |

Heston | 0.79 | 0.57 | 0 |

log-normal | 0 | 0 | 0 |

3-over-2 | 0 | 0 | 0 |

True model: Log-normal | Variance measurement | ||

Test alternative | true simulated | 2-day, 5-min. realized | Equation (1) |

Heston | 0 | 0 | 0 |

log-normal | 0.72 | 0.78 | 0 |

3-over-2 | 0 | 0 | 0 |

True model: 3-over-2 | Variance measurement | ||

Test alternative | true simulated | 2-day, 5-min. realized | Equation (1) |

Heston | 0 | 0 | 0 |

log-normal | 0 | 0 | 0 |

3-over-2 | 0.26 | 0.90 | 0 |

**Table 3.**S&P 500: Estimated parameters and standard errors from (approximate) maximum likelihood estimation of the Heston, log-normal, and 3-over-2 models.

$\widehat{\mathit{\kappa}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\epsilon}}$ | K-S p-Value | |
---|---|---|---|---|

Heston | 21.7 (3.63) | 0.022 (0.0049) | 2.98 (0.08) | 0.00 |

log-normal | 56.1 (3.36) | −3.65 (0.049) | 11.5 (0.24) | 0.68 |

3-over-2 | 2.7 (5.64) | 30.0 (62.7) | 98.2 (2.0) | 0.00 |

**Table 5.**Maximum likelihood estimated parameters and standard errors from the realized volatility measured variance with $n=192$.

Heston | IBM | MCD | CAT | MMM | MCO | XOM | AZN | GS | HPQ | FDX |
---|---|---|---|---|---|---|---|---|---|---|

$\widehat{\kappa}$ | 75.1 (7.2) | 18.2 (4.0) | 71.6 (5.3) | 37.0 (4.52) | 100.0 (6.85) | 45.4 (4.55) | 113.0 (7.92) | 41.6 (5.14) | 116.9 (12.1) | 65.2 (6.79) |

$\widehat{\theta}$ | 0.024 (0.004) | 0.052 (0.012) | 0.112 (0.009) | 0.061 (0.006) | 0.096 (0.009) | 0.052 (0.005) | 0.069 (0.005) | 0.027 (0.009) | 0.14 (0.012) | 0.082 (0.008) |

$\widehat{\epsilon}$ | 6.89 (0.4) | 4.14 (0.1) | 7.21 (0.2) | 3.89 (0.1) | 10.1 (0.4) | 3.61 (0.1) | 7.76 (0.3) | 7.97 (0.3) | 12.6 (0.6) | 6.62 (0.2) |

K/S p-value | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Lognormal | ||||||||||

$\widehat{\kappa}$ | 86.6 (6.0) | 61.1 (3.8) | 83.9 (5.1) | 76.9 (4.7) | 71.6 (4.8) | 72.3 (4.7) | 126.2 (8.6) | 58.3 (3.8) | 102.2 (7.4) | 88.0 (5.4) |

$\widehat{\theta}$ | −3.31 (0.05) | −2.96 (0.05) | −2.45 (0.04) | −3.08 (0.04) | −2.60 (0.05) | −3.10 (0.04) | −2.97 (0.04) | −2.49 (0.05) | −2.54 (0.05) | −2.63 (0.04) |

$\widehat{\epsilon}$ | 13.5 (0.4) | 11.4 (0.3) | 12.7 (0.3) | 12.1 (0.3) | 43.8 (0.3) | 11.3 (0.3) | 16.6 (0.5) | 11.6 (0.3) | 14.9 (0.5) | 12.8 (0.3) |

K/S p-value | 0.21 | 0.53 | 0.83 | 0.79 | 0.12 | 0.07 | 0.56 | 0.12 | 0.05 | 0.47 |

3-over-2(i) | ||||||||||

$\widehat{\kappa}$ | 105.0 (10.0) | 27.3 (4.2) | 77.7 (6.5) | 77.6 (7.5) | 48.8 (5.7) | 60.9 (5.8) | 152.0 (12.5) | 45.6 (6.0) | 105.8 (9.7) | 91.6 (7.5) |

$\widehat{\theta}$ | 40.0 (1.7) | 27.2 (3.0) | 16.3 (0.8) | 32.0 (1.5) | 19.2 (1.5) | 31.1 (1.6) | 30.6 (1.2) | 15.7 (1.3) | 18.7 (0.9) | 19.4 (0.8) |

$\widehat{\epsilon}$ | 95.2 (3.4) | 67.8 (1.7) | 60.7 (1.7) | 82.8 (2.5) | 61.9 (1.8) | 67.2 (1.9) | 122.5 (4.5) | 56.6 (1.6) | 71.9 (2.6) | 67.9 (2.0) |

K/S p-value | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

**Table 6.**S&P 500 ETF data: Estimated parameters, standard errors, and Kolmogorov-Smirnov p-values for realized bipower variation measured variance.

$\widehat{\mathit{\kappa}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\epsilon}}$ | K-S p-Value | |
---|---|---|---|---|

Heston | 21.4 (3.8) | 0.011 (0.004) | 2.78 (0.08) | 0.00 |

log-normal | 55.3 (3.3) | −3.9 (0.05) | 11.5 (0.2) | 0.17 |

3-over-2 | 25.4 (4.0) | 17.2 (8.8) | 142.4 (4.3) | 0.00 |

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Tegnér, M.; Poulsen, R.
Volatility Is Log-Normal—But Not for the Reason You Think. *Risks* **2018**, *6*, 46.
https://doi.org/10.3390/risks6020046

**AMA Style**

Tegnér M, Poulsen R.
Volatility Is Log-Normal—But Not for the Reason You Think. *Risks*. 2018; 6(2):46.
https://doi.org/10.3390/risks6020046

**Chicago/Turabian Style**

Tegnér, Martin, and Rolf Poulsen.
2018. "Volatility Is Log-Normal—But Not for the Reason You Think" *Risks* 6, no. 2: 46.
https://doi.org/10.3390/risks6020046