# Market Microstructure Effects on Firm Default Risk Evaluation

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## Abstract

**:**

## 1. Introduction

## 2. Non-Parametric Volatility Estimation under Microstructure Noise

## 3. Empirical Evidence

## 4. Firm Value Model in the Presence of Market Microstructure Noise

#### 4.1. Assets Value Dynamics and Default Assumptions

#### 4.2. Market Microstructure Noise on High-Frequency Equity Prices

## 5. Equity Volatility Estimation and Default Probability Computation

#### 5.1. Equity Volatility Estimation with High-Frequency Data

#### 5.2. Asset Volatility Calibration

#### 5.3. Default Probability Computation

## 6. Sensitivity Analysis

#### 6.1. Leverage Effect

#### 6.2. Average Jumps Size

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Time plot of the 1-min equity log-returns for Alcoa company over the period 10 November 2015 to 23 May 2016.

**Figure 2.**Histogram of the 1-min equity log-returns for Alcoa company over the period 10 November 2015 to 23 May 2016.

**Figure 3.**Time plot of the 5-years CDS spreads in basis points (lower panel) and corresponding equity volatility (upper panel) for our sample of 109 days after jump removal.

**Figure 4.**Asset volatility calibration and default probability computation. The plot shows how the estimated equity volatility ${\Sigma}_{t}^{E}$ can affect the resulting asset volatility calibration and default probability computation. Subplot (i) reports the underlying asset volatility calibrated for different values of ${\Sigma}_{t}^{E}$; subplot (ii) shows the absolute error on default probability when a 3y horizon is considered. The base case is ${\Sigma}_{t}^{s}=37.21\%,$${v}_{t}=25.01\%$ (red dashed line in Subplot (i)) and the underlying asset volatility is calibrated by matching the 5y default probability to highlight the behaviour of our calibration algorithm. The absolute error on the 3y-default probability is defined through the variable $D{P}_{AbsErr}:=DP\left({\Sigma}_{t}^{E}\right)-DP\left({\Sigma}_{t}^{s}\right)$ and reported in percentage (%).

**Figure 5.**Equity log-returns. Time plot of the tick-by-tick normalized equity log-returns and the ACF on three sample days. (

**A**) day with no jumps and first order autocorrelation structure; (

**B**) day with jumps and first order autocorrelation structure; (

**C**) day with no jumps and autocorrelation up to lag 15.

**Figure 6.**Intraday returns of equity log-prices. Comparison of the histogram of the normalized intraday returns and the density of $N(0,1)$. (

**A**) day with no jumps and first order autocorrelation structure; (

**B**) day with jumps and first order autocorrelation structure; (

**C**) day with no jumps and autocorrelation up to lag 15.

**Figure 7.**Heston model: default probability. The plot reports the average results obtained by computing default probabilities according to Equation (5). For each day in the sample and for each volatility estimator E we use the asset volatility ${\widehat{{v}_{t}}}^{E}$, calibrated as explained in the text, in order to compute the default probabilities for any maturity (from 1 to 5 years). Results are based on 1000 daily Monte Carlo simulations; trading noise is defined by Equation (6).

**Figure 8.**Bates model: default probability relative error. The plot shows default probability mean relative error given in Equation (9) for maturities from 1 to 5 years. Results are based on a sample of 934 days for our Monte Carlo simulations (due to jumps removal) and refer to trading noise (6).

**Figure 9.**Bates model: default probability relative error. The plot shows default probability mean relative error given in Equation (9) for maturities from 1 to 5 years. Results are based on a sample of 901 days for our Monte Carlo simulations (due to jumps removal) and refer to trading noise (7).

**Table 1.**Summary statistics of the 1-min transaction data on equity prices for the Alcoa company in the period from 10 November 2015 to 23 May 2016 (52,166 trades). ”Std. Dev.” denotes the sample standard deviation of the variable.

Variable | Mean | Std. Dev. | Min | Max |
---|---|---|---|---|

AA US Equity price | 9.02 | 1.05 | 6.15 | 11.49 |

log-return (%) | 1.68 × 10${}^{-4}$ | 1.78 × 10${}^{-1}$ | −4.66 × 10${}^{0}$ | 3.73 × 10${}^{0}$ |

**Table 2.**Summary statistics of the daily CDS premium data (basis points) for Alcoa company in the period from 10 November 2015 to 23 May 2016. ”Std. Dev.” denotes the sample standard deviation of the variable.

Variable | Mean | Std. Dev. | Min | Max |
---|---|---|---|---|

CDS premium | 433.87 | 91.09 | 329.24 | 661.63 |

Estimator | $\widehat{{\varphi}_{0}}$ | $\widehat{{\varphi}_{1}}$ | ${R}^{2}$ |
---|---|---|---|

${\Sigma}_{t}^{RV}$ | 2.5476 | 4.0915 | 0.3810 |

Std. Dev. | (0.2312) | (0.5042) | |

T-statistics | (11.0173) | (8.1155) | |

${\Sigma}_{t}^{TS}$ | 2.0615 | 4.9657 | 0.4118 |

Std. Dev. | (0.2715) | (0.5737) | |

T-statistics | (7.5916) | (8.6556) | |

${\Sigma}_{t}^{HL}$ | 2.4268 | 4.4108 | 0.3884 |

Std. Dev. | (0.2418) | (0.5350) | |

T-statistics | (10.0355) | (8.2447) | |

${\Sigma}_{t}^{K}$ | 2.5293 | 4.2330 | 0.3797 |

Std. Dev. | (0.2340) | (0.5229) | |

T-statistics | (10.8107) | (8.0944) | |

${\Sigma}_{t}^{PA}$ | 2.6140 | 4.3276 | 0.3635 |

Std. Dev. | (0.2314) | (0.5535) | |

T-statistics | (11.2956) | (7.8184) | |

${\Sigma}_{t}^{F}$ | 1.9735 | 5.1398 | 0.4236 |

Std. Dev. | (0.2749) | (0.5796) | |

T-statistics | (7.1788) | (8.8677) | |

${\Sigma}_{t}^{BV}$ | 2.5891 | 4.1263 | 0.3801 |

Std. Dev. | (0.2267) | (0.5094) | |

T-statistics | (11.4188) | (8.1010) |

Parameter | Bates | Heston |
---|---|---|

${A}_{0}$ | 100 | 100 |

r | $5\%$ | $5\%$ |

${\Sigma}_{e}$ | $37.21\%$ | $37.21\%$ |

${PD}_{5y}$ | $1.37\%$ | $1.37\%$ |

B | $43.13$ | $43.13$ |

δ | $2\%$ | $2\%$ |

${\pi}_{a}$ | $4.04\%$ | $3.49\%$ |

${v}_{0}$ | $25.01\%$ | $21.65\%$ |

k | $0.87$ | $0.74$ |

θ | $3.8\%$ | $4.24\%$ |

σ | $4.42\%$ | $4.01\%$ |

ρ | $-24.04\%$ | $-24.02\%$ |

λ | $8.88\%$ | |

${\mu}_{J}$ | $2.35\%$ | |

${\sigma}_{J}$ | $14.79\%$ |

Heston | ${\Sigma}_{t}^{\mathit{BV}}$ | ${\Sigma}_{t}^{\mathit{RV}}$ | ${\Sigma}_{t}^{\mathit{RVSS}}$ | ${\Sigma}_{t}^{\mathit{TS}}$ | ${\Sigma}_{t}^{\mathit{HL}}$ | ${\Sigma}_{t}^{K}$ | ${\Sigma}_{t}^{\mathit{PA}}$ | ${\Sigma}_{t}^{F}$ | |
---|---|---|---|---|---|---|---|---|---|

(a) | MSE | 1.79 × 10${}^{-3}$ | 1.31 × 10${}^{-2}$ | 1.65 × 10${}^{-3}$ | 3.43 × 10${}^{-4}$ | 1.05 × 10${}^{-3}$ | 4.37 × 10${}^{-4}$ | 5.10 × 10${}^{-4}$ | 5.09 × 10${}^{-4}$ |

BIAS | 2.35 × 10${}^{-2}$ | 1.13 × 10${}^{-1}$ | 2.62 × 10${}^{-2}$ | –1.47 × 10${}^{-3}$ | –9.48 × 10${}^{-4}$ | −7.69 × 10${}^{-4}$ | –4.77 × 10${}^{-3}$ | 4.57 × 10${}^{-3}$ | |

(b) | MSE | 2.42 × 10${}^{-3}$ | 2.28 × 10${}^{-2}$ | 2.30 × 10${}^{-3}$ | 3.58 × 10${}^{-4}$ | 1.66 × 10${}^{-3}$ | 4.64 × 10${}^{-4}$ | 5.05 × 10${}^{-4}$ | 5.71 × 10${}^{-4}$ |

BIAS | 2.84 × 10${}^{-2}$ | 1.48 × 10${}^{-1}$ | 3.21 × 10${}^{-2}$ | −7.66 × 10${}^{-4}$ | 5.10 × 10${}^{-4}$ | –3.72 × 10${}^{-4}$ | −4.20 × 10${}^{-3}$ | 6.06 × 10${}^{-3}$ |

Bates | ${\Sigma}_{t}^{\mathit{BV}}$ | ${\Sigma}_{t}^{\mathit{RV}}$ | ${\Sigma}_{t}^{\mathit{RVSS}}$ | ${\Sigma}_{t}^{\mathit{TS}}$ | ${\Sigma}_{t}^{\mathit{HL}}$ | ${\Sigma}_{t}^{K}$ | ${\Sigma}_{t}^{\mathit{PA}}$ | ${\Sigma}_{t}^{F}$ | |
---|---|---|---|---|---|---|---|---|---|

(a) | MSE | 2.77 × 10${}^{-3}$ | 1.63 × 10${}^{-2}$ | 2.36 × 10${}^{-3}$ | 4.69 × 10${}^{-4}$ | 1.23 × 10${}^{-3}$ | 5.44 × 10${}^{-4}$ | 5.50 × 10${}^{-4}$ | 6.75 × 10${}^{-4}$ |

BIAS | 3.16 × 10${}^{-2}$ | 1.24 × 10${}^{-1}$ | 3.19 × 10${}^{-2}$ | 7.18 × 10${}^{-4}$ | 2.08 × 10${}^{-3}$ | 2.10 × 10${}^{-3}$ | -2.45 × 10${}^{-3}$ | 8.40 × 10${}^{-3}$ | |

(b) | MSE | 3.36 × 10${}^{-3}$ | 2.78 × 10${}^{-2}$ | 3.12 × 10${}^{-3}$ | 6.37 × 10${}^{-4}$ | 1.88 × 10${}^{-3}$ | 7.18 × 10${}^{-4}$ | 7.51 × 10${}^{-4}$ | 8.91 × 10${}^{-4}$ |

BIAS | 3.40 × 10${}^{-2}$ | 1.63 × 10${}^{-1}$ | 3.71 × 10${}^{-2}$ | –6.38 × 10${}^{-4}$ | 1.54 × 10${}^{-3}$ | 2.43 × 10${}^{-3}$ | –1.65 × 10${}^{-3}$ | 1.03 × 10${}^{-2}$ |

Heston | ||||||||
---|---|---|---|---|---|---|---|---|

Mean | Median | 10 perc. | 90 perc. | Min | Max | Std Dev. | ||

(a) | ${\widehat{{v}_{t}}}^{BV}$ | 0.2082 | 0.2056 | 0.1833 | 0.2350 | 0.1614 | 0.2960 | 0.0205 |

${\widehat{{v}_{t}}}^{RV}$ | 0.1704 | 0.1699 | 0.1599 | 0.1815 | 0.1453 | 0.2027 | 0.0085 | |

${\widehat{{v}_{t}}}^{RVSS}$ | 0.2063 | 0.2045 | 0.1838 | 0.2308 | 0.1583 | 0.2922 | 0.0182 | |

${\widehat{{v}_{t}}}^{TS}$ | 0.2214 | 0.2202 | 0.2035 | 0.2410 | 0.1831 | 0.2718 | 0.0151 | |

${\widehat{{v}_{t}}}^{HL}$ | 0.2229 | 0.2195 | 0.1950 | 0.2554 | 0.1764 | 0.3385 | 0.0243 | |

${\widehat{{v}_{t}}}^{K}$ | 0.2210 | 0.2202 | 0.2021 | 0.2406 | 0.1802 | 0.2736 | 0.0155 | |

${\widehat{{v}_{t}}}^{PA}$ | 0.2238 | 0.2228 | 0.2029 | 0.2453 | 0.1802 | 0.2774 | 0.0166 | |

${\widehat{{v}_{t}}}^{F}$ | 0.2178 | 0.2169 | 0.1984 | 0.2375 | 0.1776 | 0.2684 | 0.0156 | |

${\widehat{{v}_{t}}}^{BV}/{v}_{t}$ | 0.9617 | 0.9498 | 0.8464 | 1.0853 | 0.7454 | 1.3673 | 0.0948 | |

${\widehat{{v}_{t}}}^{RV}/{v}_{t}$ | 0.7870 | 0.7845 | 0.7387 | 0.8383 | 0.6709 | 0.9364 | 0.0392 | |

${\widehat{{v}_{t}}}^{RVSS}/{v}_{t}$ | 0.9528 | 0.9448 | 0.8489 | 1.0659 | 0.7310 | 1.3495 | 0.0840 | |

${\widehat{{v}_{t}}}^{TS}/{v}_{t}$ | 1.0226 | 1.0172 | 0.9400 | 1.1134 | 0.8456 | 1.2554 | 0.0696 | |

${\widehat{{v}_{t}}}^{HL}/{v}_{t}$ | 1.0297 | 1.0140 | 0.9005 | 1.1796 | 0.8149 | 1.5637 | 0.1123 | |

${\widehat{{v}_{t}}}^{K}/{v}_{t}$ | 1.0210 | 1.0169 | 0.9335 | 1.1113 | 0.8326 | 1.2637 | 0.0714 | |

${\widehat{{v}_{t}}}^{PA}/{v}_{t}$ | 1.0337 | 1.0290 | 0.9371 | 1.1329 | 0.8326 | 1.2815 | 0.0766 | |

${\widehat{{v}_{t}}}^{F}/{v}_{t}$ | 1.0061 | 1.0018 | 0.9164 | 1.0970 | 0.8203 | 1.2396 | 0.0722 | |

(b) | ${\widehat{{v}_{t}}}^{BV}$ | 0.2062 | 0.2040 | 0.1791 | 0.2368 | 0.1485 | 0.2902 | 0.0227 |

${\widehat{{v}_{t}}}^{RV}$ | 0.1602 | 0.1600 | 0.1503 | 0.1705 | 0.1361 | 0.1846 | 0.0079 | |

${\widehat{{v}_{t}}}^{RVSS}$ | 0.2036 | 0.2023 | 0.1802 | 0.2300 | 0.1571 | 0.2942 | 0.0196 | |

${\widehat{{v}_{t}}}^{TS}$ | 0.2209 | 0.2193 | 0.2021 | 0.2413 | 0.1781 | 0.2777 | 0.0153 | |

${\widehat{{v}_{t}}}^{HL}$ | 0.2234 | 0.2180 | 0.1909 | 0.2637 | 0.1658 | 0.3779 | 0.0309 | |

${\widehat{{v}_{t}}}^{K}$ | 0.2209 | 0.2193 | 0.2007 | 0.2429 | 0.1782 | 0.2942 | 0.0164 | |

${\widehat{{v}_{t}}}^{PA}$ | 0.2235 | 0.2226 | 0.2020 | 0.2462 | 0.1791 | 0.2959 | 0.0172 | |

${\widehat{{v}_{t}}}^{F}$ | 0.2170 | 0.2156 | 0.1967 | 0.2397 | 0.1760 | 0.2849 | 0.0165 | |

${\widehat{{v}_{t}}}^{BV}/{v}_{t}$ | 0.9391 | 0.9282 | 0.8200 | 1.0757 | 0.7079 | 1.2751 | 0.0984 | |

${\widehat{{v}_{t}}}^{RV}/{v}_{t}$ | 0.7298 | 0.7296 | 0.6947 | 0.7635 | 0.6308 | 0.8285 | 0.0273 | |

${\widehat{{v}_{t}}}^{RVSS}/{v}_{t}$ | 0.9276 | 0.9216 | 0.8227 | 1.0357 | 0.7396 | 1.3023 | 0.0838 | |

${\widehat{{v}_{t}}}^{TS}/{v}_{t}$ | 1.0061 | 1.0023 | 0.9348 | 1.0789 | 0.8673 | 1.2023 | 0.0553 | |

${\widehat{{v}_{t}}}^{HL}/{v}_{t}$ | 1.0176 | 0.9911 | 0.8757 | 1.1951 | 0.7826 | 1.8007 | 0.1353 | |

${\widehat{{v}_{t}}}^{K}/{v}_{t}$ | 1.0060 | 1.0006 | 0.9283 | 1.0914 | 0.8423 | 1.2411 | 0.0627 | |

${\widehat{{v}_{t}}}^{PA}/{v}_{t}$ | 1.0176 | 1.0121 | 0.9359 | 1.1062 | 0.8570 | 1.2606 | 0.0662 | |

${\widehat{{v}_{t}}}^{F}/{v}_{t}$ | 0.9883 | 0.9834 | 0.9103 | 1.0734 | 0.8388 | 1.2148 | 0.0642 |

Bates | ||||||||
---|---|---|---|---|---|---|---|---|

Mean | Median | 10 perc. | 90 perc. | Min | Max | Std Dev. | ||

(a) | ${\widehat{{v}_{t}}}^{BV}$ | 0.2526 | 0.2497 | 0.2076 | 0.3012 | 0.1558 | 0.3792 | 0.0381 |

${\widehat{{v}_{t}}}^{RV}$ | 0.2042 | 0.2038 | 0.1893 | 0.2211 | 0.1597 | 0.2502 | 0.0125 | |

${\widehat{{v}_{t}}}^{RVSS}$ | 0.2502 | 0.2477 | 0.2202 | 0.2839 | 0.1826 | 0.3503 | 0.0254 | |

${\widehat{{v}_{t}}}^{TS}$ | 0.2703 | 0.2701 | 0.2459 | 0.2962 | 0.2018 | 0.3360 | 0.0203 | |

${\widehat{{v}_{t}}}^{HL}$ | 0.2714 | 0.2673 | 0.2361 | 0.3140 | 0.1857 | 0.3739 | 0.0304 | |

${\widehat{{v}_{t}}}^{K}$ | 0.2694 | 0.2673 | 0.2444 | 0.2979 | 0.2007 | 0.3411 | 0.0215 | |

${\widehat{{v}_{t}}}^{PA}$ | 0.2730 | 0.2710 | 0.2470 | 0.3018 | 0.2019 | 0.3498 | 0.0223 | |

${\widehat{{v}_{t}}}^{F}$ | 0.2648 | 0.2624 | 0.2396 | 0.2928 | 0.1980 | 0.3340 | 0.0215 | |

${\widehat{{v}_{t}}}^{BV}/{v}_{t}$ | 0.9391 | 0.9241 | 0.7605 | 1.1260 | 0.5392 | 1.4802 | 0.1501 | |

${\widehat{{v}_{t}}}^{RV}/{v}_{t}$ | 0.7579 | 0.7573 | 0.7169 | 0.8034 | 0.5801 | 0.8881 | 0.0355 | |

${\widehat{{v}_{t}}}^{RVSS}/{v}_{t}$ | 0.9283 | 0.9208 | 0.8262 | 1.0473 | 0.6491 | 1.2119 | 0.0860 | |

${\widehat{{v}_{t}}}^{TS}/{v}_{t}$ | 1.0027 | 1.0020 | 0.9312 | 1.0833 | 0.7840 | 1.2004 | 0.0600 | |

${\widehat{{v}_{t}}}^{HL}/{v}_{t}$ | 1.0068 | 0.9931 | 0.8884 | 1.1467 | 0.7437 | 1.4164 | 0.1036 | |

${\widehat{{v}_{t}}}^{K}/{v}_{t}$ | 0.9994 | 0.9931 | 0.9222 | 1.0891 | 0.7794 | 1.2373 | 0.0659 | |

${\widehat{{v}_{t}}}^{PA}/{v}_{t}$ | 1.0127 | 1.0071 | 0.9315 | 1.1073 | 0.7852 | 1.2410 | 0.0684 | |

${\widehat{{v}_{t}}}^{F}/{v}_{t}$ | 0.9825 | 0.9771 | 0.9027 | 1.0745 | 0.7727 | 1.2113 | 0.0668 | |

(b) | ${\widehat{{v}_{t}}}^{BV}$ | 0.2521 | 0.2485 | 0.2067 | 0.3055 | 0.1616 | 0.3794 | 0.0387 |

${\widehat{{v}_{t}}}^{RV}$ | 0.1909 | 0.1909 | 0.1787 | 0.2043 | 0.1511 | 0.2252 | 0.0106 | |

${\widehat{{v}_{t}}}^{RVSS}$ | 0.2474 | 0.2450 | 0.2160 | 0.2825 | 0.1727 | 0.3627 | 0.0270 | |

${\widehat{{v}_{t}}}^{TS}$ | 0.2712 | 0.2694 | 0.2451 | 0.2990 | 0.2019 | 0.3480 | 0.0212 | |

${\widehat{{v}_{t}}}^{HL}$ | 0.2726 | 0.2682 | 0.2318 | 0.3228 | 0.1847 | 0.3797 | 0.0347 | |

${\widehat{{v}_{t}}}^{K}$ | 0.2692 | 0.2686 | 0.2424 | 0.2978 | 0.1918 | 0.3456 | 0.0225 | |

${\widehat{{v}_{t}}}^{PA}$ | 0.2726 | 0.2721 | 0.2446 | 0.3029 | 0.1921 | 0.3568 | 0.0238 | |

${\widehat{{v}_{t}}}^{F}$ | 0.2636 | 0.2633 | 0.2368 | 0.2924 | 0.1880 | 0.3481 | 0.0224 | |

${\widehat{{v}_{t}}}^{BV}/{v}_{t}$ | 0.9378 | 0.9222 | 0.7570 | 1.1482 | 0.5435 | 1.4866 | 0.1514 | |

${\widehat{{v}_{t}}}^{RV}/{v}_{t}$ | 0.7089 | 0.7096 | 0.6710 | 0.7494 | 0.5672 | 0.8206 | 0.0322 | |

${\widehat{{v}_{t}}}^{RVSS}/{v}_{t}$ | 0.9187 | 0.9101 | 0.8112 | 1.0389 | 0.6286 | 1.4060 | 0.0940 | |

${\widehat{{v}_{t}}}^{TS}/{v}_{t}$ | 1.0070 | 1.0024 | 0.9246 | 1.0972 | 0.7306 | 1.2813 | 0.0700 | |

${\widehat{{v}_{t}}}^{HL}/{v}_{t}$ | 1.0129 | 0.9981 | 0.8711 | 1.1864 | 0.6801 | 1.4773 | 0.1275 | |

${\widehat{{v}_{t}}}^{K}/{v}_{t}$ | 0.9997 | 0.9971 | 0.9115 | 1.0918 | 0.6901 | 1.2540 | 0.0724 | |

${\widehat{{v}_{t}}}^{PA}/{v}_{t}$ | 1.0120 | 1.0086 | 0.9222 | 1.1073 | 0.6911 | 1.2946 | 0.0766 | |

${\widehat{{v}_{t}}}^{F}/{v}_{t}$ | 0.9786 | 0.9767 | 0.8952 | 1.0681 | 0.6766 | 1.2632 | 0.0719 |

Heston | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{BV}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{RV}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{RVSS}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{TS}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{HL}}$ | ${\mathit{DP}}_{\mathit{Err}}^{K}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{PA}}$ | ${\mathit{DP}}_{\mathit{Err}}^{F}$ | |
---|---|---|---|---|---|---|---|---|---|

(a) | 1 y | −13.3336 | −92.8243 | −28.5112 | 21.9938 | 78.9713 | 23.4477 | 42.2060 | 7.7918 |

1.5 y | −17.5285 | −78.4652 | −24.5641 | 9.0594 | 26.2613 | 9.1401 | 18.3708 | 0.2651 | |

2 y | −14.7317 | −62.4385 | −18.8872 | 4.7895 | 12.3708 | 4.6020 | 10.1900 | −1.1713 | |

2.5 y | −11.3451 | −47.6616 | −14.0002 | 2.8063 | 6.7021 | 2.5822 | 6.2117 | −1.3314 | |

3 y | −8.2927 | −34.8476 | −10.0211 | 1.7067 | 3.8412 | 1.5118 | 3.9022 | −1.1385 | |

3.5 y | −5.6774 | −23.9395 | −6.7715 | 1.0330 | 2.2124 | 0.8872 | 2.4191 | −0.8426 | |

4 y | −3.4884 | −14.7071 | −4.1215 | 0.5700 | 1.1696 | 0.4737 | 1.3652 | −0.5485 | |

4.5 y | −1.6097 | −6.8180 | −1.8894 | 0.2438 | 0.4832 | 0.1978 | 0.5945 | −0.2621 | |

(b) | 1 y | −13.4191 | −96.3065 | −31.8708 | 20.9369 | 168.7090 | 25.5428 | 42.4058 | 6.8763 |

1.5 y | −19.3928 | −85.0507 | −28.2278 | 8.3951 | 44.9036 | 9.7227 | 18.0795 | −0.8700 | |

2 y | −16.6626 | −69.9185 | −21.9503 | 4.3225 | 19.1746 | 4.8187 | 9.9133 | −2.1040 | |

2.5 y | −12.9684 | −54.5511 | −16.3767 | 2.4793 | 9.7422 | 2.6783 | 6.0040 | −2.0250 | |

3 y | −9.5387 | −40.4839 | −11.7690 | 1.4819 | 5.3111 | 1.5581 | 3.7556 | −1.6324 | |

3.5 y | −6.5555 | −28.1046 | −7.9772 | 0.8845 | 2.9280 | 0.9107 | 2.3219 | −1.1722 | |

4 y | −4.0377 | −17.3914 | −4.8619 | 0.4814 | 1.4919 | 0.4843 | 1.3077 | −0.7493 | |

4.5 y | −1.8674 | −8.1086 | −2.2329 | 0.2037 | 0.5952 | 0.2015 | 0.5683 | −0.3538 |

Bates | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{BV}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{RV}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{RVSS}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{TS}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{HL}}$ | ${\mathit{DP}}_{\mathit{Err}}^{K}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{PA}}$ | ${\mathit{DP}}_{\mathit{Err}}^{F}$ | |
---|---|---|---|---|---|---|---|---|---|

(a) | 1 y | 14.4309 | −87.1895 | −29.9293 | 11.2133 | 36.5447 | 10.6474 | 22.8589 | −2.1273 |

1.5 y | −6.3833 | −75.8094 | −25.6593 | 5.0568 | 15.0133 | 4.1198 | 11.6211 | −4.3575 | |

2 y | −10.3506 | −62.5783 | −20.5019 | 2.6651 | 7.5550 | 1.8140 | 6.9034 | −4.1742 | |

2.5 y | −10.1010 | −49.6830 | −15.8434 | 1.5097 | 4.1438 | 0.8191 | 4.3846 | −3.4849 | |

3 y | −8.4835 | −37.7336 | −11.7809 | 0.8810 | 2.3454 | 0.3568 | 2.8397 | −2.6973 | |

3.5 y | −6.4324 | −26.8797 | −8.2676 | 0.4938 | 1.2813 | 0.1194 | 1.7782 | −1.9495 | |

4 y | −4.2495 | −17.0514 | −5.1743 | 0.2556 | 0.6447 | 0.0189 | 1.0198 | −1.2420 | |

4.5 y | −2.0866 | −8.1234 | −2.4413 | 0.1009 | 0.2468 | −0.0116 | 0.4469 | −0.5943 | |

(b) | 1 y | 13.4257 | −91.9420 | −31.6169 | 18.4987 | 59.6240 | 13.0964 | 25.4460 | −3.3672 |

1.5 y | −6.7622 | −82.7271 | −28.0554 | 8.7867 | 24.5731 | 5.1571 | 12.4494 | −5.5782 | |

2 y | −10.6074 | −70.1849 | −22.7147 | 4.8896 | 12.5041 | 2.3203 | 7.1783 | −5.2046 | |

2.5 y | −10.3035 | −56.8420 | −17.6859 | 2.9221 | 6.9522 | 1.0776 | 4.4463 | −4.3048 | |

3 y | −8.6408 | −43.8048 | −13.2162 | 1.7902 | 3.9822 | 0.4853 | 2.8177 | −3.3196 | |

3.5 y | −6.5510 | −31.5429 | −9.3084 | 1.0624 | 2.2163 | 0.1803 | 1.7301 | −2.3910 | |

4 y | −4.3266 | −20.1699 | −5.8425 | 0.5798 | 1.1333 | 0.0436 | 0.9752 | −1.5208 | |

4.5 y | −2.1247 | −9.6830 | −2.7621 | 0.2422 | 0.4423 | −0.0050 | 0.4206 | −0.7267 |

Heston | |
---|---|

ρ | ${v}_{0}$ |

$\rho =-30.00\%$ | ${v}_{0}=20.92\%$ |

$\rho =-24.02\%$ | ${v}_{0}=21.65\%$ |

$\rho =0.0$ | ${v}_{0}=24.20\%$ |

Heston | ${\Sigma}_{t}^{\mathit{BV}}$ | ${\Sigma}_{t}^{\mathit{RV}}$ | ${\Sigma}_{t}^{\mathit{RVSS}}$ | ${\Sigma}_{t}^{\mathit{TS}}$ | ${\Sigma}_{t}^{\mathit{HL}}$ | ${\Sigma}_{t}^{K}$ | ${\Sigma}_{t}^{\mathit{PA}}$ | ${\Sigma}_{t}^{F}$ | |
---|---|---|---|---|---|---|---|---|---|

$\rho =-30\%$ | |||||||||

MSE | 2.25 ×${10}^{-3}$ | 2.18 ×${10}^{-2}$ | 2.04 ×${10}^{-3}$ | 3.53 ×${10}^{-4}$ | 1.45 ×${10}^{-3}$ | 4.23 ×${10}^{-4}$ | 4.65 ×${10}^{-4}$ | 5.28 ×${10}^{-4}$ | |

BIAS | 2.57 ×${10}^{-2}$ | 1.46 ×${10}^{-1}$ | 2.92 ×${10}^{-2}$ | −5.06 ×${10}^{-4}$ | −7.05 ×${10}^{-4}$ | −1.96 ×${10}^{-5}$ | −4.01 ×${10}^{-3}$ | 5.84 ×${10}^{-3}$ | |

$\rho =-24.02\%$ | |||||||||

MSE | 2.42 ×${10}^{-3}$ | 2.28 ×${10}^{-2}$ | 2.30 ×${10}^{-3}$ | 3.58 ×${10}^{-4}$ | 1.66 ×${10}^{-3}$ | 4.64 ×${10}^{-4}$ | 5.05 ×${10}^{-4}$ | 5.71 ×${10}^{-4}$ | |

BIAS | 2.84 ×${10}^{-2}$ | 1.48 ×${10}^{-1}$ | 3.21 ×${10}^{-2}$ | −7.66 ×${10}^{-4}$ | 5.10 ×${10}^{-4}$ | −3.72 ×${10}^{-4}$ | −4.20 ×${10}^{-3}$ | 6.06 ×${10}^{-3}$ | |

$\rho =0$ | |||||||||

MSE | 2.86 ×${10}^{-3}$ | 2.70 ×${10}^{-2}$ | 2.64 ×${10}^{-3}$ | 4.95 ×${10}^{-4}$ | 2.11 ×${10}^{-3}$ | 4.93 ×${10}^{-4}$ | 5.52 ×${10}^{-4}$ | 6.25 ×${10}^{-4}$ | |

BIAS | 3.03 ×${10}^{-2}$ | 1.62 ×${10}^{-1}$ | 3.50 ×${10}^{-2}$ | −1.62 ×${10}^{-3}$ | −1.24 ×${10}^{-3}$ | −1.64 ×${10}^{-4}$ | −4.61 ×${10}^{-3}$ | 6.55 ×${10}^{-3}$ |

Heston | ||||||||
---|---|---|---|---|---|---|---|---|

Mean | Median | 10 perc. | 90 perc. | Min | Max | Std Dev. | ||

$\rho =-30\%$ | ||||||||

${\widehat{{v}_{t}}}^{BV}/{v}_{t}$ | 0.9414 | 0.9287 | 0.8099 | 1.0820 | 0.7170 | 1.6830 | 0.1100 | |

${\widehat{{v}_{t}}}^{RV}/{v}_{t}$ | 0.7112 | 0.7105 | 0.6753 | 0.7481 | 0.6333 | 0.8161 | 0.0285 | |

${\widehat{{v}_{t}}}^{RVSS}/{v}_{t}$ | 0.9286 | 0.9186 | 0.8180 | 1.0401 | 0.7280 | 1.5650 | 0.0909 | |

${\widehat{{v}_{t}}}^{TS}/{v}_{t}$ | 1.0055 | 1.0029 | 0.9322 | 1.0815 | 0.8502 | 1.2804 | 0.0598 | |

${\widehat{{v}_{t}}}^{HL}/{v}_{t}$ | 1.0209 | 0.9977 | 0.8745 | 1.1864 | 0.7505 | 1.7650 | 0.1370 | |

${\widehat{{v}_{t}}}^{K}/{v}_{t}$ | 1.0048 | 0.9991 | 0.9250 | 1.0881 | 0.8296 | 1.2480 | 0.0651 | |

${\widehat{{v}_{t}}}^{PA}/{v}_{t}$ | 1.0180 | 1.0123 | 0.9322 | 1.1079 | 0.8298 | 1.2640 | 0.0689 | |

${\widehat{{v}_{t}}}^{F}/{v}_{t}$ | 0.9875 | 0.9822 | 0.9061 | 1.0758 | 0.7997 | 1.2254 | 0.0672 | |

$\rho =-24.02\%$ | ||||||||

${\widehat{{v}_{t}}}^{BV}/{v}_{t}$ | 0.9391 | 0.9282 | 0.8200 | 1.0757 | 0.7079 | 1.2751 | 0.0984 | |

${\widehat{{v}_{t}}}^{RV}/{v}_{t}$ | 0.7298 | 0.7296 | 0.6947 | 0.7635 | 0.6308 | 0.8285 | 0.0273 | |

${\widehat{{v}_{t}}}^{RVSS}/{v}_{t}$ | 0.9276 | 0.9216 | 0.8227 | 1.0357 | 0.7396 | 1.3023 | 0.0838 | |

${\widehat{{v}_{t}}}^{TS}/{v}_{t}$ | 1.0061 | 1.0023 | 0.9348 | 1.0789 | 0.8673 | 1.2023 | 0.0553 | |

${\widehat{{v}_{t}}}^{HL}/{v}_{t}$ | 1.0176 | 0.9911 | 0.8757 | 1.1951 | 0.7826 | 1.8007 | 0.1353 | |

${\widehat{{v}_{t}}}^{K}/{v}_{t}$ | 1.0060 | 1.0006 | 0.9283 | 1.0914 | 0.8423 | 1.2411 | 0.0627 | |

${\widehat{{v}_{t}}}^{PA}/{v}_{t}$ | 1.0176 | 1.0121 | 0.9359 | 1.1062 | 0.8570 | 1.2606 | 0.0662 | |

${\widehat{{v}_{t}}}^{F}/{v}_{t}$ | 0.9883 | 0.9834 | 0.9103 | 1.0734 | 0.8388 | 1.2148 | 0.0642 | |

$\rho =0$ | ||||||||

${\widehat{{v}_{t}}}^{BV}/{v}_{t}$ | 0.9518 | 0.9420 | 0.8545 | 1.0614 | 0.7486 | 1.2600 | 0.0809 | |

${\widehat{{v}_{t}}}^{RV}/{v}_{t}$ | 0.7775 | 0.7768 | 0.7496 | 0.8061 | 0.7093 | 0.8636 | 0.0218 | |

${\widehat{{v}_{t}}}^{RVSS}/{v}_{t}$ | 0.9403 | 0.9371 | 0.8598 | 1.0316 | 0.7810 | 1.1655 | 0.0653 | |

${\widehat{{v}_{t}}}^{TS}/{v}_{t}$ | 1.0075 | 1.0031 | 0.9481 | 1.0687 | 0.8798 | 1.2356 | 0.0497 | |

${\widehat{{v}_{t}}}^{HL}/{v}_{t}$ | 1.0201 | 0.9972 | 0.8981 | 1.1718 | 0.8000 | 1.5879 | 0.1145 | |

${\widehat{{v}_{t}}}^{K}/{v}_{t}$ | 1.0044 | 0.9992 | 0.9440 | 1.0689 | 0.8841 | 1.1871 | 0.0494 | |

${\widehat{{v}_{t}}}^{PA}/{v}_{t}$ | 1.0147 | 1.0121 | 0.9497 | 1.0812 | 0.8880 | 1.2185 | 0.0529 | |

${\widehat{{v}_{t}}}^{F}/{v}_{t}$ | 0.9905 | 0.9879 | 0.9284 | 1.0535 | 0.8628 | 1.1834 | 0.0509 |

Heston | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{BV}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{RV}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{RVSS}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{TS}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{HL}}$ | ${\mathit{DP}}_{\mathit{Err}}^{K}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{PA}}$ | ${\mathit{DP}}_{\mathit{Err}}^{F}$ | |
---|---|---|---|---|---|---|---|---|---|

$\rho =-30\%$ | |||||||||

1 y | 4.7996 | −96.9042 | −23.7409 | 22.3603 | 164.5229 | 24.3275 | 42.8046 | 6.8151 | |

1.5 y | −14.3485 | −86.5006 | −25.6429 | 8.5531 | 45.9491 | 9.0604 | 18.3006 | −0.9973 | |

2 y | −14.2628 | −71.7115 | −20.5832 | 4.2434 | 20.1667 | 4.3702 | 10.0212 | −2.2244 | |

2.5 y | −11.6035 | −56.2816 | −15.5388 | 2.3587 | 10.4761 | 2.3651 | 6.0606 | −2.1285 | |

3 y | −8.7107 | −41.9399 | −11.2325 | 1.3717 | 5.8261 | 1.3378 | 3.7860 | −1.7152 | |

3.5 y | −6.0728 | −29.2146 | −7.6546 | 0.7914 | 3.2556 | 0.7570 | 2.3270 | −1.2429 | |

4 y | −3.7607 | −18.1143 | −4.6681 | 0.4249 | 1.6958 | 0.3979 | 1.3155 | −0.7859 | |

4.5 y | −1.7491 | −8.4585 | −2.1484 | 0.1761 | 0.6865 | 0.1617 | 0.5710 | −0.3716 | |

$\rho =-24.02\%$ | |||||||||

1 y | −13.4191 | −96.3065 | −31.8708 | 20.9369 | 168.7090 | 25.5428 | 42.4058 | 6.8763 | |

1.5 y | −19.3928 | −85.0507 | -28.2278 | 8.3951 | 44.9036 | 9.7227 | 18.0795 | −0.8700 | |

2 y | −16.6626 | -69.9185 | -21.9503 | 4.3225 | 19.1746 | 4.8187 | 9.9133 | −2.1040 | |

2.5 y | −12.9684 | −54.5511 | −16.3767 | 2.4793 | 9.7422 | 2.6783 | 6.0040 | −2.0250 | |

3 y | −9.5387 | −40.4839 | −11.7690 | 1.4819 | 5.3111 | 1.5581 | 3.7556 | −1.6324 | |

3.5 y | −6.5555 | −28.1046 | −7.9772 | 0.8845 | 2.9280 | 0.9107 | 2.3219 | −1.1722 | |

4 y | −4.0377 | −17.3914 | −4.8619 | 0.4814 | 1.4919 | 0.4843 | 1.3077 | −0.7493 | |

4.5 y | −1.8674 | −8.1086 | −2.2329 | 0.2037 | 0.5952 | 0.2015 | 0.5683 | −0.3538 | |

$\rho =0$ | |||||||||

1y | −15.5280 | −94.5066 | −34.7584 | 22.5433 | 132.9700 | 18.0097 | 33.3303 | 2.8602 | |

1.5y | −18.0677 | −81.4410 | −27.5159 | 9.4160 | 40.9134 | 7.1596 | 15.0375 | −1.7585 | |

2y | −14.9654 | −65.8099 | −20.6777 | 5.0341 | 18.8418 | 3.6290 | 8.5020 | −2.2432 | |

2.5y | −11.4706 | −50.7851 | −15.1741 | 2.9833 | 10.1242 | 2.0582 | 5.2617 | −1.9516 | |

3y | −8.3639 | −37.4239 | −10.7986 | 1.8339 | 5.7829 | 1.2193 | 3.3458 | −1.5084 | |

3.5y | −5.7269 | −25.8687 | −7.2810 | 1.1117 | 3.3050 | 0.7165 | 2.0835 | −1.0672 | |

4y | −3.5034 | −15.9461 | −4.4069 | 0.6189 | 1.7562 | 0.3880 | 1.1869 | −0.6675 | |

4.5y | −1.6112 | −7.3986 | −2.0128 | 0.2692 | 0.7288 | 0.1665 | 0.5234 | −0.3087 |

Bates | |
---|---|

${\mu}_{J}$ | ${v}_{0}$ |

${\mu}_{J}=3.35\%$ | ${v}_{0}=25.04\%$ |

${\mu}_{J}=2.35\%$ | ${v}_{0}=25.01\%$ |

${\mu}_{J}=1.35\%$ | ${v}_{0}=24.94\%$ |

Bates | ${\Sigma}_{t}^{\mathit{BV}}$ | ${\Sigma}_{t}^{\mathit{RV}}$ | ${\Sigma}_{t}^{\mathit{RVSS}}$ | ${\Sigma}_{t}^{\mathit{TS}}$ | ${\Sigma}_{t}^{\mathit{HL}}$ | ${\Sigma}_{t}^{K}$ | ${\Sigma}_{t}^{\mathit{PA}}$ | ${\Sigma}_{t}^{F}$ | |
---|---|---|---|---|---|---|---|---|---|

${\mu}_{J}=3.35\%$ | |||||||||

MSE | 3.55 ×${10}^{-3}$ | 2.80 ×${10}^{-2}$ | 2.94 ×${10}^{-3}$ | 6.29 ×${10}^{-4}$ | 1.84 ×${10}^{-3}$ | 6.23 ×${10}^{-4}$ | 6.70 ×${10}^{-4}$ | 7.61 ×${10}^{-4}$ | |

BIAS | 3.36 ×${10}^{-2}$ | 1.64 ×${10}^{-1}$ | 3.50 ×${10}^{-2}$ | 2.58 ×${10}^{-4}$ | 2.12 ×${10}^{-3}$ | 5.23 ×${10}^{-4}$ | −3.99 ×${10}^{-3}$ | 7.70 ×${10}^{-3}$ | |

${\mu}_{J}=2.35\%$ | |||||||||

MSE | 3.36 ×${10}^{-3}$ | 2.78 ×${10}^{-2}$ | 3.12 ×${10}^{-3}$ | 6.37 ×${10}^{-4}$ | 1.88 ×${10}^{-3}$ | 7.18 ×${10}^{-4}$ | 7.51 ×${10}^{-4}$ | 8.91 ×${10}^{-4}$ | |

BIAS | 3.40 ×${10}^{-2}$ | 1.63 ×${10}^{-1}$ | 3.71 ×${10}^{-2}$ | −6.38 ×${10}^{-4}$ | 1.54 ×${10}^{-3}$ | 2.43 ×${10}^{-3}$ | −1.65 ×${10}^{-3}$ | 1.03 ×${10}^{-2}$ | |

${\mu}_{J}=1.35\%$ | |||||||||

MSE | 3.32 ×${10}^{-3}$ | 2.81 ×${10}^{-2}$ | 2.74 ×${10}^{-3}$ | 5.31 ×${10}^{-4}$ | 1.81 ×${10}^{-3}$ | 5.42 ×${10}^{-4}$ | 5.85 ×${10}^{-4}$ | 6.83 ×${10}^{-4}$ | |

BIAS | 3.51 ×${10}^{-2}$ | 1.65 ×${10}^{-1}$ | 3.58 ×${10}^{-2}$ | 4.81 ×${10}^{-4}$ | 2.37 ×${10}^{-3}$ | 1.15 ×${10}^{-3}$ | −3.10 ×${10}^{-3}$ | 8.59 ×${10}^{-3}$ |

Bates | ||||||||
---|---|---|---|---|---|---|---|---|

Mean | Median | 10 perc. | 90 perc. | Min | Max | Std Dev. | ||

${\mu}_{J}=3.35\%$ | ||||||||

${\widehat{{v}_{t}}}^{BV}/{v}_{t}$ | 0.9392 | 0.9267 | 0.7602 | 1.1496 | 0.4126 | 1.4478 | 0.1544 | |

${\widehat{{v}_{t}}}^{RV}/{v}_{t}$ | 0.7102 | 0.7098 | 0.6747 | 0.7501 | 0.5455 | 0.8180 | 0.0311 | |

${\widehat{{v}_{t}}}^{RVSS}/{v}_{t}$ | 0.9247 | 0.9162 | 0.8129 | 1.0481 | 0.6522 | 1.2965 | 0.0943 | |

${\widehat{{v}_{t}}}^{TS}/{v}_{t}$ | 1.0050 | 0.9998 | 0.9226 | 1.0932 | 0.7700 | 1.2898 | 0.0705 | |

${\widehat{{v}_{t}}}^{HL}/{v}_{t}$ | 1.0113 | 0.9914 | 0.8703 | 1.1860 | 0.6887 | 1.4865 | 0.1253 | |

${\widehat{{v}_{t}}}^{K}/{v}_{t}$ | 1.0047 | 1.0002 | 0.9248 | 1.0921 | 0.7424 | 1.2830 | 0.0695 | |

${\widehat{{v}_{t}}}^{PA}/{v}_{t}$ | 1.0182 | 1.0158 | 0.9288 | 1.1087 | 0.7404 | 1.2769 | 0.0733 | |

${\widehat{{v}_{t}}}^{F}/{v}_{t}$ | 0.9854 | 0.9840 | 0.9033 | 1.0720 | 0.7190 | 1.2204 | 0.0698 | |

${\mu}_{J}=2.35\%$ | ||||||||

${\widehat{{v}_{t}}}^{BV}/{v}_{t}$ | 0.9378 | 0.9222 | 0.7570 | 1.1482 | 0.5435 | 1.4866 | 0.1514 | |

${\widehat{{v}_{t}}}^{RV}/{v}_{t}$ | 0.7089 | 0.7096 | 0.6710 | 0.7494 | 0.5672 | 0.8206 | 0.0322 | |

${\widehat{{v}_{t}}}^{RVSS}/{v}_{t}$ | 0.9187 | 0.9101 | 0.8112 | 1.0389 | 0.6286 | 1.4060 | 0.0940 | |

${\widehat{{v}_{t}}}^{TS}/{v}_{t}$ | 1.0070 | 1.0024 | 0.9246 | 1.0972 | 0.7306 | 1.2813 | 0.0700 | |

${\widehat{{v}_{t}}}^{HL}/{v}_{t}$ | 1.0129 | 0.9981 | 0.8711 | 1.1864 | 0.6801 | 1.4773 | 0.1275 | |

${\widehat{{v}_{t}}}^{K}/{v}_{t}$ | 0.9997 | 0.9971 | 0.9115 | 1.0918 | 0.6901 | 1.2540 | 0.0724 | |

${\widehat{{v}_{t}}}^{PA}/{v}_{t}$ | 1.0120 | 1.0086 | 0.9222 | 1.1073 | 0.6911 | 1.2946 | 0.0766 | |

${\widehat{{v}_{t}}}^{F}/{v}_{t}$ | 0.9786 | 0.9767 | 0.8952 | 1.0681 | 0.6766 | 1.2632 | 0.0719 | |

${\mu}_{J}=1.35\%$ | ||||||||

${\widehat{{v}_{t}}}^{BV}/{v}_{t}$ | 0.9317 | 0.9196 | 0.7512 | 1.1245 | 0.5598 | 1.4828 | 0.1508 | |

${\widehat{{v}_{t}}}^{RV}/{v}_{t}$ | 0.7057 | 0.7052 | 0.6688 | 0.7443 | 0.5763 | 0.8189 | 0.0303 | |

${\widehat{{v}_{t}}}^{RVSS}/{v}_{t}$ | 0.9200 | 0.9169 | 0.8127 | 1.0360 | 0.6870 | 1.2506 | 0.0871 | |

${\widehat{{v}_{t}}}^{TS}/{v}_{t}$ | 1.0039 | 0.9992 | 0.9228 | 1.0875 | 0.7633 | 1.3245 | 0.0657 | |

${\widehat{{v}_{t}}}^{HL}/{v}_{t}$ | 1.0118 | 0.9892 | 0.8660 | 1.1991 | 0.7020 | 1.4525 | 0.1273 | |

${\widehat{{v}_{t}}}^{K}/{v}_{t}$ | 1.0021 | 0.9994 | 0.9173 | 1.0850 | 0.7610 | 1.2345 | 0.0658 | |

${\widehat{{v}_{t}}}^{PA}/{v}_{t}$ | 1.0149 | 1.0140 | 0.9262 | 1.0992 | 0.7605 | 1.2729 | 0.0699 | |

${\widehat{{v}_{t}}}^{F}/{v}_{t}$ | 0.9819 | 0.9822 | 0.8987 | 1.0669 | 0.7453 | 1.2360 | 0.0661 |

Bates | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{BV}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{RV}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{RVSS}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{TS}}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{HL}}$ | ${\mathit{DP}}_{\mathit{Err}}^{K}$ | ${\mathit{DP}}_{\mathit{Err}}^{\mathit{PA}}$ | ${\mathit{DP}}_{\mathit{Err}}^{F}$ | |
---|---|---|---|---|---|---|---|---|---|

${\mu}_{J}=3.35\%$ | |||||||||

1 y | 15.4331 | -92.3935 | −28.6628 | 17.4791 | 57.9038 | 16.4345 | 29.8876 | 0.8040 | |

1.5 y | −5.5499 | −82.9953 | −25.8303 | 7.8566 | 23.3246 | 7.4235 | 15.4241 | −2.6013 | |

2 y | −9.8511 | −70.3362 | −21.0037 | 4.1976 | 11.7265 | 3.9850 | 9.3216 | −2.9963 | |

2.5 y | −9.8283 | −56.9149 | −16.3758 | 2.4224 | 6.4631 | 2.3112 | 6.0103 | −2.6650 | |

3 y | −8.3473 | −43.8289 | −12.2420 | 1.4384 | 3.6764 | 1.3796 | 3.9382 | −2.1293 | |

3.5 y | −6.3752 | −31.5402 | −8.6199 | 0.8253 | 2.0304 | 0.7968 | 2.4951 | −1.5712 | |

4 y | −4.2349 | −20.1568 | −5.4092 | 0.4368 | 1.0308 | 0.4246 | 1.4449 | −1.0138 | |

4.5 y | −2.0879 | −9.6728 | −2.5561 | 0.1768 | 0.3994 | 0.1731 | 0.6388 | −0.4898 | |

${\mu}_{J}=2.35\%$ | |||||||||

1 y | 13.4257 | −91.9420 | −31.6169 | 18.4987 | 59.6240 | 13.0964 | 25.4460 | −3.3672 | |

1.5 y | −6.7622 | −82.7271 | −28.0554 | 8.7867 | 24.5731 | 5.1571 | 12.4494 | −5.5782 | |

2 y | −10.6074 | −70.1849 | −22.7147 | 4.8896 | 12.5041 | 2.3203 | 7.1783 | −5.2046 | |

2.5 y | −10.3035 | −56.8420 | −17.6859 | 2.9221 | 6.9522 | 1.0776 | 4.4463 | −4.3048 | |

3 y | −8.6408 | −43.8048 | −13.2162 | 1.7902 | 3.9822 | 0.4853 | 2.8177 | −3.3196 | |

3.5 y | −6.5510 | −31.5429 | −9.3084 | 1.0624 | 2.2163 | 0.1803 | 1.7301 | −2.3910 | |

4 y | −4.3266 | −20.1699 | −5.8425 | 0.5798 | 1.1333 | 0.0436 | 0.9752 | −1.5208 | |

4.5 y | −2.1247 | −9.6830 | −2.7621 | 0.2422 | 0.4423 | −0.0050 | 0.4206 | −0.7267 | |

${\mu}_{J}=1.35\%$ | |||||||||

1 y | 8.1961 | −91.5704 | −33.4203 | 14.1758 | 57.0915 | 12.3161 | 24.6874 | −3.0626 | |

1.5 y | −9.5962 | −82.6645 | −28.5201 | 6.4643 | 23.5554 | 5.4018 | 12.8792 | −4.7706 | |

2 y | −12.5535 | −70.2623 | −22.7504 | 3.4514 | 11.9553 | 2.7455 | 7.7550 | −4.4219 | |

2.5 y | −11.7049 | −56.9596 | −17.5633 | 1.9838 | 6.6331 | 1.4914 | 4.9708 | −3.6450 | |

3 y | −9.6430 | −43.9142 | −13.0488 | 1.1734 | 3.7960 | 0.8299 | 3.2388 | −2.8032 | |

3.5 y | −7.2316 | −31.6256 | −9.1501 | 0.6707 | 2.1123 | 0.4398 | 2.0399 | −2.0157 | |

4 y | −4.7442 | −20.2222 | −5.7238 | 0.3533 | 1.0806 | 0.2137 | 1.1752 | −1.2808 | |

4.5 y | −2.3185 | −9.7054 | −2.6987 | 0.1423 | 0.4222 | 0.0781 | 0.5170 | −0.6114 |

© 2016 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 ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Barsotti, F.; Sanfelici, S. Market Microstructure Effects on Firm Default Risk Evaluation. *Econometrics* **2016**, *4*, 31.
https://doi.org/10.3390/econometrics4030031

**AMA Style**

Barsotti F, Sanfelici S. Market Microstructure Effects on Firm Default Risk Evaluation. *Econometrics*. 2016; 4(3):31.
https://doi.org/10.3390/econometrics4030031

**Chicago/Turabian Style**

Barsotti, Flavia, and Simona Sanfelici. 2016. "Market Microstructure Effects on Firm Default Risk Evaluation" *Econometrics* 4, no. 3: 31.
https://doi.org/10.3390/econometrics4030031