# Correcting the Bias in the Practitioner Black-Scholes Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The PBS Method

_{i}that causes the following equality to be satisfied:

_{i}is the market price of call option i, and ${C}_{i}^{BS}(\xb7)$ is the Black-Scholes valuation of call option i.2

## 3. The Smearing Method in Prediction

## 4. Estimation of the IV Equation by NLLS on Option Price Data

## 5. Estimation of IV Equations Using Real Data

^{2}(respectively significantly negative and significantly positive in all models in which they appear) represent strong evidence of the well-known “volatility smile”.

^{2}, τ, τ

^{2}, and Kτ as explanatory variables. Model 3 is the same as Model 2 but with log(IV) as the dependent variable. Figure 1 compares predicted IV from Models 2 and 3. The two curves are broadly similar, both displaying the “smile”, although perhaps interestingly, the lowest point of the latter is somewhat to the right of that of the former. Model 3 is the model in which we are most interested, since this is the model whose estimates we use as the basis of the DGP for the Monte Carlo study reported in the next section.

## 6. A Monte Carlo Analysis of PBS with Smearing

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Data Availability Statement

## Appendix A. Data Used for Single Day Estimation in Section 5

**Table A1.**Data on 51 call options traded on 27 July 2000. C is price of option (mid-point between bid and ask); S is current price of underlying (S&P 500 index); K is strike price; rf is risk-free rate; tau is time to expiry in years; IV is implied volatility. Data Source: https://optionmetrics.com/data-products/.

t | Date | Expirydate | C | S | K | rf | tau | iv |
---|---|---|---|---|---|---|---|---|

1 | 27 July 2000 | 16 December 2000 | 40.5 | 1449.62 | 1550 | 0.0598 | 0.3890411 | 0.18459306 |

2 | 27 July 2000 | 16 September 2000 | 2.875 | 1449.62 | 1600 | 0.0598 | 0.139726 | 0.16379156 |

3 | 27 July 2000 | 19 August 2000 | 8.125 | 1449.62 | 1500 | 0.0598 | 0.0630137 | 0.16414911 |

4 | 27 July 2000 | 16 December 2000 | 7 | 1449.62 | 1700 | 0.0598 | 0.3890411 | 0.16543757 |

5 | 27 July 2000 | 19 August 2000 | 12.625 | 1449.62 | 1485 | 0.0598 | 0.0630137 | 0.16855563 |

6 | 27 July 2000 | 19 August 2000 | 157.125 | 1449.62 | 1300 | 0.0598 | 0.0630137 | 0.29000379 |

7 | 27 July 2000 | 19 August 2000 | 133 | 1449.62 | 1325 | 0.0598 | 0.0630137 | 0.26658718 |

8 | 27 July 2000 | 19 August 2000 | 26.375 | 1449.62 | 1455 | 0.0598 | 0.0630137 | 0.18138442 |

9 | 27 July 2000 | 16 September 2000 | 14.75 | 1449.62 | 1525 | 0.0598 | 0.139726 | 0.17296152 |

10 | 27 July 2000 | 16 September 2000 | 47.25 | 1449.62 | 1450 | 0.0598 | 0.139726 | 0.19113849 |

11 | 27 July 2000 | 19 August 2000 | 18.625 | 1449.62 | 1470 | 0.0598 | 0.0630137 | 0.1738262 |

12 | 27 July 2000 | 16 September 2000 | 36.125 | 1449.62 | 1470 | 0.0598 | 0.139726 | 0.18486601 |

13 | 27 July 2000 | 21 October 2000 | 6.25 | 1449.62 | 1625 | 0.0598 | 0.2356164 | 0.16558332 |

14 | 27 July 2000 | 19 August 2000 | 109.125 | 1449.62 | 1350 | 0.0598 | 0.0630137 | 0.24189121 |

15 | 27 July 2000 | 19 August 2000 | 1.625 | 1449.62 | 1545 | 0.0598 | 0.0630137 | 0.15661302 |

16 | 27 July 2000 | 19 August 2000 | 11.125 | 1449.62 | 1490 | 0.0598 | 0.0630137 | 0.16827447 |

17 | 27 July 2000 | 16 December 2000 | 31.75 | 1449.62 | 1575 | 0.0598 | 0.3890411 | 0.18067305 |

18 | 27 July 2000 | 19 August 2000 | 14.5 | 1449.62 | 1480 | 0.0598 | 0.0630137 | 0.17055967 |

19 | 27 July 2000 | 19 August 2000 | 3.6875 | 1449.62 | 1525 | 0.0598 | 0.0630137 | 0.1615991 |

20 | 27 July 2000 | 17 March 2001 | 61.375 | 1449.62 | 1575 | 0.0598 | 0.6383561 | 0.19218853 |

21 | 27 July 2000 | 16 December 2000 | 13.5 | 1449.62 | 1650 | 0.0598 | 0.3890411 | 0.17021512 |

22 | 27 July 2000 | 19 August 2000 | 1.3125 | 1449.62 | 1550 | 0.0598 | 0.0630137 | 0.15584699 |

23 | 27 July 2000 | 16 September 2000 | 63.625 | 1449.62 | 1425 | 0.0598 | 0.139726 | 0.20124521 |

24 | 27 July 2000 | 16 September 2000 | 38.875 | 1449.62 | 1465 | 0.0598 | 0.139726 | 0.18695045 |

25 | 27 July 2000 | 19 August 2000 | 16.5 | 1449.62 | 1475 | 0.0598 | 0.0630137 | 0.17231101 |

26 | 27 July 2000 | 17 March 2001 | 5.125 | 1449.62 | 1850 | 0.0598 | 0.6383561 | 0.16320811 |

27 | 27 July 2000 | 16 December 2000 | 62.625 | 1449.62 | 1500 | 0.0598 | 0.3890411 | 0.19390107 |

28 | 27 July 2000 | 16 September 2000 | 23 | 1449.62 | 1500 | 0.0598 | 0.139726 | 0.17839537 |

29 | 27 July 2000 | 19 August 2000 | 31.875 | 1449.62 | 1445 | 0.0598 | 0.0630137 | 0.18350645 |

30 | 27 July 2000 | 16 September 2000 | 1.5 | 1449.62 | 1625 | 0.0598 | 0.139726 | 0.1619217 |

31 | 27 July 2000 | 19 August 2000 | 35 | 1449.62 | 1440 | 0.0598 | 0.0630137 | 0.18568564 |

32 | 27 July 2000 | 16 December 2000 | 4.75 | 1449.62 | 1725 | 0.0598 | 0.3890411 | 0.16223686 |

33 | 27 July 2000 | 16 December 2000 | 3.125 | 1449.62 | 1750 | 0.0598 | 0.3890411 | 0.15914554 |

34 | 27 July 2000 | 16 December 2000 | 9.5 | 1449.62 | 1675 | 0.0598 | 0.3890411 | 0.16608682 |

35 | 27 July 2000 | 16 September 2000 | 19.375 | 1449.62 | 1510 | 0.0598 | 0.139726 | 0.17603378 |

36 | 27 July 2000 | 19 August 2000 | 2.0625 | 1449.62 | 1540 | 0.0598 | 0.0630137 | 0.15860899 |

37 | 27 July 2000 | 16 December 2000 | 18.125 | 1449.62 | 1625 | 0.0598 | 0.3890411 | 0.17263715 |

38 | 27 July 2000 | 19 August 2000 | 29.125 | 1449.62 | 1450 | 0.0598 | 0.0630137 | 0.18295443 |

39 | 27 July 2000 | 16 September 2000 | 81.125 | 1449.62 | 1400 | 0.0598 | 0.139726 | 0.20730415 |

40 | 27 July 2000 | 21 October 2000 | 20.625 | 1449.62 | 1550 | 0.0598 | 0.2356164 | 0.17530438 |

41 | 27 July 2000 | 16 September 2000 | 33.875 | 1449.62 | 1475 | 0.0598 | 0.139726 | 0.1846462 |

42 | 27 July 2000 | 16 September 2000 | 100.625 | 1449.62 | 1375 | 0.0598 | 0.139726 | 0.21527798 |

43 | 27 July 2000 | 16 December 2000 | 24.25 | 1449.62 | 1600 | 0.0598 | 0.3890411 | 0.1764836 |

44 | 27 July 2000 | 19 August 2000 | 64.75 | 1449.62 | 1400 | 0.0598 | 0.0630137 | 0.2074976 |

45 | 27 July 2000 | 19 August 2000 | 3 | 1449.62 | 1530 | 0.0598 | 0.0630137 | 0.15969375 |

46 | 27 July 2000 | 21 October 2000 | 65.625 | 1449.62 | 1450 | 0.0598 | 0.2356164 | 0.19772039 |

47 | 27 July 2000 | 16 September 2000 | 5.125 | 1449.62 | 1575 | 0.0598 | 0.139726 | 0.16519108 |

48 | 27 July 2000 | 16 September 2000 | 8.875 | 1449.62 | 1550 | 0.0598 | 0.139726 | 0.16814606 |

49 | 27 July 2000 | 19 August 2000 | 45.25 | 1449.62 | 1425 | 0.0598 | 0.0630137 | 0.19296223 |

50 | 27 July 2000 | 17 March 2001 | 35.625 | 1449.62 | 1650 | 0.0598 | 0.6383561 | 0.18229888 |

51 | 27 July 2000 | 16 September 2000 | 41.5 | 1449.62 | 1460 | 0.0598 | 0.139726 | 0.18801008 |

## Appendix B

`regress log_iv K K2 tau tau2 Ktau`

`predict log_ivhat, xb`

`predict uhat, resid`

`gen d1=.`

`gen d2=.`

`gen pbs_smearing=.`

`quietly{`

`local N=_N`

`forvalues j=1(1)‘N’ {`

`scalar log_ivhat_temp=log_ivhat in ‘j’`

`scalar S_temp=S in ‘j’`

`scalar K_temp=K in ‘j’`

`scalar r_temp=r in ‘j’`

`scalar tau_temp=tau in ‘j’`

`replace d1=(ln(S_temp/K_temp)+(r_temp+(exp(log_ivhat_temp+uhat))^2/2) ///`

`*tau_temp)/(exp(log_ivhat_temp+uhat)*sqrt(tau_temp))`

`replace d2=d1-(exp(log_ivhat_temp+uhat))*sqrt(tau_temp)`

`egen w=mean(S_temp*normal(d1)-exp(-r_temp*tau_temp)*K_temp*normal(d2))`

`quietly replace pbs_smearing=w if _n==‘j’`

`drop w`

`}`

`}`

## References

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1 | Examples of option pricing models for which a comparison against the benchmark of PBS is likely to be very useful are: Bakshi et al.’s (1997) SVSI-J model which allows volatility, interest rates, and jumps all to be stochastic; Heston and Nandi’s (2000) model built on the assumption of a GARCH process in the underlying index; and Duan et al.’s (2006) models built on even more general assumptions such as GJR-GARCH (TGARCH) and EGARCH. |

2 | The Black-Scholes formula (Hull 2011) also contains as arguments: current price of the underlying; strike price; time-to-expiry, risk-free rate. Since the focus here is on volatility, we suppress these arguments and express the option value as a function of only the volatility of the underlying, σ. |

3 | The log-linear form (5) is clearly a new and untested class of implied volatility function. However, since ln(·) is a monotonically increasing function, any non-monotonic pattern in the original IV function (such as the well-known “volatility smile”) is obviously reproduced (albeit with a different shape) when the log of IV is used. Other authors (e.g., Andreou et al. 2014) address the negativity problem by applying a lower bound on predicted IV, using for example $max(\sigma (\widehat{\theta}),0.01)$ in place of $\sigma (\widehat{\theta})$ in (4). On Monte Carlo evidence, we find that this approach tends to exacerbate biases in option values. For this reason, and also because the lower-bound correction is ad-hoc, we prefer the log-linear approach. |

4 | The data is extracted from OptionMetrics http://www.optionmetrics.com/. |

5 | For the positively skewed error, we simulate ${\epsilon}_{i}=0.0282\times ({\chi}^{2}(3)-3)/\surd 6$. For the negatively skewed error, we apply the same formula with the sign reversed. Note that these skewed distributions have the same mean and variance as the normal error N(0, 0.0282 ^{2}) used for the symmetric case. |

6 | The Wald test can be applied following estimation of a NLLS model in STATA by using the test command immediately after the nl command. |

**Figure 2.**Histogram of residuals from log-linear model (Model 3 in Table 1). Normal density superimposed.

**Table 1.**Estimation Results. Implied volatility (IV) equations from various models: (1) Linear regression with IV as dependent variable and no explanatory variables; (2) Linear regression with IV as dependent variable and standard set of explanatory variables; (3) log-linear regression with log(IV) as dependent variable and standard set of explanatory variables; (4) non-linear least squares estimation of log(IV) equation with option price as the dependent variable. Hausman test tests for systematic difference between estimates from Models 3 and 4. Data is from 51 options traded on 27 July 2000 (See Appendix A for complete data set).

Variable | Model 1 | Model 2 | Model 3 | Model 4 |
---|---|---|---|---|

IV Mean Only | IV (Standard PBS) | Log(IV) (Log-Linear PBS) | NLLS (Log IV) | |

K | −0.00414 *** | −0.0168 *** | −0.00626 *** | |

(−8.91) | (−8.85) | (−4.55) | ||

K^{2} | 0.00000131 *** | 0.00000517 *** | 0.00000159 ** | |

(7.98) | (7.72) | (3.27) | ||

τ | 1.104 *** | 3.978 *** | −0.256 | |

(4.34) | (3.83) | (−0.38) | ||

τ^{2} | 0.0262 | −0.136 | −0.421 ** | |

(0.49) | (−0.62) | (−3.48) | ||

K * τ | −0.000668 *** | −0.00222 ** | 0.000594 | |

(−3.76) | (−3.06) | (1.24) | ||

constant | 0.182 *** | 3.426 *** | 11.72 *** | 4.003 *** |

(50.33) | (10.42) | (8.73) | (4.10) | |

σ | 0.0259 | 0.00692 | 0.0282 | |

n | 51 | 51 | 51 | 51 |

R^{2} | 0 | 0.936 | 0.954 | |

Hausman χ^{2}(5) (p-value) | 516.0 | |||

(0.0000) |

**Table 2.**Monte Carlo Results. Models correspond to Models 1–4 in Table 1. The data generating process (DGP) combines Equation (15) and design matrix presented in Appendix A. Distribution of error is explained in footnote 5. There are 10,000 Replications for each result set. Means over 10,000 replications taken: bias (of predicted option prices); mean absolute error (MAE); mean squared error (MSE). Results were obtained using simulate command in STATA with random number seed 87654321.

(1) PBS MEAN | (2) PBS IV (Standard PBS) | (3) PBS LogIV SMEARING | (4) NLLS | |
---|---|---|---|---|

Symmetric error: | ||||

In sample: | ||||

BIAS | 0.0972 | 0.0127 | 0.0005 | −0.0031 |

MAE | 2.4102 | 0.6714 | 0.6428 | 0.6324 |

MSE | 9.5759 | 0.9086 | 0.8262 | 0.7435 |

Out of sample: | ||||

BIAS | 0.0937 | 0.0092 | −0.0033 | −0.0067 |

MAE | 2.4147 | 0.7640 | 0.7413 | 0.7681 |

MSE | 9.6238 | 1.2787 | 1.1951 | 1.2787 |

Negatively skewed error: | ||||

In sample: | ||||

BIAS | 0.0965 | 0.0129 | 0.0007 | −0.0028 |

MAE | 2.4081 | 0.6269 | 0.5983 | 0.5891 |

MSE | 9.5111 | 0.8620 | 0.7798 | 0.7017 |

Out of sample: | ||||

BIAS | 0.0979 | 0.0143 | 0.0018 | −0.0015 |

MAE | 2.4132 | 0.7096 | 0.6891 | 0.7118 |

MSE | 9.5551 | 1.2064 | 1.1290 | 1.1948 |

Positively skewed error: | ||||

In sample: | ||||

BIAS | 0.0974 | 0.0127 | 0.0005 | −0.0033 |

MAE | 2.4137 | 0.6510 | 0.6188 | 0.6113 |

MSE | 9.6282 | 0.9629 | 0.8786 | 0.7908 |

Out of sample: | ||||

BIAS | 0.0959 | 0.0112 | −0.0013 | −0.0049 |

MAE | 2.4176 | 0.7385 | 0.7119 | 0.7120 |

MSE | 9.6708 | 1.3516 | 1.2623 | 1.3573 |

**Table 3.**Further Monte Carlo Results. Proportion of replications in which null hypothesis is rejected by Wald Test. Null Hypothesis is that each parameter equals its true value under the DGP. Nominal test size 0.05, with 10,000 replications.

Distributional Assumption | P(Reject H_{0}) with Log-Linear Regression | P(Reject H_{0}) with NLLS |
---|---|---|

Symmetric error | 0.052 | 0.473 |

Negatively skewed error | 0.053 | 0.396 |

Positively skewed error | 0.054 | 0.399 |

© 2019 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**

Yin, Y.; Moffatt, P.G. Correcting the Bias in the Practitioner Black-Scholes Method. *J. Risk Financial Manag.* **2019**, *12*, 157.
https://doi.org/10.3390/jrfm12040157

**AMA Style**

Yin Y, Moffatt PG. Correcting the Bias in the Practitioner Black-Scholes Method. *Journal of Risk and Financial Management*. 2019; 12(4):157.
https://doi.org/10.3390/jrfm12040157

**Chicago/Turabian Style**

Yin, Yun, and Peter G. Moffatt. 2019. "Correcting the Bias in the Practitioner Black-Scholes Method" *Journal of Risk and Financial Management* 12, no. 4: 157.
https://doi.org/10.3390/jrfm12040157