Correcting the Bias in the Practitioner Black-Scholes Method
Round 1
Reviewer 1 Report
See attached report.
Comments for author File: 
Comments.pdf
Author Response
We are grateful to the Reviewer for the careful reading of the paper, and for the suggestions for improvement.
Major Issues
The question being raised here is whether the log-linear form for the implied volatility function can produce features such as the “smile”, which amounts to a non-monotinic relationship between strike price and IV. The answer is “yes”. Since ln(×) is a monotonically increasing function, any non-monotonic pattern in the original IV function (such as the smile) is reproduced (albeit with a different shape) when the log of IV is taken. Two sentences have been inserted at the start of footnote 4, explaining this point clearly. The question being raised is whether the use of one day of data is sufficient to capture the possibility of non-linear effects. The answer is again yes. As Table 1 reveals, for the data from the chosen day, the coefficients of K and K2 are always (respectively) significantly negative and significantly positive, and this is fully consistent with the well-known “volatility smile”. We have inserted a paragraph (third paragraph of Section 5) in which we make this point as further justification for the choice of data set used as an example.Minor Issues
We now include a footnote to the very first sentence of Section 1, in which we provide examples of option pricing models for which a comparison against the benchmark of PBS is likely to be useful. The problem in equations (2) and (7) is the “hats” indicating estimation of parameters are not appearing as intended in the pdf version of the document. They appear correctly in the word version. There seems to be an error in the process of compiling the pdf version.Reviewer 2 Report
See the enclosed file
Comments for author File: Comments.pdf
Author Response
We are grateful to the Reviewer for the careful reading of the paper, and for the suggestions for improvement.
We have altered the final sentence of the Abstract in the way suggested. The problem in equations (2) and (7) is the “hats” indicating estimation of parameters are not appearing as intended in the pdf version of the document. They appear correctly in the word version. There seems to be an error in the process of compiling the pdf version. Lines 140-143 contain a verbatim quote from the well-known paper by Christoffersen and Jacobs (2004). We consider it important to include this quote since it precisely conveys the motivation for our paper. Since it is a quote, the wording cannot be altered. See response to 3. “of” changed to “or”. dot has been centered.See reponse to point (2) above. Changed to “…and all of the n residuals”. We thank the reviewer for pointing this out. The function CBS(×) indeed requires an i subscript for option i, since the characteristics of option i such as K and t are not included as arguments to the function. An appropriate subsctipt has been inserted in (8), (9) and (10). Thanks for this also. sigma(theta) has been replaced with sigma_i(theta) in (11), (12) and (13). The Berkowitz paper has been removed from the reference list.
Round 2
Reviewer 1 Report
I appreciate the fact that the authors have taken my observations into account.
I would have expected more mathematical details regarding the difference of shapes between the "polynomial" and the "log-polynomial" approach, and its financial meaning. Notably, the behavior around the minimal IV can be very different between a parabola and an exponential function, and the position of this minimum itself can be affected. In my opinion it could have been interesting to make a financial interpretation of these features.
However, the authors have globally addressed the issues I raised in my report, therefore I think it is fair to accept their paper for publication in its present form.
Author Response
Thank you for your further comments. We have responded by Inserting a graph (Figure 1) which compares the predicted IV against strike from the two models (linear and log-linear). We remark that the two curves are broadly similar, although - as you anticipated - the latter has a minimum that is somewhat further to the right. We remark that this may be interesting but we are unable to provide a financial interpretation.
