Discounted Optimal Stopping of a Brownian Bridge, with Application to American Options under Pinning
Round 1
Reviewer 1 Report
This manuscript provides a development in solving American options for a case when the final price matches the strike price. It is, therefore, a Brownian bridge process, for which an optimal stopping problem with the particularity of employing discounted prices was solved. On the other hand, I must highlight that this work is limited to a finite-horizon.
The manuscript has been written to depict the solution development, and includes all the required stages, namely a brief framework, problem statement, resolution and validation. Yet, the latter two are obtuse and not clearly communicated. This is probably why the readers will gain knowledge of the work motivation only while reading through the solution development. Nevertheless, it is worthwhile to stress out that, on one hand, this work has an immediate employment in financial mathematics for stock prices analysis and, on the other, it offers a solution for a problem that can be solved with other techniques.
Abstract does not provide the much needed framework, nor is stylistically correct or well written. While this must be well-known by the Authors, I recall that an abstract must elucidate the motivation, the problem statement, the method, results and conclusions. It certainly should not be a short summary.
The title is adequate and appealing but not particularly informative. It should be enhanced to provide more complete information of the work specificity.
The theme is current, but not a hot-topic. This is clear both for the marginal developments in Brownian Bridge models and for the plethora of more widely used alternatives for the same potential employment (that I can anticipate).
The problem is well defined, from the mathematical point of view, but the wider problem that is deemed to be solved is not. Furthermore, there is not an adequate problem definition without assessing current methods shortcomings.
Regarding the introductory chapter, one can take into account that there are not many references on this and, therefore, it would not be easy to perform a broader literature review. However, it would certainly be possible to go deeper in the past achievements, lines of research and limitation, adding a personal dimension of critical analysis and better framing the following work.
The redaction is weak and the text is not well organized.
Research methods are adequate and sufficiently explained, the research beneath the manuscript is substantial and replicable. These are three major strengths of this manuscript. Furthermore, it clearly fits into the Journal scope.
Conclusions are significant and supported by the attained results.
Despite not bringing major innovation to the community, I believe that the former reasons and the fact that this manuscript brings, indeed, a solution that hadn’t been achieved yet, make this manuscript meritorious as a research article.
On the other hand, organization and redaction are clearly a weak point. It is obviously less stringent in sections 4 to 6 but, as a whole, the manuscript should be, firstly re-organized, and then thoroughly proof-read and partially rewritten.
All things considered, I believe that this manuscript is a good scientific contribute and shall be published, but some improvements are needed, as aforementioned.
Author Response
Please see the attachment
Author Response File: 
Author Response.pdf
Reviewer 2 Report
1. Summary:
The manuscript explores the exponentially discounted optimal stopping problem for a Brownian bridge with a gain function linked to an American put option and a finite horizon. The problem is solved using standard steps in optimal stopping theory. A recursive fixed point algorithm is proposed to compute the solution of the free boundary equation. The manuscript tests the optimal stopping boundary with real data. In particular, the method is compared with the optimal exercised time based on a geometric Brownian motion. The manuscript provides also confidence curves for this optimal stopping boundary. Proofs of propositions and theorems are also presented.
2. Comments:
The topic of this work is interesting and the results are convincing. The manuscript reads well and its structure is clear. I have a single short comment regarding the presentation:
The title of the article gives the impression that it is a manuscript with a strong focus in mathematical finance. However, the introduction section starts talking about the abstract problem and relegates the finance motivation for later on. I suggest to start the introduction section with a motivation of the problem for a general audience.
3. Recommendation: This work constitutes an interesting extension of previous works and the results indeed add more to the understanding of the problem. In my opinion the manuscript can be published after the authors address my short comment.Author Response
Please see the attachment
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
I thank the Authors for the manuscript enhancement.
While I still believe that writing isn’t perfect, it had a major improvement, as well as the organization did.
I recommend accepting the manuscript.
Reviewer 2 Report
The authors addressed my comments and in my opinion the manuscript is
ready for publication.
