Deep Equal Risk Pricing of Financial Derivatives with Non-Translation Invariant Risk Measures^†

Round 1

Reviewer 1 Report

The Abstract and whole manuscript should fill out all Journal requirements. The manuscript should be improved in this case. Firstly, in the Abstract, the manuscript goal is not given. Although the Authors have explained what they study, it needs to be depth and not to miss the article's purpose. Now, the Abstract does not encourage reading.

The Introduction introduces the subject, but the Authors do not indicate the literature gap or utility of the proposed approach. It is especially important in the case of the leaded study. The manuscript aim (actually two purposes) should be transparent and directly given. Obviously, the manuscript goal in the Abstract and Introduction should be the same. (See the previous remark to the manuscript goal – section Abstract).

The Literature review –  the Authors have done quite a depth literature review but do not refer critically to the previous study results. In this part, there is a lack of joins with the manuscript goal and literature gap. The literature review is too general.

The Financial market setup -  this title needs to be revised, especially if the text, mainly the mathematical framework, was quite deeply explained. The theoretical setup for ERP derivatives valuation is too general. What did the authors want to show in this section? 

Section 4 and 5 – is acceptable, but the Authors should remember previous remarks about the manuscript's aim. 

The Conclusions – should be depth, including all previous remarks: aim, literature gap and utilities of the proposed approach.

Author Response

We thank the referee for their feedback.

• We took this opportunity to revise the abstract and clarify the objective of the study in the abstract.
• We changed section 3’s title based on the referee’s suggestion to “Mathematical setup for the financial market”.
• We judge the main contributions were very clearly outlined in the introduction and thus performed no further modification: “The main contribution of this manuscript is twofold. The first is to propose a modification of the deep reinforcement learning approach illustrated in Carbonneau and Godin (2021b) and Carbonneau and Godin (2021a) to handle non-translation invariant risk measures within ERP naturally and without excessive additional computational burden. […] The second contribution consists in exploring equal risk prices of options generated when using typical non-translation invariant risk measures.”
• The connexion between our study and the literature have already been outlined clearly in the introduction and thus were not repeated in the literature review section.
• The conclusion already recalls the aim “studies the class of semi-$\mathbb{L}^{p}$ risk measures in the context of equal risk pricing (ERP) for the valuation of European financial derivatives”/literature gap “the use of semi-$\mathbb{L}^{p}$ risk measures as the objective functions measuring residual hedging risk is shown to have several preferable properties over the use of the $\text{CVaR}_{\alpha}$” and utility “The latter conclusion is highly important in the context of ERP as it demonstrates that in the case where options are not or cannot be used within the hedging strategy, the ERP methodology used in conjunction with the semi-$\mathbb{L}^{p}$ class of risk measures can produce reasonable option prices”, and thus no modifications have been made.

Reviewer 2 Report

I have the following major comments to improve the article.

1. The abstract is not effectively communicating the essence of research and missing key components. I strongly suggest to re-write the entire abstract while clearly summarizing the results of the study and mentioning significance and implications of the paper.

2. The literature review requires significant improvements. For example: the term of structure is not well defined, often the authors jump from one topic to another without a logical flow making it challenging for readers. In addition, there is a lack of synthesis, it fails to produce a critical analysis of the existing literature and does not highlight gaps in the knowledge which is crucial to mention the significance and a need to carry out research.

3. The paper focuses primarily on the methods and techniques used in derivatives pricing and hedging, without discussing broader issues in the form of market dynamics, regulatory considerations, or practical implications. While expanding the scope to include these aspects in either introduction section or literature review section would provide a more comprehensive overview of the field and to strengthen its contribution to the academic discourse.

4. It is also essential to include and cite recent papers year 2022, and 2023 related to derivative pricing and strategies, and the methods, while critically evaluating their results and comparing the results achieved in this study.

5. Some equations need to be numbered, for sub-equations it is preferred to use for example: (3.1a), and (3.1b) etc, respectively.

6. The conclusion section is missing two important aspects: the limitations of the study, and future research domain/scope.

Moderate editing required.

Author Response

We thank the referee for their feedback.

• We have modified the abstract to clarify the study’s objective.
• We added references to several recent studies Marzban et al. (2021), Marzban et al. (2022), Alfeus et al. (2022), Lütkebohmert, Ma et al. (2022) and Wu and Jaimungal (2023).
• It is common practice in our field to leave equations which weren’t directly referenced unnumbered, and as such we did not perform this modification.
• We have added the following avenue for research in the conclusion: “A potential avenue for further research would consist in considering objective functions reflecting a risk-reward trade-off rather than simply considering hedging risk minimization. This could for instance be achieved by using rank dependent expected utility functions as suggested in Wu and Jaimungal (2023).”
• The term structure is a well-understood term for any person knowledgeable in finance, so we chose not to add any precision about this. We simply dismissed the comment “the authors jump from one topic to another without any logical flow” as we politely disagree.
• We left discussions about regulatory implications and market dynamics out of scope; readers of this literature are already well-aware that the implication of developing option pricing schemes are a better ability to mitigate/transfer risk at more reasonable cost.

Reviewer 3 Report

A highly commendable piece of work - also see attached document for further comments 

Comments for author File: Comments.pdf

Author Response

We wish to thank the referee for their feedback and positive comments.

Round 2

Reviewer 1 Report

I accept corrections made.